text:
Constantinos Xenofontos
email: xenofontos.c@unic.ac.cy

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

Explanations as tools for evaluating content knowledge for teaching: A cross-national pilot study in Cyprus and Greece

Keywords: Mathematical knowledge for teaching, instructional explanations, disciplinary explanations, prospective teachers

In this paper, as a proxy for their mathematical knowledge for teaching, we examine and Greek (n=12) first-year undergraduate teacher education students' written explanations regarding a linear equations-related scenario. Qualitative analyses identified four broad themes permeating most accounts, irrespective of nationality, which were interpretable as either disciplinary explanations or instructional explanations. The themes concerned (a) definition of unknowns, (b) inverse operations, (c) presentation of objectives, and (d) rote procedure. The analyses indicated the weakness of students' explanation, whether from a disciplinary or an instructional perspective, which was independent of their country of origin.

Introduction

The question regarding what kind of knowledge is needed to teach mathematics has preoccupied a number of scholars, while, at the same time, the association of mathematics teacher knowledge with instructional quality and student learning is considered to be complex [START_REF] Charalambous | Perspectives on priority mathematics Education: Unpacking and understanding a complex relationship linking teacher knowledge, teaching, and learning[END_REF]. Over the last years, several conceptualizations of this kind of knowledge have been proposed. Drawing upon [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] seven tentative categories for teacher knowledge, [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF], for instance, developed the mathematics knowledge for teaching framework, which divides subject matter knowledge and pedagogical content knowledge into three further sub-categories each. From a different perspective, grounded in data-driven analyses of videotaped lessons by prospective elementary teachers, [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: the Knowledge Quartet and the case of Naomi[END_REF] proposed the knowledge quartet, a framework comprising four units, namely, foundation, transformation, connection, and contingency. These units, the authors claim, can be found in every mathematics lesson and can be used for evaluating teacher knowledge during the course of teaching. More recently, [START_REF] Davis | Profound understanding of emergent mathematics: broadening the construct of teachers' disciplinary knowledge[END_REF] have written on profound understanding of emergent mathematics, arguing that teachers' knowledge could be productively interpreted as a complex evolving form which is tacit and is better understood as a learnable disposition than a domain to be mastered.

Several of scholars (i.e. Copur-Gencturk & Lubienski, 2013;[START_REF] Phelps | Assessing mathematical knowledge for teaching: The role of teaching context[END_REF][START_REF] Hill | Developing measures of teachers' mathematics knowledge for teaching[END_REF] have urged for the development of effective measures of the knowledge required for teaching mathematics. Various approaches to this task have been have been undertaken [START_REF] Charalambous | Perspectives on priority mathematics Education: Unpacking and understanding a complex relationship linking teacher knowledge, teaching, and learning[END_REF], including, paper-and-pencil tests, the use of lesson videos inviting teachers to critique and/or predict, as well as inviting post-instruction reflections on both lesson plans and the actual realization of the plan. Adaptations of such approaches, however, should be carried out with careful sensitivity to the cultural context, as teacher knowledge, of both inservice [START_REF] Andrews | Teaching linear equations: Case studies from Finland, Flanders and Hungary[END_REF] and pre-service teachers [START_REF] Xenofontos | The cultural dimensions of prospective mathematics teachers' beliefs: Insights from Cyprus and England[END_REF], is conditioned by cultural expectations with respect to what mathematics is valued and how it is presented. For example, [START_REF] Andrews | Opportunities to learn in the Budapest mathematics classroom[END_REF] has highlighted how some cultures construe competent mathematics teachers as those who ensure their students complete a large number of tasks every lesson, while others see competence in the completion of a few.

An important element of a teacher's didactical repertoire is the explanation. Explanations, which typically draw on students' prior knowledge [START_REF] Leinhardt | Seeing the complexity of standing to the side: Instructional dialogues[END_REF], are manifestations of teachers' depth of content knowledge [START_REF] Inoue | Rehearsing to teach: Content-specific deconstruction of instructional explanations in preservice teacher training[END_REF]. Of [START_REF] Leinhardt | Instructional explanations: A commonplace for teaching and location for contrast[END_REF] four classifications of explanation, two are particularly relevant here; disciplinary explanations are domain specific and conform to the epistemological expectations of the relevant discourse, and instructional explanations, which are intended to teach some aspect of a particular subject matter to others, are "jointly built through a coherent discourse surrounding a task or text that involves the whole class and the teacher working together" (Leinhardt, 2001, 340). Therefore, in accordance with prior research that explanations may be a useful tool for measuring prospective teachers' knowledge (see, for example, [START_REF] Charalambous | Prospective teachers' learning to provide instructional explanations: How does it look and what might it take[END_REF][START_REF] Inoue | Rehearsing to teach: Content-specific deconstruction of instructional explanations in preservice teacher training[END_REF], we present the results of a pilot study focused on the use of a single task as a means of evaluating Greek and Cypriot beginning teachers' didactically-related mathematical understanding. The extent to which we consider our tool as effective is discussed in another paper of this TWG (see [START_REF] Andrews | Beginning teachers' perspectives on linear equations: A pilot quantitative comparison of Greek and Cypriot students Paper presented to the Tenth Congress of European Research in Mathematics Education[END_REF]. Here, we focus on the identification of similarities and/or differences between the two cohorts' written explanations, which, we believe could offer better insights into the mathematical knowledge beginner teachers bring to teacher education and how we, as teacher educators, could build on or deconstruct this knowledge.

The study

Participants were first-year undergraduate students reading for a degree in elementary education at a private university in Cyprus. With Greek being the language of instruction, the programme includes both Greek-Cypriot and Greek students. Data collection took place before students had experienced any university instruction and involved 21 Greek-Cypriot and 12 Greek students. Students were shown, with no additional text, the solution to the equation presented below, and asked to write a short account indicating how they would explain it to someone who had missed the lesson in which it had been taught. Such a task has the advantage that it allows for both disciplinary and instructional explanations [START_REF] Leinhardt | Instructional explanations: A commonplace for teaching and location for contrast[END_REF].

x + 5 = 4x -1 5 = 3x -1 6 = 3x 2 = x Data were analysed as single set by means of a constant comparison process [START_REF] Fram | The constant comparative analysis method outside of grounded theory[END_REF]. These analyses, as reported in Andrews and Xenofontos (2017, TWG11), yielded seven themes with respect to how participants explained the solution. Of these seven, four themes were present in most accounts and are here considered in depth. In the following, we present the results of this process, discussing each theme in relation to first the Cypriot and then the Greek students. All names are pseudonyms.

