text:
Aibhín Bray
email: braya@tcd.ie

Elizabeth Oldham
email: eoldham@tcd.ie

Katriona O'sullivan

A study of readiness for the integration of 21 st century practices in mathematics classrooms

Keywords: Teaching practices, 21 st century learning, mathematics education

This poster explores the complex process of integration of 21 st Century (21C) teaching and learning practices into mathematics classrooms, reporting on mathematics teachers' perceptions of the predictors for, and barriers to, their integration in European post-primary schools. Data are drawn from teachers' responses to a questionnaire for an Erasmus+ project that addresses readiness for the integration of 21C practices. Responses from 52 Irish, Swedish, Estonian and German practitioners listing mathematics as one of their teaching subjects are considered. For quantitative data, descriptive and inferential statistics were used; a directed content analysis approach was taken for qualitative data. Findings indicate that system restrictions and resources are major barriers, and that classroom management and teacher beliefs impact on confidence with and frequency of use of 21C practices. We propose that this work form the basis of a broader study.

The perceived importance of a '21 st Century' (21C) approach to teaching and learning is well documented [START_REF] Dede | Comparing frameworks for 21st century skills[END_REF][START_REF] Voogt | A comparative analysis of international frameworks for 21st century competences: implications for national curriculum policies[END_REF]. In terms of mathematics pedagogy however, while there is considerable research into the use of technologies for teaching and learning (e.g., TWG15 and TWG16), the broader field of 21C practices in the classroom is less considered. This research explores responses to a survey instrument developed for an Erasmus+ project, Teaching for Tomorrow (TfT). TfT is a partnership between institutions in four countries (Ireland, Sweden, Estonia and Germany) that are working to develop a model of 21C teaching and learning across subject areas. The poster reports on the responses of 52 teachers who name mathematics as one of their teaching subjects. The aim is to identify what they see as the predictors for and barriers to usage of 21C practices in the classroom, with a view to larger-scale research.

The theoretical framework underpinning the model for 21C practices being developed by TfT draws on the work of [START_REF] Ravitz | Using project based learning to teach 21st century skills: Findings from a statewide initiative[END_REF], which emphasises a project-based, collaborative, and student-led pedagogic approach. "Readiness for integration" is taken as involving confidence in using and encouraging, and frequency of using, the 21C practices of: Critical thinking, Collaboration, Communication, Creativity & Innovation, Use of Technology, Self-direction, Global and Local Connections.

The questionnaire used to gather data was developed by the Irish partners, with items drawn from the validated instruments of [START_REF] Euler | Report about the survey on inquiry-based learning and teaching in the European partner countries[END_REF], [START_REF] Ravitz | Using project based learning to teach 21st century skills: Findings from a statewide initiative[END_REF], and the OECD (2010). It involved 4 main sections: (1) Background information; (2) Teachers' beliefs about the nature of teaching and learning (direct transmission versus constructivist); (3) Orientation towards, usage of, and barriers to 21C teaching and learning; and (4) Confidence with and frequency of integration of 21st skills in practice. Apart from section 1 and an open-ended item in the Barriers section, all items used 5-point Likert-type scoring system.

Thematic Working Group 19

Proceedings of CERME10

Multiple regressions were performed to identify whether the categories of beliefs, opinions and usage, and barriers had a significant bearing on teachers' confidence with, and frequency of, integration of 21CL practices in the mathematics classroom. Also, t-tests and one-way ANOVAs were used to compare the mean ratings across the four participating countries. Directed content analysis was undertaken for the qualitative data.

Results indicate that teachers' mean orientation towards 21C practices is quite high, with respondents tending to agree that 21C teaching and learning has a positive impact on student motivation. However mean levels of confidence are less positive, and mean frequency of usage is rather low, pointing to a lack of readiness for integration. Respondents' mean scores for self-reported direct transmission beliefs are lower than those for constructivist beliefs, the latter being predictors of confidence in 21C practices.

In the qualitative analysis, students' and teachers' direct transmission beliefs are reported by respondents as barriers to the integration of 21C practices, with "teacher inertia and general reluctance to move from traditional methods" emerging as a common issue. Barriers at the system level, particularly those associated with time, and curriculum and assessment, also appear important. In addition, both quantitative and qualitative analysis reflects that classroom management issues act as barriers to teachers' implementation of 21C practices: "Students are not used to 21CL, because most of the time they do not have to do it, so at first it takes a lot of time."

In order to encourage teachers to integrate 21C practices in the mathematics classroom, it is essential to address some of the barriers identified. The features of the TfT model, outlined above, are intended to provide guidance for teachers and students, a structured approach to the development of 21C activities, and relevant assessment practices.

It should be noted that although the samples from each country are small and not representative, and that there were variations in the criteria for participant selection, the results across counties show surprising commonality. Thus, we propose to conduct a larger study, involving representative groups, uniformly selected in each country, to see if such trends arise outside the confines of TfT.

Proceedings of CERME10


domain: math shs

text:
Merav Weingarden
email: merav.weingarden@gmail.com

Einat Heyd-Metzuyanim

Talli Nachlieli
email: talli.nachlieli@gmail.com

The Realization Tree Assessment tool: Assessing the exposure to mathematical objects during a lesson

Keywords: Teaching practices, Realization tree, commognition, cognitive demand, lesson assessment tools

We present a new tool -Realization Tree Assessment (RTA) for assessing the mathematical quality of lessons and the ways in which the whole classroom discussion expose students to mathematical concepts. The tool, built upon the commognitive framework, depicts the different realizations of a mathematical object treated in a lesson, and then uses different shades to signify who articulated the realizationthe teacher or the students. We exemplify the tool on two lessons implementing an identical Hexagon pattern generalization task. The RTA visualizes the manner in which one lesson gave students sufficient opportunities to "same" different algebraic expressions, while the other lesson did not. We show how this visual presentation of the mathematical ideas complements existing assessment tools, particularly, the Instructional Quality Assessment and Accountable Talk. We conclude by discussing the potential of the tool as an aid for lesson planning.

Introduction

Recent years have seen increasing efforts to train teachers to teach explorativelyprovide students with opportunities to engage with cognitively demanding tasks, problem solve, and participate in rich mathematical discussions [START_REF] Schoenfeld | What Makes for Powerful Classrooms, and How Can We Support Teachers in Creating Them? A Story of Research and Practice, Productively Intertwined[END_REF]. Within such efforts an important role lies in the tools that are used to examine lessons enacted by the trained teachers [START_REF] Boston | Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers' classrooms[END_REF][START_REF] Schoenfeld | Thoughts on scale[END_REF]. Scoring and evaluation tools (such as the Instructional Quality Assessment tool or TRU math) can be used both for evaluating lessons and thereby examining the effectiveness of the training program, as well as tools for teachers' professional development. A common difficulty with these tools, however, lies in operationalizing their criteria for evaluating the quality of mathematical ideas dealt with in the lesson. In this paper, we propose an analytical toolthe Realization Tree Assessment tool which is based on the "commognitive" framework [START_REF] Nachlieli | Growing mathematical objects in the classroom -The case of function[END_REF][START_REF] Sfard | Thinking as communicating[END_REF]. This tool enables drawing a succinct yet sufficiently meaningful picture of the mathematical concepts surfaced in a specific lesson in such a way that lessons can be both compared with each other as well as planned ahead more accurately.

Theoretical background

Tools for the examination and evaluation of classroom instruction can be categorized into three types: scoring tools such as Instructional Quality Assessment, or IQA [START_REF] Boston | IQA Mathematics Lesson Observation Rubrics and Checklists[END_REF] and the Teaching for Robust Understanding of Mathematics summary, or TRU math [START_REF] Schoenfeld | What Makes for Powerful Classrooms, and How Can We Support Teachers in Creating Them? A Story of Research and Practice, Productively Intertwined[END_REF]; "coding and counting" tools, such as Accountable Talk [START_REF] O'connor | Scaling down" to explore the role of talk in learning: From district intervention to controlled classroom study[END_REF]; and qualitative analytical tools (e.g. commognition, [START_REF] Sfard | Thinking as communicating[END_REF]). Scoring and coding measures have the benefit that they are quantifiable. They thus enable both the comparison of teachers with each other, as well as comparison of within-teacher change from lesson to lesson, for example, as a result of professional development. Scoring tools, however, have a drawback. They are heavily based on extensive training of scorers for the development of inter-rater reliability. This, because of their high-inference nature. Coding and counting tools, which are based on coding talk moves, necessitate lower inferences and are therefore easier for achieving reliability. However, these tools are mostly good for capturing non-mathematical aspects of the discourse.

The difficulty in assessing the surfacing of "important mathematical concepts" (Boston, 2012a) or "important content and practices" [START_REF] Schoenfeld | Thoughts on scale[END_REF] in a lesson is not surprising, given that the definition of "mathematical concepts" has been under much dispute for decades [START_REF] Sfard | Thinking as communicating[END_REF]. To our aid, we draw on commognition (ibid), which we have extensively used in the past as a qualitative tool for describing learning-teaching processes. In the present work, we simplify this tool, to attune it with the demands of coding and scoring schemes that seek to evaluate lessons in a relatively short period of time, for the goal of comparing large sets of lessons.