Thematic Working Group 11

Results

Definition of unknowns

A number of Cypriot students wrote statements interpreted as explicit definitions of the unknown. For example, Ekaterini wrote that the "known numbers are the ones that don't include a letter, as for example, 5 and -1. Unknown numbers are the letters or the numbers that are accompanied by a letter, for example, 4x and x". In similar vein, Carissa wrote, "I separated the known from the unknown numbers: Unknown x + 5 (known) = unknown 4x -1 (known)". Both comments, we argue, are structurally equivalent in their assertions that unknowns are represented by letters and the knowns by numbers. Other students' definitions tended to the implicit. For example, Hermione wrote that "the first thing to do is to separate x from the numbers. In other words, to set apart the known from the unknown so that we can find what x is". Likewise, Alexandra commented that their teacher had given "two expressions, each with an unknown (x) and which are equal to each other. She asked us to find out what the value of x is". While neither student explained explicitly what they meant by the unknown, their comments present a tacit understanding of the unknown as the number represented by the letter x that has to be found.

In ways similar to their Cypriot colleagues, Greek students offered either explicit or implicit definitions of the unknown. With respect to the former, Nina wrote that "we separate the known from the unknown numbers: known: 5, -1 and unknown: x, 4x", while, in a slightly more fluid manner, Paraskevi commented that "I separate the known from the unknown. The known numbers are all numbers, unknown is whatever is a letter, for example, x, y, z, w". Students explicitly distinguished between x, as a representation of the missing number, and numerical values separated from the unknowns. With respect to the latter, Callisto, having written of the need to separate knowns from unknowns, wrote that "all the 'x' are taken to the right and whenever 'x' is on the left part of the equation, you change the sign. On the left part, the known numbers enter while the unknown enter the right". Similarly, Daphne commented that "the first unknown x is moved to the second part of the equation but instead of a plus we make it a minus". In both cases, students offer instructions in which the x terms are afforded a treatment that distinguishes them from numerical terms in ways that indicate their understanding of the nature of the unknown and its significance in the equation solving process.

Presentation of objectives

The accounts of almost all Cypriot students included some statement regarding the objectives of equation solving. In most cases this was implicit and typically represented in statements concerning the separation of knowns from unknowns. The briefest such statements were as in Carissa's "I separated the known from the unknown numbers" and Ioanna's "we set apart the known and unknown numbers". In such statements can be seen an understanding that the identification of the unknown was the objective. Even when such students wrote longer statements, the message was the same, as in Chloe's "we separated known and unknown numbers, that is, 5 and 1 and 4x and x".

Here, the distinction between Chloe's and the other two students' statements is that she incorporated an implicit definition of the unknown, but offered no more than them with respect to her objectives. Other Cypriot students offered accounts with a slightly less implicit objective in that they also discussed the separating of the knowns from the unknowns but in so doing explicitly mentioned the role of x, which Chloe did not. For example, Medea wrote that the "first step is to set apart the known and the unknown numbers, that is, to bring 'x' to one side and the numbers to the other", while Stamatia commented that "we separated the known from the unknown numbers, that is, the 'x' and the 'simple' numbers.

Four students wrote statements indicative of an explicit objective. Two of these were brief, as in Stefania's "the aim is to find which number equals x" and Irene's "the question in this equation is to find 'x' and what value it has". The other two students wrote longer statements, as with Hermione's "the first thing to do is to separate x from the numbers. In other words, to set apart the known from the unknown so that we can find what x is". In all three statements, albeit expressed differently, can be seen an expression concerning the identification of the unknown, x. Finally, of the four students who offered an explicit objective, Alexandra's statement also offered the only evidence of a relational understanding of the equals sign. She wrote that "the teacher gave us two expressions, each with an unknown (x) and which are equal to each other. She asked us to find out what the value of x is".

For the Greek students, the same three categories of response were identified. For example, with respect to implicit goals embedded in statements about separating the knowns from the unknowns, Pantelis wrote that "we part the known from unknown numbers", while Moira wrote that "I would tell the student that we separate the known from the unknown numbers and then make the calculations". As far as the second category is concerned, whereby students offered implicit objectives alongside an explicit mention of the role of x, Callisto commented that "we separate the known from the unknown numbers. For assistance, you can underline the 'x' from the numbers", while Paraskevi wrote that "I separate the known from the unknown. The known numbers are all numbers, unknown is whatever is a letter, for example, x, y, x, w". Finally, one Greek student offered an explicit objective. In this respect Panorea wrote that she "would explain to the student that it's an equation whereby we are trying to find the unknown x. The data we have are the numbers. After, I would explain how to find x".

Rote procedure

Few Cypriot students did not offer a rote rule for solving the equation and of those that did all alluded, either explicitly or implicitly, to the redistributive 'change the side and change the sign'. With respect to those who discussed the rule explicitly, one of the more detailed accounts was offered by Irene, who wrote that Whatever moves to the other side changes 'sign'. The 'sign' is (+) or (-). We have x + 5 = 4x -1 and want to separate 'x' from the numbers, so we have to say 5 = 3x -1. Why? 1x went to the right part and changed signs becoming -1x (since x is now 1x) therefore 4x -1x = 3x (4 -1 = 3). 'x' in common. Now we have to justify 6 = 3x. We have 5 = 3x -1. Since (-1) is a number, it must shift to the left part of the equation and become plus, that is 5+1= 6. (It was -1 and since crossing over the = it becomes plus). So 6 = 3x. These, and other, students seemed both confident and clear as to the process involved in solving the equation. They offered rules whereby objects were moved from one side of the equals sign to the other along with a change of sign. Of course, the direction of such movement was determined by the solution presented to them but in no case did students offer any justification for the changing of the sign. Interestingly, and unique among students, Elina offered a no less unwarranted but general account. She wrote that At first we get an equation with letters and numbers. The first thing to do is to move all numbers to one side and all letters on the other. If a number has a minus (-) sign or plus (+) once they are moved to the other side their sign changes; in other words, from minus (-) it becomes plus (+). That's how we deal with similar calculations.

Several Cypriot students offered accounts, typically short, in which the movement and the changing of signs were implicit. For example, Hermione wrote that "[a]s you can see x and 4x have been joined and the same is true of 5 and 1. Therefore, x has become 3x and then 5 + 1 is 6". In these, and other, cases neither the movement across the equals sign nor the changing of the sign were made explicit. Our interpretation is that students were familiar with the process and saw such properties as givens rather than something in need of either explanation or warrant.