Realization trees

Mathematical learning, says [START_REF] Sfard | Thinking as communicating[END_REF], is a process whereby students gradually become able to communicate about mathematical objects. These objects are produced by discourse (or communication), and are made up of different "realizations" (ibid, p. 165). The term realization is used by Sfard instead of the more common term "representation", to emphasize the fact that nothing is, in fact, "there" to be represented. All mathematical objects are products of human discourse and come to life by being different realizations being "samed" and alienated from human agency so that they are talked about as existing of themselves. For example, the signifier ½, the process of dividing a pizza into two pieces, and the process of shading 3 circles out of 6, are all samed into the object "one half". Children often learn each of these realizations separately and only later come to relate to them all to one object. This is the heart of a process Sfard calls "objectification". Objectification, or talking about mathematical signifiers as "standing for" mathematical objects that "exist" in the world, is a major and necessary accomplishment for advancing in the mathematical discourse. Sfard used the term "realization tree" to illustrate the fact that realizations are usually hierarchical. A half is made of different realizations (1/2, 0.5, 50%, 3/6 etc.) but the whole numbers making up these realizations also have endless realizations (3 apples, 3 fingers, etc.). [START_REF] Nachlieli | Growing mathematical objects in the classroom -The case of function[END_REF] used realization trees to visually explain the complexity of the object function and to relate to the historical development of this object, as well as to make explicit students' development of the discourse of function. Before moving to explain our use of realization trees as tools for assessing the conceptual quality of a lesson, let us briefly describe the two other tools that have been serving us for quantifying and comparing mathematics lessons.

IQA

The IQA (Instructional Quality Assessment tool) has been designed by [START_REF] Boston | Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers' classrooms[END_REF]Boston, 2012a) to evaluate the cognitive demand of mathematical lessons. This, based on the "task framework" put forward by Stein and her colleagues (1996), which differentiates between the cognitive demand of a task, the way it is presented to the classroom, and the way students eventually engage in it. Every rubric in the IQA is scored on a scale from 0 to 4. For reasons of space, we will concentrate here only on two rubrics: AR-2 (implementation) and AR-X (mathematical residue). Regarding the implementation rubric, 1 means students engage only in rote memorization and producing facts, 2 means they engage in the application of procedures explicitly taught, 3 means cognitive demand is not lowered but mathematical reasoning is not sufficiently explicated, and 4 means full engagement in a cognitively demanding mathematical task. 'Mathematical concepts' or 'ideas' are mentioned almost in every rubric in the IQA. For example, in the rubric that refers to the mathematical residue, the highest score should be given when: "The discussion following students' work on the task surfaces the important mathematical ideas, concepts, or connections embedded in the task" (Boston, 2012b, p. 20). However, IQA does not provide any clear guidance on this matter, besides giving a few examples of high and low level lessons.

Accountable Talk

Accountable Talk coding [START_REF] Resnick | How (well structured) talk builds the mind[END_REF]) is a tool originating in sociolinguistic analysis of classroom talk [START_REF] O'connor | Aligning Academic Task and Participation Status through Revoicing : Analysis of a Classroom Discourse Strategy[END_REF]. It provides teachers with a set of specific talk moves they can make during whole classroom discussions, to hold students accountable to the community, to knowledge and to reasoning. Our version of Accountable Talk coding (Heyd-Metzuyanim, Smith, Bill, & Resnick, 2016) includes eight codes for teacher moves (e.g. press for reasoning, revoice, restate, agree/disagree) and four codes for students' moves (e.g. student-agree, student-justification). These moves track the amount in which teachers attempt to make students' thinking public, help students to reason mathematically, and hold them responsible for attending to the reasoning of others. Though the manual does contain examples of mathematical statements, Accountable Talk's basic framework does not deal specifically with content. It has no clear indicator of what consists as more important or "conceptual" reasoning, and what does not.

The study

In what follows, we first describe the setting of the study on which we developed the Realization Tree Assessment tool (RTA). We then describe the results of analysis using the IQA and AT, showing what could be achieved by them and what was missing or difficult to agree upon. We follow this by describing the RTA results for the data, showing where they agree, complement and elaborate on the findings obtained by the IQA and AT.

Setting

The study reported here was performed in the context of a project for training Israeli teachers to implement explorative instructional practices in middle school mathematics classrooms, using methods inspired by Smith & Stein's (2011) "Five Practices for Orchestrating Productive Mathematics Discussions". In this report, we focus on two teachers: Dani and Sivan. Dani was teaching a 7th grade classroom in a school serving a community of middle-high socio-economic background. Sivan was teaching an 8th grade classroom in a school serving a community mostly from a low-middle socio-economic background. Both teachers participated in training sessions where the instructor planned together with each of them separately a lesson according to the "5 Practices". In both cases, the lesson centered around an identical task: the Hexagon Task. The main session in the task was to write description that could be used to compute the perimeter of any train in the pattern of hexagons (See Figure 1):

The reason this task was used, was that it has proved in a previous study [START_REF] Heyd-Metzuyanim | Change in Teachers' Practices Towards Explorative Instruction[END_REF] to be very productive for teachers who are beginning to implement the "5 Practices". We observed, video recorded, and transcribed both lessons. In addition, Dani and Sivan were both interviewed before and after the lessons, and their lesson planning sessions were recorded. In what follows, we present the IQA and AT measures of the two lessons, as well as what was still missing from them for a full understanding of the task implementation.

Findings

Accountable Talk in the two lessons. Both Dani and Sivan's lessons were conducted over a double period (90 Minutes) and both included work in groups (or pairs) where the teacher was walking between the groups, followed by a whole classroom discussion. The two whole classroom discussions took similar time (in Dani's classroom 28 minutes and in Sivan's lesson 26 minutes).

Overall, there were many more AT moves in Dani's lesson (98) than Sivan's (46). In particular, Dani's lesson had much more student talk moves coded as AT moves, either as student agree/disagree (N_Dani=22, N_Sivan=0), or as student justifications (N_Dani=20, N_Sivan=11). Dani was also higher than Sivan in pressing for students' reasoning (N_Dani=23, N_Sivan = 14). The overall picture drawn from the AT measure is, thus, that Dani's lesson had more accountability to reasoning and to the community than Sivan's lesson. Using the AT measure alone, however, does not enable learning about what mathematical concepts were dealt with, and which mathematical ideas surfaced through the discussion.

IQA scoring of two lessons. According to the IQA, Dani's lesson got higher scores then Sivan's lesson on all the rubrics, except the potential of the task, which was given in both cases by the teachers' trainer. In the Implementation rubric, we scored Dani's lesson as a 4, since multiple solutions were found and presented by the students; the teacher did not lead the students towards any particular solution; solutions were linked to each other both by the teacher and by the students; and there was no proceduralization of the task. In contrast, Sivan's implementation scored a 2. Though students generalized the Hexagon pattern into a expression, this was not done through the visual Hexagon's representation, only through the table; connections were not made with other algebraic expressions; in particular, students seemed to be well rehearsed in producing a table, algebraic expression from it and a graph of that expression, thus the task was proceduralized.

In the mathematical residue rubric, the results of the scoring were similar. Dani's lesson received a 4 since: the mathematical idea of equivalence of algebraic expressions was driven through the different algebraic solutions student presented. Evidence for students' understanding could be seen in one of the girls' exclamation "so they're all the same!" In contrast, Sivan's lesson scored a "2" on the mathematical residue rubric, since although the discussion dealt with some mathematical ideas, it did not touch upon the main idea behind the Hexagon task. The teacher did not focus on the different algebraic expressions but rather on the different representations of a linear function (graph, table and algebraic expression). However, as will be shown later, even this idea was not treated fully and appropriately.

Of all the Academic Rigor rubrics, we found the "Mathematical Residue 1 " most difficult to operationalize. It appeared Dani and Sivan had different ideas regarding the mathematical goals of their lessons and this had consequence for the way they led the lesson. While Dani seemed to be well aligned with the goal of showing the equivalence of algebraic expressions, Sivan seemed as though she was mostly aiming at ideas related to linear functions (which are, indeed, part of the 8 th grade curriculum). We therefore searched for a tool that would aid in explicating the mathematical ideas explored in the two lessons. For this end, we developed the RTA.

Realization Tree Assessment tool

The first step in RTA is examining the task and explicating the mathematical object(s) that can be surfaced through engagement with the task. This includes the different realizations that are reasonable to expect from students at a certain grade level. In our case, we built our realization tree based on a lesson plan provided by the Institute for Learning (http://ifl.pitt.edu/index.php/educator_resources), where the different solutions, expected from middle schoolers for this task were drawn out. This produced a "blank" tree, with nodes as seen in Figures 2 and3. We then proceeded to shade the tree nodes with four different colors, as follows: Shade no 4: the student's explanation was complete and accurate; Shade no. 3: the student's explanation was not complete and accurate but the teacher helped explicating the idea; Shade no. 2: the student did not articulate the realization, but the teacher did; Shade no. 1: The realization was partially mentioned, but neither the student nor the teacher explained it fully. 1 The Mathematical Residue rubric appears in our manual as "under development". Finally, if the realization was not mentioned at all, but was hypothesized to be relevant to the lesson and the grade level according to the lesson plan, it was shaded white (no. 0).

As can be seen in Figures 2 and3, the main branch of our realization tree ("algebraic expression"), branches out on the multiple realizations of the algebraic expression . This, in accordance with the potential of the task to explore the different ways in which the visual representation of the hexagon sides can be generalized into a pattern and expressed algebraically.