As with their Cypriot colleagues, Greek students typically offered a 'change the side change the sign' rule. Also, as before, the extent to which this was presented explicitly varied. For example, with respect to an explicit account, Daphne wrote that Since we have two unknowns on both parts of the equation, the first unknown x is moved to the second part of the equation but instead of a plus we make it a minus. Then we subtract it from 4x to get 3x. After getting -5 in the second part, we move it to the first, therefore 1 is not a minus but plus. Then we add 5 and 1 to get 6.

In their accounts can be seen an understanding of the 'change the side change the sign' rule for solving equations. In neither case, however, can be seen evidence of a relational understanding of the equals sign as would be represented in statements justifying the described actions. There was no sense, for example, that students saw these actions as a consequence of adding or subtracting equivalent amounts from each side. In addition, as with the Cypriot students, one student offered a generalised account of the same process. In this respect, Callisto wrote that "[a]ll the 'x' are taken to the right and whenever 'x' is on the left part of the equation, you change the sign. On the left part, the known numbers enter while the unknown enter the right. Afterwards, we add the known to the known and the unknown with the unknown".

However, the majority of Greek students offered implicit summaries of the solution with which they had been presented. Typical of others, Panorea wrote that she would "shift the given numbers to one side and the unknown, x (or 3x) on the other side (e.g., 3x) and that way I would find the unknown x". Similarly, Paraskevi wrote that "[a]t this stage, we carefully observe when to change (+) or (-). When they move from the 1 st part to the 2 nd , the sign changes from let's say (+) to (-) and vice versa. Once we reach this point 6=3x". In such statements can be seen different but equally incomplete accounts of the rote rule. For example, Panoreas's account highlighted the bringing together of the knowns and unknowns respectively, leaving the reader to infer the changing of the sign, while Paraskevi's emphasised the changing of the signs at the expense of her detailing the changing of the sides of like objects. In other words, both left significant elements for their readers to infer.

The inverse operation

In almost all cases, having articulated some sense of a rote rule, students from both countries invoked an operation reversal to explain the final line of the solution, whereby 6 = 3x became 2 = x. In this respect, eleven of the 21 Cypriot students offered, albeit implicitly, an understanding of the inverse operation necessary to transform the line 6 = 3x to 2 = x. Typical of these were the comments of Chloe, who wrote that "after reaching 6 = 3x, we divided 6 by 3 so that x will be by itself. I found x to be 2" and Hermione, who wrote "then I divide 6 by 3 and x is equal to 2". Of these eleven students, two students offered more extended but no less implicit suggestions, as in Stamatia's comment that "we take number 6 and equate it '=' with 3x. Then we divide 6 by 3, i.e. 6/3 equal to x1, "6/3 = x. 6/3 makes 2 and so x1 is equal to 2, 'x = 2' ". In these cases, students' comments about dividing by three seemed to us to reflect, at least implicitly, a recognition of the structural significance of the unknown's coefficient that necessitated division.

A further seven students offered accounts indicative of a better developed understanding of the role of the coefficient and its function with respect to inverse operations. For some this could be seen in the ways they reiterated earlier comments about isolating the unknown. This was seen in, for example, the writing of Perikles, who added that "arriving at 6 = 3x I divided 3x by 3 to get x alone, and 6 on the other side with the 3 before the unknown. So the answer is x = 2. Others in this group were more explicitly aware of the coefficient and its significance, as in Medea's comments that "once the basic calculation was made, we divide both parts by the unknown's coefficient and then get the result" or Vassiliki's "I take the number in front of the x and divide both sides of the equation by it". In such comments, focused on the coefficient of the unknown, can be seen.

The Greek students' comments could be categorised similarly, although a smaller proportion, three of the eleven, offered entirely implicit statements. Of these, typical was Crino's "we have reached 6 = 3x and consequently, we divide 6 by 3 to get the result". Seven students, also a higher proportion than with the Cypriot cohort, presented accounts with an explicit reference to the coefficient, as with Ivy's comment that he (the invisible teacher) "divided by the coefficient of the unknown 3 to get 2 = x" and Paraskevi's note that "once we reach this point 6 = 3x, I must divide by the coefficient of the unknown to see what the value of x is, that is to say, to divide both the 1 st part and the 2 nd by 3". Interestingly, two of these seven students offered accounts, albeit whose intentions were clear, that employed an incorrect vocabulary, as with Daphne's "finally, we divide 3 by the fraction" or Nikoleta's "we divide it by the denominator of x, namely, 3 and we arrive at the final result that 2 = x". In such cases students' intended meaning was clear.

Discussion

Our analyses provide substantial insights, both encouraging and discouraging, into the conceptualisation of mathematics these beginning teachers bring to their teacher education programme. In this respect, both sets of students appeared procedurally competent, recognising the equation for what it is and typically understanding how it had been solved. However, from a disciplinary perspective [START_REF] Leinhardt | Instructional explanations: A commonplace for teaching and location for contrast[END_REF], students' explanations showed very little awareness of the epistemological underpinning of mathematical knowledge, almost without exception warranting their chosen procedures on the basis of their personal authority as teachers. Moreover, from the perspective of research on the solving of equations, students indicated no relational understanding of the equals sign, a prerequisite for learners confident solving of equations of the form above with the unknown on both sides [START_REF] Alibali | A longitudinal examination of middle school students' understanding of the equal sign and equivalent equations[END_REF][START_REF] Filloy | Solving equations: The transition from arithmetic to algebra[END_REF]. From an instructional perspective [START_REF] Leinhardt | Instructional explanations: A commonplace for teaching and location for contrast[END_REF], students' explanations typically alluded to a presentation of objectives and the importance of the unknown and its role in equation solving. However, the presentation of such a rule would typically allow little opportunity for learners to understand the reasoning behind it. Interestingly, although their rote rules never alluded to operations performed on both sides, many students were aware of inverse operations in relation to the unknown's coefficient. In sum, and drawing on Skemps's (1976) distinctions, our data showed that both Cypriot and Greek students were locked into an instrumental rather than relational understanding of mathematics, perspectives which their respective curricular traditions would not have encouraged. Finally, when framed against the mathematical knowledge for teaching literature, it seems that at this stage of their careers both sets of students showed evidence of Rowland et al.'s (2005) foundation knowledge, albeit problematic, but, as yet, none of transformation, connection or contingency. Moreover, if the goal of teacher education is to facilitate beginning teachers' profound understanding of emergent mathematics [START_REF] Davis | Profound understanding of emergent mathematics: broadening the construct of teachers' disciplinary knowledge[END_REF] then much is still to be done.

Proceedings of CERME10


domain: math shs

text:
Mourat Tchoshanov
email: mouratt@utep.edu

Maria Cruz Quinones

Kadriya Shakirova

Elena Ibragimova

Liliana Shakirova

A cross-national study of lower secondary mathematics teachers' content knowledge in the USA and Russia

Keywords: Comparative studies, teacher knowledge, lower secondary mathematics

This study presents a quantitative analysis of middle school mathematics teachers' content knowledge in two countries. The sample comprises lower secondary mathematics teachers from the US (grades 6-9, N=102) and Russia (grades 5-9, N=97). The instrument was designed to assess teacher content knowledge based on the cognitive domains of knowing, applying, and reasoning, as well as addressing the lower secondary mathematics topics of number, algebra, geometry, data and chance. Results indicate significant differences in teacher knowledge between the countries in content as well as in cognitive domains. The study results may inform the field on priorities placed on lower secondary mathematics teachers' knowledge in USA and Russia.

Introduction

The motivation for the study is based on the 8th-grade mathematics portion of the TIMSS-2011 results [START_REF] Mullis | The TIMSS 2011 International Results in Mathematics[END_REF]. We identified two countries ranked closely to each other: Russia -in the 6th position and the USAin the 9th position. At the same time, a difference in the US and Russian students' performance was revealing: the average score of Russian students in the content domain was 539 and of the US students 509, with Russian students gaining higher scores on Number (534 vs. 514), Algebra (556 vs. 512) and Geometry (533 vs. 485) whereas US students outscored Russian students in the domain of Data and Chance (527 vs. 511). Russian students also outperformed the US students in each cognitive domain: Knowing (548 vs. 519) Applying (538 vs. 503),and Reasoning (531 vs. 503). These data triggered the following question: to what extent does US and Russian lower secondary mathematics teachers' knowledge differ by content and cognitive domains?

Cross-national studies of teacher knowledge

Conducting cross-national studies allow comparing, sharing, and learning about issues in an international context which in turn helps researchers understand their own context, teaching practice, teacher knowledge, and student learning [START_REF] Stigler | Cross-cultural studies of mathematics teaching and learning: Recent finding and new directions[END_REF]. During the last decade, the number of cross-national studies on teacher education is increasing in order to understand differences in student performance on international tests such as TIMSS, PISA [START_REF] Wang | Comparative studies on U.S. and Chinese mathematics learning and the implications for standards-based mathematics teaching reform[END_REF]. Scholars have addressed these differences focusing on characteristics such as teachers' perceptions of effective mathematics teaching [START_REF] Cai | How do exemplary Chinese and U.S. mathematics teachers' view instructional coherence[END_REF], teacher knowledge [START_REF] Tatto | The mathematics education of future primary and secondary teachers: Methods and findings from the Teacher Education and Development Study in Mathematics (TEDS-M)[END_REF][START_REF] Tchoshanov | Examination of lower secondary mathematics teachers' content knowledge and its connection to students' performance[END_REF], among others. Few cross-national studies focused on teacher knowledge. A large-scale study conducted by the University of Michigan examined the mathematical content and pedagogical content knowledge of pre-service teachers from 17 countries including USA and Russia [START_REF] Tatto | The mathematics education of future primary and secondary teachers: Methods and findings from the Teacher Education and Development Study in Mathematics (TEDS-M)[END_REF]. The nature of mathematics teacher knowledge, conceptual representation, and curriculum materials were examined by [START_REF] Ma | Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States[END_REF] to explain differences in students´ performance in the U.S. and China. [START_REF] An | The pedagogical content knowledge of middle school, mathematics teachers in China and the U[END_REF] studied the PCK of middle school teachers in the U.S. and China. They found that mathematical PCK differs among the countries since Chinese teachers emphasize developing procedural and conceptual knowledge through traditional teaching practices while their counterparts in the U.S. focus on promoting creativity and inquiry through activities designed to develop students' understanding of mathematical concepts. [START_REF] Sorto | Teacher knowledge and teaching in Panama and Costa Rica: A comparative study in primary and secondary education[END_REF] administered a survey that measured teachers' content knowledge in Costa Rica and Panama and found that teachers in both countries focus more on knowing rules and procedures than on making connections and reasoning.

In the last several decades, the field of mathematics education is expanding its knowledge-base in understanding the role of teacher characteristics in student learning and achievement. The major shift in the field had happened with [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] work on teacher knowledge that proposed an alternative approach to the educational production function perspective, which was concerned with examining proxies of teacher knowledge such as coursework/certification and its impact on student achievement [START_REF] Charalambous | Perspectives on priority mathematics education unpacking and understanding a complex relationship linking teacher knowledge, teaching, and learning[END_REF]. Research on teacher knowledge initiated by work of [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] has focused on teacher knowledge as a major predictor of student learning and achievement. Recently, the field benefited from numerous studies [START_REF] Hill | Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students[END_REF][START_REF] Baumert | Teachers' mathematical knowledge, cognitive activation in the classroom, and student progress[END_REF] that substantially advanced the conceptualization of teacher knowledge and its association with student performance.

Following up on this conceptualization, some scholars (Izsak, Jacobson, & de Araujo, 2012) examined different facets of teacher knowledge without explicitly emphasizing its connection to student learning. Other scholars stressed the importance of the kind of knowledge a teacher possesses because it impacts his/her teaching [START_REF] Steinberg | Teacher's knowledge and structuring content in mathematics[END_REF]. Another line of research (e.g., [START_REF] Baumert | Teachers' mathematical knowledge, cognitive activation in the classroom, and student progress[END_REF][START_REF] Hill | Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students[END_REF][START_REF] Tchoshanov | Relationship between teacher content knowledge, teaching practice, and student achievement in middle grades mathematics[END_REF] specifically targets the effects of different types of teachers' knowledge on student achievement. Additionally, scholars have advanced the field by examining teacher knowledge in variety of domains including Number Sense [START_REF] Ma | Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States[END_REF][START_REF] Izsac | Measuring mathematical knowledge for teaching fractions with drawn quantities[END_REF]), Algebra (McCrory et al., 2012), Geometry and Measurement [START_REF] Nason | Facilitating growth in prospective teachers' knowledge: Teaching geometry in primary schools[END_REF], and Statistics [START_REF] Groth | Preservice elementary teachers' conceptual and procedural knowledge of mean, median, and mode[END_REF]. However, the field lacks cross-national research that provides a comprehensive analysis of the various facets of teacher knowledge (including content and cognitive domains) and its connection to student performance.