Figure 2 describes the RTA for Dani's lesson. It shows that three realizations were explained fully and completely by students, three were explained by students, but the teacher filled in some gaps in these explanations, and one realization was explained only by the teacher. This full treatment of the "algebraic expressions" branch led students to endorsement of the narrative that "they all (all the algebraic expressions describing the pattern) equal to ", thus to the saming of different realizations, which was the goal of the lesson, as expressed both by Dani and by the teacher trainer.

In contrast, the RTA for Sivan's lesson (see Figure 3) is much lighter and sparser. It shows that only three realizations were treated in the lesson, and none of them was fully explained by the students. Moreover, the main branch of the tree -the "algebraic expressions" branch, is particularly empty.

Only the realization was treated, and even that one was not explained accurately by the teacher or the students. The relative "emptiness" of Sivan's RTA corresponds well with the relatively low IQA and AT scores her lesson received. Still, it puzzled us, since Sivan was prepared in the PD very specifically for a lesson that was envisioned as similar to that of Dani. "What went wrong?", we asked ourselves. In order to answer that, we went back to the planning session, as well as to the post-lesson interview with Sivan, conducted right after the lesson. We found that, despite the PD instructor's conviction that she and Sivan were "on the same page", Sivan, in fact, had different goals for the lesson. She was focused on connecting the lesson to the previously learned unit on linear functions, where she had taught students to connect the concept of "slope" with the term " " in , as well as connect it with the visual slope of a linear graph:

"I wanted the students to see that every time it rises by four so that they will connect it with the slope that we have done with functions… I deliberately divided the board into three sections, to show the different stages in reaching the function itself -the graph that combines all the various representations of the function". (Sivan, Post-lesson interview)

It appears, then, that Sivan had a different mathematical object in mind (though probably only tacitly) when she planned the lesson -the "linear function" object2 . Within the linear function object, the "slope" attribute of that object was her focus of attention. This could have been an appropriate goal for the lesson, had it been explicated and thought through. In particular, the following realization tree (see Figure 4) could have been appropriate for discussing slope and linear functions. However, the Hexagon task, especially as written for this lesson, was probably not the optimal task for talking about "slope". This, since it depicts a situation where the function is discreet and cannot be described using a linear line. In practice, Sivan neglected very early the connection to the Hexagons drawing. Thus even the "rate of change" (which could have been visualized as the addition of four sides with the addition of each hexagon) was not connected to the "slope" on the graph.

Discussion

Our goal in the present report was to present a new analytical tool for the evaluation of mathematical lessonsthe RTA. Though this tool does not give a numerical value such as scoring and "coding and counting" tools do, it still enables relatively easy qualitative comparisons between lessons. We have used this tool to enable comparison between two more lessons that were performed on the Hexagon task, and the results give a quick overview of the mathematical opportunities to learn in each lesson. The RTA can also serve as an aid for determining the quality of mathematical content (or "mathematical residue") that is sought after in coarser grained assessment tools such as the IQA. In addition, the RTA can give us information about the potential of the task to engage students in explorative mathematical learning and about the relation between this potential and the actual implementation of the task in the classroom.

In the two cases reported here, the application of the RTA was done post-hoc, after the lessons were planned, implemented and recorded. However, we believe there is much potential for using this tool as an aid for planning lessons and training teachers for explorative mathematics instruction. Such a tool is particularly needed in light of previous findings which point to the difficulty of teachers to explicate to themselves the mathematical goals of the lesson (Heyd-Metzuyanim, Smith, Bill, & Resnick, submitted). We also believe that drawing realization trees with teachers will help them plan tasks and whole classroom discussions that provide sufficient opportunities for explorative participation. Often, when teachers talk about explorative instruction, their focus lies on the social or socio-mathematical norms of the classroom, such as students talking and listening to each other

Figure 1 :

 1 Figure 1: The Hexagons Pattern

Figure

  Figure 2: The RTA of Dani's lesson Figure 3: The RTA of Sivan's lesson

Figure 4 :

 4 Figure 4: Alternative Realization Tree for discussing slopes and rate of change

Proceedings of CERME10

Though she named it inaccurately simply a "function", we understood from the context and from the curriculum she was referring to linear functions.Thematic Working Group 19Proceedings of CERME10

 (Heyd-Metzuyanim, Munter, & Greeno, submitted). We believe no less emphasis should be put on the nature of mathematical objects that students get exposed to, and on the paths for objectification that are opened through sufficiently rich mathematical discussions.


text:
Joana Villalonga Pons

Paul Andrews
email: paul.andrews@mnd.su.se

Self-scaffolding students' problem solving: Testing an orientation basis

Keywords: Mathematical problem solving, scaffolding, orientation basis, Catalonia

It is known not only that young students have difficulty solving mathematical problems but also that appropriate scaffolding can support them in the process. In this paper we describe the development and pilot implementation of a device for self-scaffolding grade 6 Catalan students' mathematical problem solving. Called an orientation basis (OB), the OB, which addresses cognitive, metacognitive and affective aspects of problem solving, drew on research describing the actions of an expert problem solver. The evidence indicated that the OB, while in need of refinement, had a positive impact on the problem solving behaviours of many participant students.

Introduction

Mathematical problem solving is difficult, both for students [START_REF] Mason | Thinking mathematically[END_REF][START_REF] Pólya | How to solve it[END_REF] and teachers trying to create appropriately conducive classroom environments [START_REF] De Corte | Fostering students' self-regulation skills in mathematical problem solving[END_REF][START_REF] Schoenfeld | Reflections on problem solving theory and practice[END_REF]. When children solve problems without being conscious of the relationship between their actions and their solutions their ability to transfer their solution processes to new situations will be limited [START_REF] Coltman | Scaffolding learning through meaningful tasks and adult interaction[END_REF]. However, appropriate adult intervention can help children become aware of these processes [START_REF] Coltman | Scaffolding learning through meaningful tasks and adult interaction[END_REF]. Such interventions, known as scaffolding, build on what learners already know in order to close the gap between current learner competence and task objective [START_REF] Bruner | Vygotsky: A historical and conceptual perspective[END_REF][START_REF] Greenfield | A theory of the teacher in the learning activities of everyday life[END_REF][START_REF] Wood | The role of tutoring in problem solving[END_REF]. Moreover, given time, scaffolding can be provided by peers and, ultimately, students themselves [START_REF] Holton | Scaffolding and metacognition[END_REF]. In this paper we describe the development and use of a device, which we have called an orientation base (OB), for use in Catalan sixth grade classrooms. The OB's role is to support the transition from where teachers scaffold learner's problem solving to where students scaffold their own.

Problem solving

As a human activity, problem solving can be understood as an example of goal-directed behavior [START_REF] Schoenfeld | What is mathematical proficiency and how can it be assessed? Assessing Mathematical Proficiency[END_REF]. It is a dynamic, but not necessarily linear, activity requiring the organization and activation of multiple skills and strategies [START_REF] Mason | Thinking mathematically[END_REF][START_REF] Pólya | How to solve it[END_REF]. At the heart of problem solving lies an appropriate mathematical knowledge, an awareness and experience of solution strategies, self-regulatory or metacognitive competence and a belief not only that the problem is worth solving but also that the solver can solve the problem [START_REF] De Corte | Self-regulation: A characteristic and a goal of mathematics learning[END_REF][START_REF] Schoenfeld | What is mathematical proficiency and how can it be assessed? Assessing Mathematical Proficiency[END_REF][START_REF] Schoenfeld | Reflections on problem solving theory and practice[END_REF].

A key aspect of the above, at least from the perspective of providing scaffolding for the learner, is the encouragement of students' metacognitive competence. For example, expert solvers spend more time understanding and analyzing the problem and solution process than calculating, and they continuously reflect on the state of the problem solving process [START_REF] De Corte | Self-regulation: A characteristic and a goal of mathematics learning[END_REF], behaviours that are typically absent with weak problem solvers [START_REF] De Corte | Fostering students' self-regulation skills in mathematical problem solving[END_REF]. Such students need scaffolded support with respect to interpreting a task, identifying its sub-objectives and planning a strategy [START_REF] De Corte | Self-regulation: A characteristic and a goal of mathematics learning[END_REF][START_REF] Mason | Thinking mathematically[END_REF]. They need to learn how to reflect on their existing knowledge and thought processes; that is they need to learn how to evaluate and regulate their own thinking [START_REF] Sanmartí | Evaluar para aprender. 10 ideas clave[END_REF]. This regulative competence is not acquired automatically but emerges over time [START_REF] De Corte | Fostering students' self-regulation skills in mathematical problem solving[END_REF]. Thus, with support in understanding how things work, students can become more efficient and self-regulated problem solvers [START_REF] Schoenfeld | Reflections on problem solving theory and practice[END_REF].