Methodology

The proposed study is based on the assessment framework used by TIMSS [START_REF] Mullis | The TIMSS 2011 International Results in Mathematics[END_REF]. In this section, we will describe the study participants, the instrument as well as data collection and data analysis procedures.

Participants

The sample of this study consisted of lower secondary mathematics teachers from the US (grades 6-9, N=102) and Russia (grades 5-9, N=97). The US teacher-participants were selected from urban public middle schools in the Southwestern part of the country. Teacher sample demographic information was self-reported by participating teachers. In terms of gender distribution, 55% of teacher participants were females and 45% -males. Most of the US participants (64%) had 1-5 years of teaching experience. Additionally, 62% of the teacher sample received their teaching certificate through traditional teacher preparation programs and 38% of participating teachers were certified through alternative programs. The Russian teacher-participants were selected from urban public secondary schools in the Volga region. Russian participating teachers had attained a secondary mathematics teacher preparation Specialist's degree, which allowed them to teach in secondary schools (grades 5-11). The majority of participating teachers were females (89%). The sample was composed of 78% of teachers who have more than 10 years of teaching experience.

Instrument

The instrument used in this study was the Teacher Content Knowledge Survey which was developed using TIMSS framework [START_REF] Mullis | The TIMSS 2011 International Results in Mathematics[END_REF]. It was designed to assess teacher content knowledge based on the three cognitive domains: Knowing, Applying, and Reasoning. The TCKS survey consisted of 33 multiple-choice items addressing main objectives of lower secondary mathematics curriculum: Number, Algebra, Geometry, Data and Chance. The instrument was piloted for construct and content validity as well as checked for the reliability. The alpha coefficient technique was utilized to evaluate the reliability of the teacher content knowledge survey. "The value of the coefficient of .839 suggests that the items comprising the TCKS are internally consistent" (Tchoshanov, 2011, p. 148). Examples of the TCKS items in Algebra domain across different cognitive types (Knowing, Applying, and Reasoning) are presented below. Use the diagram above (see Figure 1) to answer the questions that follow.

Knowing

Which of the following equations best describes the function y3?

A.

y = ax 2 + bx + c B. y = ax 2 + bx + 1 C. y = ax 2 + 1 D. y = x 2 + 1.

Applying

The function y3 is translated 4 units left and 7 units down. Which of the following equations best describes the new function?

A.

y = ax 2 + 11x + 28 B. y = ax 2 + 4x + 7 C. y = ax 2 + 8ax + c D. y = x 2 + 28x + 11.

Reasoning

The diagram shows a family of functions in the form y = ax 2 + bx + c. Which of the following statements best describes the changes in the values of the coefficients as the graphs transform from y1, to y2, to y3?

Data Collection and analysis

Each teacher was given 90 min to complete the survey. In correspondence with the research question, data analysis was performed using non-parametric techniques (chi-square). This statistic was selected to measure the variance between independent groups of the same (not normal) distribution with arbitrary sample sizes of each group. The selection of this test was also based on the ranked nature of data for content and cognitive domains of teacher knowledge and student performance.

Results

In this section, we first analyze teacher knowledge data by content domain, then we examine teacher data by cognitive domain, and finally we discuss parallels between student and teacher performance within and between countries. However, there is a statistically significant difference between Russian and US teachers' knowledge on Algebra domain (in favor of Russian teachers; Chi-square 6.311 p<.05) and Data and Chance domain (in favor of US teachers; Chi-square 8.003 p<.05) (see Table 5). This finding closely parallels the US and Russian students' performance on TIMSS on Algebra domain (in favor of Russian students) and Data and Chance domain (in favor of US students).

Also, this study reported that there is no significant difference between Russian and US teachers' knowledge on Knowing and Applying cognitive domains (Chi-square 1.707 p>.05 and Chi-square 0.008 p>.05) whereas there is a statistically significant difference on Reasoning domain (in favor of Russian teachers; Chi-square 19.117 p<.05) (see Table 6). This finding parallels the US and Russian students' performance on TIMSS' cognitive domain.

Discussion and conclusion

This study confirms the differences between Russian and the U.S. lower secondary in-service teachers' knowledge in the content domain as it was reported by the TEDS-M study that was focused on pre-service teachers [START_REF] Tatto | The mathematics education of future primary and secondary teachers: Methods and findings from the Teacher Education and Development Study in Mathematics (TEDS-M)[END_REF]. At the same time, this study expands the examination of in-service teachers' knowledge to the cognitive domain.

Teacher preparation could be considered as the main factor contributing to the differences between Russian and US teachers' knowledge. Overall, there is a tangible difference in secondary teacher preparation curriculum between the two countries: in average, Russia offers about 240 credit hours in teacher preparation programs compare to 120 credits in the USA. Furthermore, cross-national curriculum analysis shows that Russian lower secondary mathematics teachers have more extensive content preparation compare to their American counterparts. A number of contact hours for mathematical content knowledge, as well as pedagogical content knowledge and specialized mathematics knowledge offered at selected teacher preparation programs (e.g., the University of Texas at El Paso, USA and Kazan Federal University, Russia) in two countries, are presented in table 7. Numbers depicted in the table are compatible with the findings of the TEDS-M study [START_REF] Tatto | The mathematics education of future primary and secondary teachers: Methods and findings from the Teacher Education and Development Study in Mathematics (TEDS-M)[END_REF]. Close examination of secondary teacher preparation curriculum in Russia shows that more emphasis is placed on an analytic and algebraic component of mathematics curriculum and less emphasis -on statistic and probability component compare to the US curriculum. Moreover, item analysis of standardized tests for the lower secondary schools in USA and Russia revealed the difference in selection and composition of algebra problems as well as problems related to data and chance in the test: while in Russia more emphasis is placed on algebraic problems and less emphasis on data and chance problems, in the USAthe emphasis is equally distributed among algebraic problems and data and chance problems. We observed another noticeable difference in the role of proof in the academic mathematics component of the teacher preparation program which could explain the difference in the reasoning domain of the teacher knowledge: Russian curriculum places a heavy emphasis on proof across the mathematics coursework including school mathematics whereas the US curriculum uses proof in selected mathematics courses primarily in academic mathematics coursework.