Scaffolding Scaffolding in an educational context

Drawing on [START_REF] Bruner | The ontogenesis of speech acts[END_REF] initial observations with respect to the ways that parents scaffold their infants' learning, [START_REF] Wood | The role of tutoring in problem solving[END_REF] argued that knowledgeable adults can scaffold students' problem solving activity. Here, the adult seeks to reconcile implicit theories of the task components, the necessary steps to solution, and the child's capabilities [START_REF] Stone | The metaphor of scaffolding: Its utility for the field of learning disabilities[END_REF]. In this way, acknowledging a socially imitative process, six ways of assistance were differentiated; recruiting the child's interest, reducing the degrees of freedom, maintaining goal direction, highlighting critical task features, controlling frustration and modelling preferred solutions paths [START_REF] Wood | The role of tutoring in problem solving[END_REF]. Recent work has continued this theme, examining how teachers can best provide (temporary) support that enables learners to complete tasks they would otherwise not have been able to complete independently [START_REF] Smit | A conceptualisation of whole-class scaffolding[END_REF][START_REF] Van De Pol | Scaffolding in teacher-student interaction: A decade of research[END_REF]. In this process, whereby the learner becomes incrementally independently functional [START_REF] Smit | A conceptualisation of whole-class scaffolding[END_REF], both teacher and learner actively share and build common understanding [START_REF] Stone | The metaphor of scaffolding: Its utility for the field of learning disabilities[END_REF][START_REF] Van De Pol | Scaffolding in teacher-student interaction: A decade of research[END_REF].

Scaffolding strategies

Scaffolding is not a 'technique' that can be applied in every situation in the same way (van de [START_REF] Van De Pol | Scaffolding in teacher-student interaction: A decade of research[END_REF]. As in the construction industry, where each scaffold is unique to a specific building, learning scaffolding can be provided at different ages and in a variety of ways, addressing learners' knowledge gaps as part of an ongoing progress [START_REF] Wood | The role of tutoring in problem solving[END_REF]. Significantly, effective scaffolding is thought to comprise three components, involving the six processes of feeding back, giving hints, instructing, explaining, modelling and questioning [START_REF] Van De Pol | Scaffolding in teacher-student interaction: A decade of research[END_REF], which are  Contingency: Support should be adapted to the student's current level of performance.

 Fading: Support is gradually withdrawn over time.

 Transfer of responsibility: Task completion is gradually transferred to the learner. Moreover, effective scaffolding not only promotes learners' cognitive and metacognitive activities but also positive affect. Finally, acknowledging different agents in the process, whether they are informed adults, a group of learners or the individual student, scaffolding is progressively relocated to the learner, whereby the external dialogue of scaffolding becomes the inner dialogue of metacognition [START_REF] Holton | Scaffolding and metacognition[END_REF].

Orientation basis for problem solving

One means of encouraging self-scaffolding of students' problem solving-related self-monitoring skills is to use an orientation basis (OB) [START_REF] Sanmartí | Evaluar para aprender. 10 ideas clave[END_REF]. Here we understand a problem solving-related OB to be a necessary sequence of actions based on the problem solving behaviour of experts. An orientation basis leads the learner to a solution in ways that structure an emergent independence and problem solving autonomy. An OB is not a 'one size fits all' tool but tailored according to learners' requirements and achievements. At every age and according to the learner's needs, an OB can be presented through different statements. In this paper we discuss the development and implementation of an OB for grade 6 Catalan pupils. At this age, pupils are typically expected to have acquired a minimum background in problem solving. However, experience has shown that they lack regulative and problem solving competence, especially in understanding and analyzing the problem, and planning and implementing a solution process. Therefore, drawing on [START_REF] Pólya | How to solve it[END_REF] problem solving principles, the OB depicted in Table 2 was developed. Each of Pólya's four dimensions comprised three particular actions, which can be tracked in the right hand column. The inclusion of each action was a consequence of earlier observations of the problem solving behaviours of grade 6 Catalan pupils and the problem solving strategies found in the literature (e.g. De [START_REF] De Corte | Fostering students' self-regulation skills in mathematical problem solving[END_REF][START_REF] Mason | Thinking mathematically[END_REF]. The OB shown in Table 2, translated from the original Catalan, was designed to be a contingent, hint-giving, feedback tool focused on facilitating both fading and transfer of responsibility (van de Pol et al., 2010). As indicated above, the aim of this paper is to present an initial evaluation of the efficacy of the OB shown in Table 2 for scaffolding grade 6 students' mathematical problem solving.

The study

The participants were students in a 6 th -grade class of a Barcelona primary school. Their teacher was an experienced generalist primary school teacher. Such teachers, who receive relatively little subject knowledge instruction during their pre-service education, typically acquire their mathematical knowledge in practice, a situation much criticised [START_REF] Gálvez | Cambios y dilemas en la formación del profesorado (19612011). Cincuenta años de historia de España en perspectiva europea[END_REF]MECD, 2012). Data were collected during a regular, 50 minute, lesson at the end of the second quarter of the academic year 2015-2016. They derived from 22 students' initial use of the OB as they tried to solve the mathematical problem posed in Figure 1, which was originally posed in Catalan. Before solving the problem, the teacher explained the purpose of the OB carefully and together with the class discussed and clarified the meaning and purpose of each element. This ensured, as far as is practicable, that students understood its vocabulary and overall purpose. Students were each given a copy of the OB's rubric, which included a grid in which they recorded their engagement with the OB as well as a paper copy of the problem on which their solution was to be written. Students were instructed to solve the problem, using the OB to guide their activity, and then record the OB actions they addressed. They were also told that their teacher would not intervene in the problem solving process but check, as they worked, that they completed their OB tracking.

Results

Table 3 shows the data from all 22 students' use of the OB as they worked on the problem. It can be seen that only one student, Student 21, failed to engage with the OB, while all others used it in varying degrees. Nine students obtained correct solutions for both parts of the problem, a further five managed just one part and eight failed to complete either, including the one who failed to complete any OB actions. Four students indicated some difficulty with respect to understanding some OB actions. In this respect, all four found A3, 'I have identified and understood the data', difficult to understand. The only other action that caused uncertainty was A6, 'I have checked that my strategy fits the data'. Thus, in the light of an OB being necessarily adaptive (Sanmarti, 2007), these issues would be addressed in the next iteration of its development. Importantly, even when faced with uncertainty, each of these students was able to continue the problem solving process to at least the next step. Student 12, the only student who found two statements difficult, completed 11 of the OB's stages but failed to complete either part of the problem. Importantly, from the perspective of the OB's development, Student 9 completed all the OB's actions but failed to solve either part of the problem, pointing, perhaps, to the need of cognitive interviews to determine in depth the nature of the difficulties encountered in completing the task.

Looking at the data qualitatively, it can be seen that students' solution attempts tended to show that they took the OB seriously. Students were able to connect OB actions to their own activity, and did not confirm those actions until after they had been completed.

Figure 2: A solution of the problem and related OB tracking

For example, Figure 2 shows a student solution and his OB tracking. The picture confirms that he had read the problem and understood what the first question required. For example, his arithmetical operations and note, 'on each side there are 6 squares', indicate not only that he had identified and understood the required data but that he had also played with the data which let him to prepare a strategy for the first part of the problem. In short, the solution the student presented corresponded with the OB actions he claimed to have completed. Even when students failed to complete any OB action, there was evidence of its having influenced their solution attempts. For example, Figure 3 shows how the single pupil who failed to complete any OB action attempted to address the OB's first action.

Student response to the OB

Several students attempted to communicate with the OB, particularly when uncertain as to its intentions. Figure 4 shows, in the underlining of the word quefer, uncertainty as to its meaning and, essentially, an invitation for someone to explain. In similar vein, students annotated their OB in ways indicative of doubt or just a desire to comment on their response, both cognitively and affectively. Figure 5 shows comments inserted alongside the ticks indicating the student's completion of the various actions. The top two comments are the same and translate as, 'yes, but it takes me a great effort'. The lower comment, while similar in its intention, translates as, 'regular, because it takes me a great effort'.

Discussion

In this paper we have outlined the development and trial of an orientation basis, designed to support 6 th grade-students' problem solving-related self-scaffolding. Derived from the literature the four dimensions, and their respective actions, provided evidence suggesting that the OB has a role to play. The four dimensions and the means of their operationalisation make real for students the actions that guide problem solving [START_REF] Holton | Scaffolding and metacognition[END_REF]. The evidence supports earlier findings that appropriate scaffolding may have a beneficial impact on cognition, metacognition and affect (van de [START_REF] Van De Pol | Scaffolding in teacher-student interaction: A decade of research[END_REF]. However, with respect to the extent to which the OB for problem solving is contingent, exploits fading and encourages transfer of responsibility (van de Pol et al., 2010) is variable. With respect to contingency, our view is that students were able to connect OB actions to their own activity and those who were affected by typically persisted until at least the next step. Also, students took the OB seriously, indicating initial support for both fading and transfer of responsibility, although a longitudinal study would allow these to be better examined. The dimensional structure and the ways in which students use the actions embedded within it point towards a productive cycle of refinement. Despite its linearity, based on the behaviours of an expert problem solver, students' engagement with the final dimension confirmed not only the cyclic nature of problem solving but also the role of the OB in supporting students' awareness of it. Finally, the OB comprised short statements written in the first person. Our view is that it helps learners' not only understand what problem solving expects of them but also anticipate possible actions.

Finally, this paper has reported on the first iteration of an emergent study. Since the completion of this first task students have solved two further problems using the OB. Their teacher has commented, anecdotally, that students are becoming more familiar with and confident in their use of the OB. Therefore, a longitudinal analysis of students' OB-related problem solving would seem an appropriate next step. As found with previous studies, the impact of scaffolding is difficult to evaluate (van de [START_REF] Van De Pol | Scaffolding in teacher-student interaction: A decade of research[END_REF] and this will remain a key objective of future work.

Figure 1 :

 1 Figure 1: The Problem translated from Catalan

Figure 3 :

 3 Figure 3: Solution of the pupil who failed to complete explicitly any OB action

Figure 4 :Figure 5 :

 45 Figure 4: A student's doubts with respect to the meaning of the fourth action

Table 2 : The orientation basis (OB)

 2

Proceedings of CERME10

Total 5 7 12 11 9 12 5 4 12 10 12 11 3 12 4 12 12 3 1 12 0 12 Correct 0 2 2 1 1 2 0 1 0 0 2 0 0 2 1 2 2 1 0 2 0 2 Student Table 3: OB-related data for each student. An asterisk shows a difficult but completed OB action

Was the OB effective and did students take it seriously?

As is typical of classroom interventions, eliciting evidence of their efficacy and their being taken seriously is not straightforward. With respect to its efficacy, it is interesting to compare the number of completed OB actions with the number of completed problems, as shown in Table 4. A Fisher exact probability, p = 0.008, indicated not only that the figures of Table 4 were unlikely to have been due to chance but, importantly, that students who failed to complete the OB tended not to complete the tasks. Indeed, Table 4 shows that a necessary but not sufficient condition for the completion of both tasks was that students completed seven or more OB actions.

Table 4: Number of completed OB actions and number of successfully completed problems

When viewed as four dimensions rather than as individual actions the data offer further interesting insights. For example, while probably not surprising, the data of Table 3 show that as they move down the OB, the number of students completing each dimension gets smaller. With respect to the first dimension, 'I understand the problem', 21 students began its three actions, of which 19 (90.5%) completed all three. With respect to the second dimension, 17 began with the first action, of whom 12 (70.6%) completed all three. The third dimension, 'I apply my action plan', was begun by 13 students (one more than completed the second dimension), of whom 9 (69%) completed the dimension. Finally, of the 12 students who began the final dimension, 'I review my task', 11 completed (91.7%) it. These figures tell two stories. The first is that students who begin working on a dimension can typically be expected to complete it. The second is that once they reach the final dimension students seem almost guaranteed to complete it. In other words, an in-depth examination of the four dimensions can also inform future developments of the OB; the first and last dimensions seem less problematic with high completion rates in comparison with the middle two.


text:
Helén Sterner
email: hse@du.se

Long-term learning in mathematics teaching and problematizing daily practice

Keywords: Collaborative learning, long-term learning, mathematical communication, mathematical reasoning, mathematics teaching

 when working with key mathematical issues. I will use the data and results from the developmental research to design another study on long-term learning.

Introduction

Changing mathematics teaching is a complex process that requires the improved alignment of theory and practice [START_REF] Sowder | The mathematical education and development of teachers[END_REF]. To that end research has failed to focus on answering questions about how mathematics teaching can change as a result of collaborative teacher learning projects [START_REF] Sowder | The mathematical education and development of teachers[END_REF]. In this paper, I present an earlier study [START_REF] Sterner | Problematisera görandet, Lärares lärande om kommunikation och resonemang i matematikundervisningen i en organiserad praktikgemenskap[END_REF] as a background for a discussion of potential ways of conducting further research on understanding what a developmental research project can achieve in three years after its completion. The previous study addressed a school developmental project in mathematics in a middle sized community in Sweden. Figure 1 illustrates the background of the study and a possible direction for further research. The first section of this paper focusses on the learning process and results of a working group (i.e. the reflection group) that formed part of the developmental research study (Fig. 1). The second section comprises questions about possible ways to conduct further research three years after the study's completion. The bulk of research in the field of teacher learning and development has indicated the A community mapping of mathematics teaching failure of teachers to learn how to promote and support teaching and student learning (e.g. [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF][START_REF] Opfer | Conceptualizing teacher professional learning[END_REF][START_REF] Sowder | The mathematical education and development of teachers[END_REF]. New issues emerged during and after the previous study, including ones addressing what happened in mathematics teaching after the completion of collaboration in the reflection group and how research on mathematics teaching can integrate the significance of context. Among other questions, whether we can listen to teachers' voices (Potari, Figeiras, Mosvold, Sakonidis & Skott 2015, p. 2972-2973), and what comes to mind when teachers listen to their own narratives three years after the completion of work in a reflection group are of particular importance (Fig. 1).

Background

As Figure 1 illustrates, results from a pilot study (November 2012 -February 2013) revealed teachers' difficulties with describing the concepts of mathematical communication and reasoning, as well as with applying those concepts in their teaching. Based on the results, the main study [START_REF] Sterner | Problematisera görandet, Lärares lärande om kommunikation och resonemang i matematikundervisningen i en organiserad praktikgemenskap[END_REF] was designed as a collaborative development initiative in a working group called reflection group. The author and five mathematics teachers in grades 1-6 collaborated on the key issue of mathematical communication and reasoning in relation to their teaching of algebra. Since the reflection group met monthly for a year the study can be characterised as developmental research, which [START_REF] Jaworski | Inquiry community in an activity theory frame[END_REF] have defined as:

Research which both studies the developmental process and, simultaneously, promotes development through engagement and questioning (p. 353).

The developmental work in this study addressed change achieved in an ongoing investigative process which occurred in parallel with the active creation of the participants' meaning making related to the key issue. However, as [START_REF] Goodchild | A quest of `good´ research[END_REF] pointed out, transformations through such dialectic cyclical processes of research and development are complex. In a literature review, Sowder (2007, p. 158) outlined 10 important issues facing mathematics teachers' development. Three of those issues were of specific importance to the study at hand and constituted the underlying questions addressed in the developmental research study:

1. How do teachers learn from their professional communities about teaching mathematics? 2. What can teachers learn from investigating their own teaching of mathematics?

3. What can be learned from research on teacher change?

In this paper, I discuss theoretical frameworks for understanding mathematics teachers' sustainable and long-term learning by describing what teachers can learn in an organised community of practice [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF]) addressing a key mathematical issue. The teachers in the reflection group wanted to develop their understanding of communication in mathematics teaching in order to stimulate mathematical reasoning in their own teaching of algebra.

Methodology

The study derived from a developmental project that adopted the perspective of collaborative learning among mathematics teachers and a researcher when designing tasks and environments to investigate students' learning of mathematical reasoning related to algebra. A socio-cultural approach was adopted and the focus of the study was the learning process of the reflection group.

The theoretical perspective employed was that of communities of practice [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF], in which learning is an aspect of participation in a social practice, whose participants engage in the negotiation of meaning [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF]. The theory of communities of practice [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF] focusses on meaning making, participants' learning, and their reification of the key issue in a social context. Negotiating meaning is a central and dynamic process when teachers participate and reify. In that sense the reflection group's joint enterprise [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF] was its members to understand more about communication and reasoning in their own mathematics teaching.

The process in the reflection group concentrated on two interacting parts: participation and reification [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF]. Framing the case as a community of practice shed light on the teachers' negotiation of meaning. At the same time the negotiation of their experience with teaching and learning related the key issue of mathematical communication and reasoning to their teaching of algebra.

The developmental research cycle method [START_REF] Goodchild | A quest of `good´ research[END_REF]) and the theory of communities of practice [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF] together shaped the methodology of the study (Fig. 2). Developmental research and choice of methodology were intended to provide duality between a developmental and a research process over time and to enable a participant's perspective. Figure 2 illustrates my interpretation of Goodchild's (2008, p. 8) This study draws upon the idea that mathematics teachers' professional development should be based on their own classroom practice and students' learning (e.g., Broodie 2014; Goodchild 2014; [START_REF] Goodchild | Critical alignment in inquiry-based practice in developing mathematics teaching[END_REF][START_REF] Kazemi | Teacher Learning in Mathematics: Using Student Work to Promote Collective Inquiry[END_REF][START_REF] Matos | Mathematics teachers' professional development: Process of learning in and from practice[END_REF]. The reflection group constituted a learning community that reflected on their teaching practices as well as on their students' mathematical communication and reasoning related to algebra. Working collaboratively, the mathematics teachers developed a shared repertoire [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF]) of the key issue, mathematical communication and reasoning in relation to their teaching of algebra. Developmental research represents a methodology based on interacting cycles of research and development [START_REF] Goodchild | A quest of `good´ research[END_REF]. As illustrated in Figure 2, the developmental research cycle constitutes the largest ellipse that spans the entire study since a cyclical process clearly exists between development and research. The ellipse on the left, representing the developmental cycle (A-E), illustrates the work performed in and organisation of the reflection group [START_REF] Sterner | Problematisera görandet, Lärares lärande om kommunikation och resonemang i matematikundervisningen i en organiserad praktikgemenskap[END_REF]. The developmental process appears as a cycle between a practical experiment and a thought experiment (Fig. 2). Every meeting of the reflection group were started at phase C (i.e. common reflections, challenges and questions and activities completed by the students when working on mathematical reasoning in algebra). This meetings were recorded. The ellipse on the furthest left, representing mathematics teaching (Fig. 2), illustrates the teachers' own practice in which they attempt to align and adjust common mathematical tasks and make individual reflections. Figure 3 illustrates systematic reflections in the process and the three levels of reflection in the reflection group. The discussions in the reflection group provided empirical data that nurtured the research cycle. In Figure 2 the research process appears as a cyclical process between global and local theories. My interpretation of global theories [START_REF] Goodchild | A quest of `good´ research[END_REF]) is comparable to a theory-guided design research approach [START_REF] Gravemeijer | Educational development and developmental research in mathematics education[END_REF][START_REF] Gravemeijer | Design research from the learning design perspective[END_REF] that in turn produces new theories [START_REF] Gravemeijer | Educational development and developmental research in mathematics education[END_REF][START_REF] Goodchild | A quest of `good´ research[END_REF]). The research process guides the developmental cycle by means of local theories, which nurture the research cycle in the form of thought experiments and new questions. Reflecting together in the reflection group (phase C, Fig. 2) and the challenges of group members' own teaching resulted in problematizing questions.