We are cognizant of the limitations concerning the convenient sampling technique that influences generalizability of the study results. Moreover, there is no cluster matching between teachers participating in the study and students tested in TIMSS. However, the study main results suggest that student performance on international tests could be explained by teacher knowledge. The study also presents opportunities for comparing, sharing, and learning about issues in cross-national context in US and Russian teacher education, training, and development. Moreover, the reported cross-national study on teacher knowledge may inform the field on priorities placed on lower secondary mathematics teachers' knowledge in USA and Russia by content and cognitive domains.

Figure 1 .

 1 Figure 1. Diagram to the TCKS item in Algebra domain

A

  . a is increasing, b = 0, and c is increasing B. a is increasing, b = 0, and c is decreasing C. a is decreasing, b is increasing, and c = 0 D. a is decreasing, b is decreasing, and c = 0.

Table 1 . US teachers´ means scores by content domain

 1 The results reported on teacher content knowledge show that the US teachers' highest mean score was obtained on Number domain -623 and lowest on Geometry domain -514 (see Table1).

	Content Domain	Mean	SE	SD	Conf. level (95%)
	Number	623	20.3129	205.1512	40.296
	Algebra	563	23.2356	234.6679	46.093
	Geometry	514	25.4349	256.8802	50.456
	Data and Chance	593	20.9738	211.8252	41.606

Russian teachers' highest mean score was obtained on Algebra domain -728 and lowest on Data and Chance domain -387 (see Table

2

).

Table 2 . Russian teachers´ means scores by content domain

 2 Moreover, we found that in the cognitive domain the US teachers' highest mean score was obtained, as expected, on Knowing -734 and lowest on Reasoning -495 (see Table3).

	Cognitive Domain Mean	SE	SD	Conf. level (95%)
	Knowing	734	19.7673	197.6733	39.2226
	Applying	505	20.7101	207.1015	41.0934
	Reasoning	495	23.8130	238.1303	47.2502

Table 3 . US teachers´ means scores by cognitive domain

 3

	Cognitive Domain Mean	SE	SD	Conf. level (95%)
	Knowing	760	14.2486	135.1745	28.3117
	Applying	504	12.7961	121.3950	25.4257
	Reasoning	593	17.7406	168.3028	35.2503

Russian teachers' highest mean score was obtained, as expected, on Knowing domain -760 and lowest, unexpectedly, on Applying domain -504 (see Table

4

).

Table 4 . Russian teachers´ means scores by cognitive domain Moreover

 4 , we identified that there is no significant difference between Russian and US teachers' knowledge on Number and Geometry domains (Chi-square 0.347 p>.05 and Chi-square 1.293 p>.05) (see Table5).

	Content Domain	Number	Algebra	Geometry	Data and Chance
	Russia	656	728	586	387
	USA	623	563	514	593
	Chi-square (df=1) 0.347	6.311*	1.293	8.003**

Table 5 . Russian and US teachers' knowledge by content domain (* p<.05, **p<.01)

 5

Table 6 . Russian and US teachers' knowledge by cognitive domain (* p<.05, **p<.01)

 6

	Cognitive Domain Knowing	Applying	Reasoning
	Russia	760	504	593
	USA	734	505	495
	Chi-square (df=1) 1.707	0.008	19.117**

Table 7 . Contact hours in Mathematics related disciplines in teacher education programs in Russia and United States

 7

	Country	Mathematics Content	Pedagogical Content	Specialized Mathematics
		Knowledge	Knowledge	Knowledge
	Russia	1857	278	380
	United States	442	72	87

Proceedings of CERME10


text:
Mona Nosrati
email: mona.nosrati@matematikksenteret.no

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

Ten years of mathematics education: Preparing for the supermarket?

Keywords: Purpose of school mathematics, Norway, Sweden, student perspectives

In this paper we investigate Norwegian and Swedish upper secondary students' perspectives on the purpose of school mathematics. Students were group interviewed in various schools in both Norway and Sweden and the video recordings of those interviews fully transcribed. In each country transcriptions were subjected to a constant comparison analytical process that, coincidentally, yielded the same two dominant themes that we report here. Firstly, all students spoke about how learning mathematics facilitates being able to manage shopping, personal finances and other functional aspects of the real-world. Secondly, students spoke about how learning mathematics would facilitate their getting a job, including some Swedish students who saw mathematical success as a high status qualification preferred by employers.

Introduction

Despite evidence that large scale assessments of mathematics achievement like the Organisation for Economic Cooperation and Development's (OECD) Programme of International Student Assessment (PISA) have prompted curricular changes in many OECD member countries [START_REF] Breakspear | The policy impact of PISA: An exploration of the normative effects of international benchmarking in school system performance[END_REF], the nature of school mathematics is not an unproblematic given and varies considerably cross-culturally [START_REF] Andrews | Understanding the cultural construction of school mathematics[END_REF]. Indeed, all curricula, which are based on a culture's conception of an ideal person [START_REF] Cummings | The institutions of education: Compare, compare, compare![END_REF], are rooted in substantially more than what can be inferred from official documents. Consequently, the ways in which teachers conceptualise and present mathematics varies from one cultural context to another, inevitably influencing the beliefs that students form about mathematics and its purpose [START_REF] Cobb | Two children's anticipations, beliefs, and motivations[END_REF].

Students' beliefs about mathematics and its teaching have been the focus of much research for more than four decades, following [START_REF] Erlwanger | Benny's concept of rules and answers in IPI mathematics[END_REF] case study of Benny, a twelve year-old student who had come to see mathematics as invented rules, each for a particular type of problem, that work like magic. Benny's beliefs about mathematics effectively laid the ground for the field, not only from the perspective of what students believe about mathematics and its teaching but also the ways in which teachers' beliefs and practices are implicated in the construction of such beliefs. In short, the significance of this work draws on the premise that students' beliefs are informed by, and inform how they respond to, their mathematics-related learning opportunities [START_REF] Erlwanger | Benny's concept of rules and answers in IPI mathematics[END_REF][START_REF] Callejo | Approach to mathematical problem solving and students' belief systems: two case studies[END_REF]. In this regard, "beliefs constitute, for the believer, current knowledge about the world" (Cobb, 1986, p.4).