Analysis and results

The analysis of the reflection group's discussions involved three steps. The first two continued throughout the developmental research process and constituted tools used for reflection in the reflection group (Fig. 3). The third step of analysis occurred following the completion of work in the reflection group. All three analyses were based on Wenger's (1998) concepts of participants' meaning making, reification and shared repertoire related to the key issue. The first two analyses and the preliminary results motivated the reflection group to negotiate their meanings of the key issue and wielded questions about what the group needs to discuss in terms of mathematical communication and reasoning. The reflection group returned to the preliminary results of analyses in order to identify further opportunities for development [START_REF] Goodchild | Critical alignment in inquiry-based practice in developing mathematics teaching[END_REF]). As a participant researcher, I provided reflection to the group members with "findings of the research" and problematizing questions based on their own thoughts and questions.

During the ongoing data analysis from the reflection group discussions, new questions emerged when participants problematized daily mathematics teaching practices and became aware of new questions and challenges in their practice. The key principle in that process was reflection on three levels, as illustrated in Figure 3 alignment and imagination) served as the participants' means for aligning and changing the discussions and activities of the reflection group. Those alignments and changes derived from participants' negotiation of meaning and reification of the key issue. The following dialogue from the initial analysis reveals that participants' shared repertoire concerns their frustration with failing to understand the meaning of reasoning in mathematics teaching.

Majken: There is, generally speaking, no resistance among the students to conducting mathematical reasoning, but when we tell them to do so, they have no idea what it means.

Irma: We need to provide them with tools that enable them to practise mathematical reasoning.

Majken: But how can we do it, when we don't know the meaning of mathematical reasoning ourselves?

The dialogue led to consider textual content of mathematics as a science and in teaching from Lampert ' (1990; 2001) and the National Council of Teachers of Mathematics ( 2008). [START_REF] Lampert | When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching[END_REF][START_REF] Lampert | Teaching problems and the Problems of Teaching[END_REF] described the science of mathematics as the formulation of assumptions followed by investigations to verify or refute them. When it comes to learning from a participant perspective, [START_REF] Lampert | Teaching problems and the Problems of Teaching[END_REF] has outlined how she stimulated students' mathematical reasoning by encouraging them to make a mathematical assumption (conjecture) about, for example, a strategy or a solution. She also stressed the importance of advancing a plausible mathematical justification for the assumption that can be explored and verified. This dynamics exemplifies how reflection group members returned to the analysis and its results.

The textual content of strategies for mathematical reasoning in teaching suggested by from [START_REF] Lampert | When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching[END_REF][START_REF] Lampert | Teaching problems and the Problems of Teaching[END_REF] and the National Council of Teachers of Mathematics ( 2008) can be global theories transformed into local ones (Fig. 2). During discussions and activities participants aligned and developed local theories into a practical experiment (Fig. 2), which they sought to align for implementation in their own mathematics teaching. The teachers attempted to support their students in using the strategies for mathematical reasoning and to conceptualise mathematical reasoning as a cyclic process of exploration, conjecture and justification. The quote bellow is from the reflection group. The students have worked with equations and to concretize that 𝑥 can have different values, the students used boxes with different amounts of beans.

Irma: […] the strategies for "the reasoning cycle" (conjecture, justification and exploration) helped both me (in grade 4) to understand the students' mathematical thinking. The students worked with the equation 3𝑥 + 3 = 2𝑥 + 5 and the students had to determine the value of 𝑥. I saw differences between students who made a wild guess and students who argued for their assumptions e.g. [… if we imagine that 𝑥 represent the boxes with beans. In each box there is same number of beans, we don´t know the amount yet. We need to balance the left and right side… if we reduce the same amount of boxes (2𝑥) from the both side of the equal sign, what will happen then?] Irma gives a student example of an initial mathematical reasoning. Later on the reflection group discussed situations from their own mathematical teaching in terms of how and when mathematical reasoning occurred and interpreted why. In the reflection group the negotiation of meaning centered on teachers' awareness of stimulating students to "become involved in the reasoning cycle of" exploration, conjecture and justification [START_REF] Lampert | Teaching problems and the Problems of Teaching[END_REF]the National Council of Teachers of Mathematics 2008). The teachers reflected on and interpreted their own teaching and used a thought experiment as a form of individual experience and reflection (Fig. 3). Ongoing analysis revealed how participants' discussions and shared repertoire about the key issue changed over time. As a participant researcher, my strategy was to focus on questions that arose in the reflection group and search for mathematical education theories that problematized the teachers' challenges and questions [START_REF] Goodchild | A quest of `good´ research[END_REF] in thought experiments (Fig. 2).

Results and conclusions

I investigated how the reflection group developed their meaning making and shared repertoire related to mathematical communication and reasoning, which promoted a change in the members' ways of communicating about mathematics teaching in relation to students' mathematical communication and mathematical reasoning. Four relevant changes in the mathematics teaching were identified in the reflection group's discussions and learning. The changes ranged from understanding communication and reasoning to identifying, interpreting, applying and practising that reasoning. Teachers in the reflection group also changed their approach to discussion. In the initial stage, they achieved consensus, but gradually adopted a positive yet critical approach in which they problematized the process of learning in and from daily practice [START_REF] Sterner | Problematisera görandet, Lärares lärande om kommunikation och resonemang i matematikundervisningen i en organiserad praktikgemenskap[END_REF]. The three levels of reflection (individual and shared reflections and the researcher's reflection on the preliminary outcome in the reflection group) resulted in discussions that promoted new and meaningful ways to communicate mathematically and stimulate mathematical reasoning in algebra. This methodology could be a way of linking the activities of students and teachers.

Ultimately, in response to Potari et al.'s (2015, p. 2,972) 'How can we link students' activity to teachers' activity', the present study demonstrates the importance of linking research and development in order to enable teachers to learn about their own mathematics teaching and students' learning. Moreover it provides a response to Sowder's (2007, p.158) questions; 'How do teachers learn from their professional communities about teaching mathematics' and 'What can teachers learn from investigating their own teaching of mathematics' by indicating the combined method of the developmental research cycle [START_REF] Goodchild | A quest of `good´ research[END_REF]) and the theory of communities of practice [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF], along with reflection on three levels (Fig. 3) allowed using the results and questions that emerged in the reflection group.

Implications and further research

The main study, between March 2013 and January 2014 (Fig. 1) focused on a group's learning process, the group's meaning making of the key issue. The third question from Sowder (2007) 'What can be learned from research on teacher change' found a partial answer. Results of the study demonstrate what can happen in the change process when a reflection group begins to work actively. On that note, other questions are whether mathematics teachers' activities and shifts in collaborative learning change their mathematics teaching and whether teachers' meaning making and their shared repertoire about communication and reasoning in mathematics teaching influence their teaching and persisted three years later. Since I am curious about teachers' learning from a long-term, sustainable perspective, one question I will continue to carry with me comes from the last meeting in the reflection group, when one of the teachers, Clara said:

Clara: I'm worried about myself. It's very easy to sit back and fall into old habits when we no longer meet for reflection. What will my teaching be like now?

As Clara suggests, a question not answered in this study is whether teachers' activities and the shift in their approach in the discussion can change their mathematical teaching shortly after and also three years later.

Possible new routes and issues three years after the completion of the reflection group

The research in the reflection group involved the group's process of learning about the key issue of communication and reasoning. The teachers' meaning making and shared repertoire [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF] about that issue shifted from understanding to identifying, interpreting, applying and practising mathematical reasoning. The present study does not provide answers about what happened in the light of the grey ellipse representing mathematics teaching in Figure 2 or what happened to the teachers' thoughts and their mathematical teaching three years later. What questions will arise when the five teachers listen to the interviews they gave in 2014, after the completion of work in the reflection group and what thoughts will they have on hearing their own narratives? Will it be possible to use the same theory of communities of practice [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF] to analyse the teachers' individual reflections when they listen to their own voices from those interviews? Further research is necessary to understand sustainable, long-term learning in this case whether the mathematics teachers' activities and shifts in their collaborative learning actually changed their mathematics teaching over time. What roles, if any, do teachers' discussions changed meaning making and changed shared repertoire about mathematical communication and reasoning play in their teaching in a long-term sustainable perspective?

Figure 1 :

 1 Figure 1: The study's chronological development.

  schematic figure of the developmental research cycle.

Figure 2 :

 2 Figure 2: Interpretation of the developmental research cycle (Goodchild 2008, p. 208)

Figure 3 :

 3 Figure 3: Systematic reflections among participants in the reflection group

  , since an essential component of developmental research is participants' interpretation (Kvale & Brinkman 2009). Wenger's (1998) modes of belonging (i.e engagement, Individual reflection before during and after implemented lessons. Common reflections and discussions in the reflection group. The researcher's reflection on the analysed discussions in the reflection group.

Proceedings of CERME10


text:
Fredrick Odindo
email: f.odindo@uea.ac.uk

Teacher learning in teaching the topic of quadratic expressions and equations in Kenyan high schools: Learning Study design

Keywords: Learning Study, variation theory, problem-solving, teamwork

Performance in mathematics national examinations in Kenya has been weak and raises questions about the pedagogical approaches adopted by teachers. The purpose of this paper is to report on teachers' experiences of the application of a problem-solving teaching strategy in a Learning Study design to teach factorization of quadratic expressions. The study follows a qualitative and interpretive research approach with data collected through questionnaires, classroom observations, video replays and reflection sessions after lessons. The findings reveal teachers' appreciation of the application of problem-solving teaching in a Learning Study design, saying it helped them to observe the difficulties students experience in learning algebra and the shortcomings in their lesson preparation. In addition, they noted the importance of reflection in that it helped them improve the second lesson.

Introduction

Teaching is an activity that involves engagement of the teacher, the student and the content [START_REF] Loef Franke | Mathematics teaching and classroom practice[END_REF][START_REF] Stigler | The Teaching Gap, Best Ideas from the World's Teachers for Improving Education in the Classroom[END_REF]. In mathematics teaching, this engagement needs to be effective to create a productive learning environment for both teachers and students [START_REF] Loef Franke | Mathematics teaching and classroom practice[END_REF]. However, [START_REF] Stigler | The Teaching Gap, Best Ideas from the World's Teachers for Improving Education in the Classroom[END_REF] note that levels of engagement among the three is not uniform due to different classroom cultures and teaching strategies in different countries. They observed, for example, that in some US classrooms, teaching strategies are so procedural and teacher-directed that students are passive recipients of knowledge, and that there is little interaction between student and content. [START_REF] Stigler | The Teaching Gap, Best Ideas from the World's Teachers for Improving Education in the Classroom[END_REF] finding concerning US mathematics classrooms seems to be relevant in other classroom cultures in countries such as Kenya (CEMASTEA, 2010). Based on past performances, the Government of Kenya initiated an in-service education programme for high school teachers of mathematics and science in 1998. Among the strategies employed was Lesson Study approach, which implicitly applies constructivist theory [START_REF] Elliot | Lesson and learning study and the idea of the teacher as a researcher[END_REF]. However, a survey conducted to check the extent of the implementation of the programme, reported that about 65% of the teachers were not implementing the programme and were applying the teacher-directed method of teaching (CEMASTEA, 2010). Teachers cited lack of clear guidelines and workload among reasons. Students continued to post weak results in mathematics in 2012 and 2013 from the Kenya national examinations (KNEC, 2014). This study adopts a Learning Study (LS) design which explicitly applies Variation Theory to support classroom learning. Little is known about LS in Africa, and the outcome of this present study may support the further development of LS in Kenya in particular.

The purpose of this study is to solicit teachers' views and experiences of the application of a problem-solving teaching strategy in a LS design to teach factorization and solution of quadratic equations, one of the topics poorly performed by students (KNEC, 2014). My intention is to eventually extend the approach to other mathematics topics. The research question of the present study is "What are the teachers able to learn about the students learning of factorization of quadratic expressions in a Learning Study design?"

Variation theory

Variation Theory is a theory of learning which asserts that to learn something entails experiencing it in a variety of ways. The theory proposes that learning is always directed towards an object, which could be a skill or a concept referred to as the object of learning in the 'what' and 'how' aspects of learning [START_REF] Marton | Learning and Awareness[END_REF]. The 'what' aspect is the content to be learnt while the 'how' aspect is concerned with the process of learning that enhances a student's ability to apply the learned concept in a new environment [START_REF] Elliot | Lesson and learning study and the idea of the teacher as a researcher[END_REF][START_REF] Lo | Variation Theory and the Improvement of Teaching and Learning[END_REF]. The theory postulates that learning takes place when students focus on a critical feature of the object of learning. For example, suppose the solution of simultaneous linear equations by elimination were the object of learning: a critical feature to be discerned can be the collating of the equations so that one variable has the same numerical coefficient in both equations. To achieve this, [START_REF] Lo | Variation Theory and the Improvement of Teaching and Learning[END_REF] and [START_REF] Marton | Necessary Conditions of Learning[END_REF] propose that teachers need to create learning opportunities by explicitly or implicitly offering patterns of variation in which some parts remain invariant as others vary. Four patterns of variation are identified, namely: contrast, separation, generalization and fusion. For example, to discern the concept of quadrilateral, a type of quadrilateral would be kept invariant and contrasted with other polygons such as triangles and pentagons. To discern a particular type of quadrilateral, for instance a kite, it is kept invariant as other types such as rectangle, trapezium are varied. The kite is separated from the whole and focused on. To generalize that the total sum of interior angles of a quadrilateral is 360°, each type of a quadrilateral is varied with its sum of interior angles calculated. To compare properties of different types of quadrilaterals, such as a kite and a rhombus, the two and their properties are brought into focus simultaneously, the fusion pattern of variation.

Learning Study

Learning Study (LS) provides a framework for supporting learning in the classroom by applying aspects of Variation Theory, in which all the three categories of persons participating in the lesson (the teachers, the students and the researchers) could learn in the process [START_REF] Marton | Necessary Conditions of Learning[END_REF][START_REF] Pang | Using the learning study grounded on the variation theory to improve students' mathematical understanding[END_REF]. LS adopts a Lesson Study organizational structure in which a group of teachers prepare a lesson together, then one teaches the lesson while others observe as they collect research data and thereafter converge for a reflection session [START_REF] Pang | Using the learning study grounded on the variation theory to improve students' mathematical understanding[END_REF]. LS focuses on the object of learning which points to the beginning of the learning process with learning 'what' and learning 'how' aspects [START_REF] Lo | Variation Theory and the Improvement of Teaching and Learning[END_REF]. In this study, the topic of quadratic expressions and equations is the 'what' aspect while the 'how' aspect is addressed by small group discussion approach to learning. For ease of monitoring the learning process, the object of learning is categorized into: (a) lived object of learning 1, (b) lived object of learning 2, (c) intended object of learning and (d) enacted object of learning. Part (a) is concerned with prior experiences and awareness that students have about the concept and is monitored through a student's diagnostic pre-test or interview whose outcome is considered in lesson preparation. Part (b) is the acquired experiences after the teaching of the lesson and is monitored through a student's post-test or interview. Part (c) is the planned lesson and part (d) is the taught lesson [START_REF] Pang | Using the learning study grounded on the variation theory to improve students' mathematical understanding[END_REF]. Parts (c) and (d) are monitored through a post-test or interview and reflection session.

Methodology

The study on which this paper is drawn follows a qualitative and interpretive research approach in a LS design conducted in two classes in Kenya. Three teachers planned the lesson together as explained earlier. Prior to the first lesson, a diagnostic pre-test was given to the students in both classes; a post-test identical to the pre-test was given to each class at the end of each lesson. The teachers will be addressed by pseudonyms as Dominichead of mathematics, Peterteacher of the first lesson and Johnteacher of the second lesson. The two classes were for third year high school students (age 16-18 years) comprising 68 students altogether.

Data were collected through classroom lesson observation, pre-post questionnaires, video recordings of the lessons and transcribed reflection sessions. Teaching was approached through a small group discussion by the students. The object of learning was factorization of a quadratic expression with a unit coefficient of x 2 (i.e. x 2 + bx + c). The critical feature was the identification of factors of the constant term of a quadratic expression that sum to the coefficient of x, often expressed in textbooks as "sum and product".

The items in the questionnaire, whose outcomes were considered in the preparation of the lesson, included: (1) Why is

6 5 2   x x
called a quadratic expression? (2) What do we consider in attempting to factorize a quadratic expression such as the one given above? (3) How many factors do we expect from a factorized quadratic expression? Frequent student responses included: (1) the given expression is called a quadratic expression because it has unknowns, (2) we consider like terms, (3) two factors (considered the correct answer). The questionnaire responses were scored 1 for a correct answer and 0 otherwise. Based on the students' responses, the teachers prepared for the intended object of learning in a 40minutes lesson, incorporating a hands-on activity intended to raise students' curiosity, and to motivate them to discuss in small groups as the approach was new to them. The first task on the activity was to form a rectangle using the pieces of paper shown in Figure 1, and to find the product of the sides of the rectangle formed. This was intended to lead to the factorization of The second task was to explain the relationship between the numerical terms from their expression of the area, (x + 2) (x + 3); and the coefficient of x and the constant term in the quadratic expression.

First lesson

The first lesson was taught by Peter in his class of 40 students. Prior to this lesson, students had been taught the expansion of quadratic factors of the form (p + q) (p + r). Peter introduced the lesson by asking the students to identify the coefficients of x 2 and x in the expression x 2 + 4x + 3. Whereas all the students could correctly identify the coefficient of x as 4, most of them were unable to correctly identify the coefficient of x 2 . Students gave responses that included: x × x, x and 2presumably from the exponent 2. The teacher asked them to discuss in pairs and seek a correct solution. After a while, a student correctly identified the coefficient as 1, but could not explain her answer. The teacher explained why it is 1.