Over the succeeding years, research into students' mathematics-related beliefs has addressed a variety of themes, which have been well summarised by Op 't Eynde and his colleagues [START_REF] Op 't Eynde | Framing students' mathematics-related beliefs: A quest for conceptual clarity and a comprehensive categorization[END_REF][START_REF] Op 't Eynde | Epistemic dimensions of students' mathematics-related belief systems[END_REF], which they present under three broad headings; beliefs about mathematics education, beliefs about the self as a mathematician and beliefs about the mathematics class context. Within these three broad headings, which space prevents us from discussing in depth, are located most aspects of mainstream beliefs-related research. However, a strand that has received less explicit attention the literature, particularly in the Scandinavian context, has been the beliefs students hold about why they are compelled to learn mathematics for so many years.

The purpose of school mathematics

In general terms, the reasons why children spend so many years studying mathematics are not always clear, although typically they seem to be tied to ensuring either the social, economic and political mobility of the individual citizen or the nation's economic growth through preparing the next generation of employees [START_REF] Schoenfeld | The reading and math wars[END_REF]. Such perspectives are philosophically rooted in either personal emancipation or the reproduction of the existing social order respectively. That being said, while much philosophical research has addressed the general aims of education, "very little sustained work of this kind appears to have been carried out in mathematics education", to the extent that "there is sufficient disagreement, lack of clarity and modesty to warrant further enquiry" (Huckstep, 2000, p. 8). It is on this that we next focus, confirming Huckstep's concerns.

Various scholars have considered the purpose of school mathematics. For example, [START_REF] Niss | Goals of mathematics teaching[END_REF] discusses the justification and the goals of mathematics education from both theoretical and historical perspectives. [START_REF] Davis | Why teach mathematics to all students? For the Learning of Mathematics[END_REF] adopts a largely philosophical standpoint in suggesting three distinct purposes, which he describes as the teacherly, the rhetorical and the hermeneutic. The teacherly purpose for teaching mathematics derives from a set of largely forgotten reasons tied to the belief that "knowledge of mathematics is necessary for every citizen of today's world" (p. 18). The rhetorical purpose derives from an "explanatory fantasy that is currently preferred to organize and structure experience" (p. 19). The hermeneutic purpose lies in beliefs that "there are moral and ethical imperatives that operate in the human and in the more-than-human realms" (p. 21). In a slightly more prosaic manner, [START_REF] Noyes | Rethinking school mathematics[END_REF] writes of six broad purposes; mathematics for the academy, for employment, for general education, for citizenship, for the information age and, finally, critical mathematics education. [START_REF] Ernest | Mathematics and values[END_REF], in a not unrelated manner, describes seven purposes; functional numeracy, work-related knowledge, advanced specialist knowledge, problem posing and solving, mathematical confidence, social empowerment through mathematics and, finally, appreciation of mathematics itself. Finally, [START_REF] Watson | Red herrings: Post-14 'best' mathematics teaching and curricula[END_REF], in a manner that seems to summarise both Ernest and Noyes and mirror [START_REF] Davis | Why teach mathematics to all students? For the Learning of Mathematics[END_REF], proposes three broad purposes; mathematics as a set of useful skills and procedures, a support for the burgeoning mathematicsrelated professional, and a tool that facilitates successful societal participation. In sum, as [START_REF] Huckstep | The utility of mathematics education: Some responses to scepticism[END_REF] indicated, mathematics is taught for various reasons tied to cultural norms and values.

Interestingly, despite research discussing the purpose of mathematics from an outsider perspective, few studies have engaged with the insider; what do recipients of mathematics teaching think is the purpose of what they are taught? In this respect, it is salient to note that [START_REF] Brown | Primary student teachers' understanding of mathematics and its teaching[END_REF], when investigating beginning primary teachers' understanding of mathematics and its teaching, found that "a sense of bafflement about the purpose of school mathematics permeated many accounts" (Brown et al., 1999, p. 305). That is, these students had completed school and still had no idea as to the purpose of school mathematics. In the light of such uncertainty and acknowledging the lack of research in the field, particularly in the context of Scandinavia, we examine the beliefs about the purpose of school mathematics of both Norwegian and Swedish upper secondary students. As in Brown et al.'s (1999) study, these students are close to the end of their school careers and should have formed clear ideas as to the purpose of mathematics.

The study and its methods

The data on which this paper is based derive from a comparative interview study of upper secondary students in Norway and Sweden. In broad terms the study set out to explore students' perspectives on their many years of school mathematics. Students were interviewed in pairs or threes, and the interviews were structured around the following four questions, with follow-up questions where appropriate:

1. How would you describe a typical mathematics lesson at school? 2. What do you think is the purpose of compulsory school mathematics?

3. What do you think mathematics as a subject has to offer to those who engage with it? 4. If you could say something about the nature of mathematics education to those in charge of the educational system, what would it be?

It is the answers to question 2 that we will be focusing on in this paper.

The Norwegian data derived from 17 interviews involving 42 students from three schools. Two schools, one in Oslo and one in Trondheim, were high-achieving academic schools, while the third was a relatively low-achieving vocational school in Oslo. The Swedish data derived from 18 interviews involving 50 students from four schools. These schools, from various parts of Stockholm, all offered a range of vocational and academic tracks. Consequently, we make no claims about schools' representativeness nor do we seek to generalise. All participants were fully aware of the purpose of the research and of their rights to withdrawal.

Interviews, undertaken at a time chosen by the students, were video recorded on laptop computers, a decision justified in four ways. Firstly, video, especially when participants talk over each other, simplifies transcriptions. Secondly, video captures non-verbal communication. Thirdly, due to their classroom ubiquity, laptops were expected to create less disruption than tripod-mounted video cameras. Fourthly, laptops record data directly to their hard-drives, simplifying data storage and analysis. All interviews were transcribed and, in each country, subjected to a constant comparison analysis whereby each transcript was read and categories of response identified. With each new category, previously read episodes were re-read to determine whether the new category applied to them also. The two data sets were analysed separately to ensure the cultural integrity of the findings.

Results

The data from both countries yielded a variety of themes. However, two closely related themes dominated the analyses in both contexts and it is these we turn to in this paper. These were related to being able to function in the real world and getting a job, and were present in every single interview in one form or other. Below we give a more detailed description of the two themes as well as exemplifying interview extracts from both academic (A) and vocational (V) students. Other themes, relating to mental training, appreciation, and uselessness were also present in the data (though much less so). However, it is beyond the scope of this paper to discuss those here.