Peter then asked the students to factorize the expression

6 5 2   x x
. After about five minutes, he asked the students to form eight groups of five members each and he distributed the pieces of paper in Figure 1 to each group. He explained the first task of the activity and allowed 15 minutes for the task. After 15 minutes, only two groups had formed the rectangle. Peter allowed a further 10 minutes for discussion before calling upon groups to present their work. The teacher grouped the students' work in categories as shown in Table 1.

Category

No One group from category one and the category two group were asked to present their work shown in Figure 2 and Figure 3 respectively. Student 2:

The width has two pieces of x each giving an area of x × x = x 2 . The length has a piece of x at the bottom and the upper part has two pieces of x plus the big piece whose length is x giving a total of 4x 2 . Total area, A = x 2 + 4x 2 .

Due to pressure of time, Peter summarized the lesson by explaining task two procedurally, thus, to factorize a quadratic expression such as

6 5 2   x x
, identify the factors of the constant term, 6, that sum to 5, the coefficient of x. Thereafter he told the students to factorize x 2 + 3x + 2 as homework, before administering the post-test. The results of the pre-test and the post-test are shown in Table 2.

Items

Percentages of correct responses

Diagnostic pre-test Post-test Question 1 3 30

Question 2 3 10

Question 3 70 43

First lesson's reflection session

Peter felt that his introduction took more time than he had expected: as he remarked,

Peter: I would have taught how to get the coefficient first before I look at this lesson.

Other highlights during the reflection include: Dominic:

The students had some fear I do not know what they were fearing [...]

John: I think we should have done some peer teaching. We forgot how the students would present their work and this became unexpected challenge.

Researcher: Peter gave students a long time for discussion because he wanted everybody to obtain the correct answer not knowing that the strength of learning at times is in the few mistakes made by students.

Based on the reflections, the post-test results and the fact that the lesson was not implemented as planned, the teachers modified the lesson, which was retaught in John's class (second lesson).

Second lesson

John introduced his lesson by asking students to expand the expression, (x + 2) (x + 1), which they did to obtain x 2 + 3x + 2. He then asked them to identify the coefficients of x 2 and x from x 2 + 3x + 2. The majority answered the question correctly. John then asked the students to factorize the expression

6 5 2   x x
. After about five minutes, he asked the students to form seven groups of four members each. The class had 28 students present. He distributed the pieces of paper shown in Figure 1 to each group and stated the tasks for the activity as: (1) Form a rectangle with all the pieces of paper given and work out the area of the rectangle formed. (2) Find the relationships between the constant terms of the factors of your worked-out area and i) the coefficient of x in the expression 6 5

2   x x
, ii) the constant term in the same expression 6 5

2   x x
. This was part of modification made on the lesson. After 15 minutes, he stopped the group work and asked some groups to discuss their work with the whole class. One group from category one and the category two group were asked to present their work. The representative of the category one group explained the working thus, Student 1:

Width = 1 + 1 + 1 + x = 3 + x, Length =1 + 1 + x = 2 + x Area, A = L × W = (2 + x) (3 + x) = x 2 + 5x + 6
The representative of category two explained her work referring to a figure similar to Figure 3.

Student 2:

The two strips above are multiplied to obtain x 2 and the four pieces on one side (pointing at the width with 1 unit by x units strip and the three 1 unit by 1 unit) are counted and multiplied by x to obtain 4x. Area, A = x 2 + 4x.

After the presentation, John discussed task two with the whole class. With the help of the students and referring to student 1's expression, John simultaneously presented the factors of 6, {(1 × 6), (2 × 3), (-1 × -6), (-2 × -3)} and the addends of 5, {(0 + 5), (1 + 4), (2 + 3) and so on}. He introduced the second activity that asked the students to factorize, x 2 + 3x + 2. The students correctly factorized the expression. The teacher summarized the lesson and administered post-test questionnaire. Both pre-post results are shown in Table 4. The teachers prepared the lesson to apply the generalization of patterns of variation and invariance. This was realized fully in the second lesson.

Items Percentages of correct responses

Varied Invariant Discernment x 2 + 5x + 6
Working out the area of rectangles formed.

Factorization of a quadratic expression with a unit coefficient of x 2 depends on the factors of the constant term that sum to the coefficient of x

x 2 + 3x + 2 The two different expressions were varied to help the students to generalize the process of factorizing a quadratic expression with a unit coefficient of x 2 such as x 2 + bx + c. The students applied the cuttings to form the rectangles whose areas represented the factorizations of the given quadratic expressions, that is, x 2 + 5x + 6 and x 2 + 3x + 2.

same after the second lesson through the post-test result. This supports [START_REF] Hiebert | Learning to learn to teach: an "experiment" model for teaching and teacher preparation in mathematics[END_REF] suggestion that lessons should be treated as experiments with explicit preparation of all activities. The teachers learned that the LS design through its post-test aspect helps monitor students' learning progress, which enables a teacher to address students' errors/misconceptions in the immediate subsequent lessons.

Figure 1 :

 1 Figure 1: Paper cuttings for a hands-on activity aiming at the factorization of 6 5 2   x x

.

Figure

  Figure 2: Correct working Figure 3: Incorrect working The Figure 2 representative explained thus, Student 1: Length, L = 1+1+1+x = (3+x) and width, W = 1+1+x = (2+x). Area, A = L×W = (3+x) (2+x)The Figure3representative explained,

Table 1 : The categories of groups and how they carried out the task

 1

Table 2 : The percentage correct responses of the pre-test and post-test from the first lesson

 2

  Table 3 shows how groups carried out the tasks, with same categories as in Table 1.

	Category	No. of Group(s)	Rectangle	Area
	One	3	Correct rectangle	Correct working
	Two	1	Correct rectangle	Wrong working
	Three	3	Correct rectangle	No area worked out

Table 3 : The categories of groups from the second lesson and how they carried out the task

 3

Table 4 : The percentage correct responses of the pre-test and post-test from the second lesson

 4

Table 5 : Generalization pattern of variation and invariance applied in the enacted object of learning

 5

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Acknowledgment

This study is supported by a University of East Anglia doctoral studentship. I thank the teachers and students who participated in data collection, the Principal of the school and the entire school community that cooperated with the research team. I sincerely thank my supervisors, Elena Nardi and Tim Rowland, for their support in the writing of this paper.

Discussion and conclusion

The activities proved challenging as students took time to discuss and explore ways of factorizing the expression x 2 + 5x +6 [START_REF] Lester | Learning how to teach via problem solving[END_REF]. At the end of the discussion, eight groups out of 15 did not factorize the expression, and one group even failed to form the rectangle, as shown in Table 1 andTable 3. Peter even allowed more discussion time but still some groups could not form the rectangles. Peter therefore, did not apply fully the patterns of variation and invariance. [START_REF] Marton | Necessary Conditions of Learning[END_REF] cautioned teachers to take control of students' own work during learning to implement the planned patterns of variation. The effect was reflected in post-test results of Table 2. A comparison of pre-post results (lived objects of learning 1 and 2) of the two lessons, Tables 2 and4, show that at the pre-test, Peter's class had higher scores in questions 2 and 3 but at the post-test, John's class, where the pattern of variation was fully applied, had a notable improvement in all the questions than Peter's class. Question 2 that addressed the critical feature had a slight improvement of 7% in Peter's class compared with 71% in John's class.

The content of the lessons addressed the "what" aspect of the object of learning. Eight groups had difficulty expressing the sides of their rectangles in algebraic form, thus failing to factorize the expression x 2 + 5x + 6 as shown in Tables 1 and3. The expected algebraic expressions for the sides of the rectangle fall within the topic of formation of algebraic expressions taught in the first year of high school according to the Kenyan mathematics syllabus for high schools. Also, in the introduction of the first lesson, the students could not identify the invisible 1 as the coefficient of x 2 from the expression x 2 + 4x + 3, which suggested that students either did not understand multiplicative identity property of real numbers or they did not know the term coefficient. These cases show that students could not recall what they had been taught earlier. These were learning moments for teachers to realize that the problems that students experience in quadratic expressions and equations originate from the introductory contents of the algebra.

The planning gaps identified during the lessons contributed to the time management issues especially, the first lesson where students did not work on the second activity as was planned; and they also had a short time to discuss the second task of the first activity. The effect was reflected in the post-test outcome from the first lesson's class as shown in Table 2. From these observations teachers learned that LS design is helpful in focusing the teachers in every aspect of the lesson that improves students learning. They also learned that a good implementation of a lesson by applying patterns of variation can improve students' learning as suggested in the second lesson's post-test result Table 4. The teachers appreciated the implementation of the second lesson where students were able to generalize the conditions for factorization of a quadratic expression with a unit coefficient of x 2 .

Peter:

Yeah the lesson was good. I am sure now they are aware that they can use that formula without the cuttings and factorize any quadratic expression.

Dominic:

The fact that the two activities were discussed helped the students to see the relationship and I am sure they can now factorize the quadratic expression without any problem.

The teachers also learned the need for explicit preparation of all activities and for good time management, as they stated during the reflection session after the first lesson and confirmed the Thematic Working Group 19

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