Mathematics enables one to function in the real world: Norway

A large number of students emphasised the importance of mathematics in daily life, which typically seemed to consist of going to the supermarket or calculating one's salary and taxes. In respect of the former, Line's (A) comment was typical; "maths is a lot about everyday life… you wouldn´t be able to go to the store and buy goods if you didn´t have some maths". Similarly, Robin (V) commented that "when you are going to buy food and stuff, you need to know how much things cost and stuff".

With respect to both shopping and income, Tania's (A) comments were not atypical. 'daily life'. For example, Amalie (A) commented that she hadn't "thought that much about it" before adding that "you do at least have to know some maths… like in daily life, then maths is useful". Interestingly, she then offered the observation that "when we are at high-school level then I don´t exactly think that all the maths I learn will be useful".

In sum, the Norwegian students were confident that knowing mathematics would support their functioning in the real world, although there were occasions when the manner of this support was vague and imprecise. It was also interesting to see Amalie's and Ruben's comments that while mathematics was a real-world support, much of what they learned was unnecessary in this respect.

Mathematics enables one to function in the real world: Sweden

In every Swedish interview students spoke about how they saw mathematics as preparing them to face a world beyond school, mathematics as supporting their real-world functionality. Within this utilitarian strand two major themes were identified. The first concerned personal finance and the management of money. Here, almost all students spoke about understanding interest, as with Jacob's (V) comment that "percentages in terms of interest and loans and things like… that's very good to know because you might not really figure out in your head how much you can spend". In similar vein Pedram (A) said that "amortization…, interest rates, interest costs, to be able to figure it out, it is important". Others spoke more generally about the management of personal finances. For example, Mark (V) commented about the need to manage "larger sums, as well as your salary", while Kenneth, in the same vocational track interview, added that "it becomes much easier to make financial plans… if one has several years of mathematics". Finally, several students spoke of the need to avoid overextension, as in Göte's (A) concerns with respect to "SMS loans and stuff, there are many who do not know how much you lose there" and Omar's (A) worries that "there are too many adults today who do not really understand interest and how it works".

The second major theme relating to being able to function in the real world related to the mundane world of everyday shopping. Comments typical of others were like those of Alice (A), who said that "it's something you use in everyday life, it's typical when you go shopping, it's just the basics", or Mark (V), who said that "it's always good to have a base in maths. So, if you go shopping and have a hundred, then you cannot buy for more than a hundred". Others offered a slightly different perspective. For example, Hanna (A) raised the overall mathematical expectation of such transactions by commenting that "if you've gone shopping and there is a discount and know how to do that or how to pay what you're gonna pay in. It is like useful to use maths like this", while André (V) added that "you always look at the prices and compare them to the prices in another store".

Knowing mathematics enables one to get a job:

A number of students spoke about how they saw their learning of mathematics in relation to their future careers. In most cases students spoke hypothetically as few seemed to have considered their own particular career aspirations. For example, Sarah (A) said that I think that maybe we need to spend as much time on maths as we do because we don´t know who ends up as engineers or who maybe works with something that you maybe don´t need a lot of maths for... you can´t know in advance, so everyone has to learn everything... so that you can be what you want.

In similar vein, Kristine (A) commented, after acknowledging "that is a bit of a difficult question", that it depends a lot on what you plan to do after high school... if you have planned to take a fairly high education then it is clear that there is a lot of maths; if you, for example, are going to be a chemist or a physicist or... medical studies too... or generally medicine, if you are to calculate dosages for a pain killer it is a bit bad to get a really huge dose… but if you plan to work at Mc Donalds for example, then it is not so important.

The comments of both students indicate an understanding that different professions will require different levels of mathematical competence. However, neither of them sees mathematics as an important component of education for everyone for its own sake. However, having observed that working in McDonalds does not require much by way of mathematical competence, Kristine added that "but how do we know from the start who will be a physicist and who will work at McDonalds?"

Other students noted that they do mathematics for as long as they do because it is a formal entry requirement for higher education (even if that higher education is unrelated to mathematics). In this respect, Andreas (V), in a comment typical of others, noted that "actually it is an entrance requirement for the college I plan to go to later... but apart from that I don´t see any reason to know such advanced maths". In such a comment, and those of Sarah and Kristine, lies a tacit concession that everyone has to spend twelve years learning mathematics just in case it might be useful, a conclusion summarised well by Gina (A), who said, "I think maybe you don´t notice the importance until later".

In a related manner, Malin (A) spoke of how mathematics supports the learning of other subjects. She commented that you need to use maths a lot in other subjects, so you have it in chemistry and physics and stuff... so maths is a very important tool… so it is not... if you look at the maths as a subject... just a subject, then it can be a lot of numbers and a bit like... but if you put it in the context of other science subjects, then it is the most important tool you have.

Knowing mathematics enables one to get a job: Sweden

Very few students did not, during the course of their interviews, refer to the ways in which learning mathematics would enhance their employment prospects. Within the Swedish data two major subthemes were identified. The first of these concerned students' beliefs about the generic ways in which mathematical knowledge would prepare them for work. At its most basic, according to Andreas (V), "any job requires quite some math knowledge, in all cases, plus or minus, often times. … and that applies to the majority of all jobs". More specifically, as in the comments of Thomas (V), "you get better chance of getting a better job with like higher level of mathematics", while Alice (A) believed that "when one has read maths, one can study further and get … more opportunities for the future, one can say, more career choices". Others still, saw beyond their first job, as with Kenneth's (V) comment that "even when you think you have found out what you want to be, you might want to switch careers later and then maybe you need more knowledge in maths".

From the perspective of particular career routes, Dennis (V) commented that "as an electrician you need so this current, voltage, and everything like that, it requires maths for", while Roxana (A) suggested that "if you're going to become an engineer for example you're gonna read a lot of mathematics".

The second, slightly cynical, theme found students discussing how being mathematically qualified gave them an advantage over those who were not. For example, Ted (V) commented that "I think that many who have not read maths will not get a job, for the employer will not hire those who cannot work in the same way", while Jacob felt that the mathematically qualified "is more knowledgeable, that way he can solve the problem the less skilled cannot; he can see the big picture of things better". Hanna's (A) view was that such advantages stemmed from the fact that it might be like easy for that (mathematically well-qualified) person to think logically, to think outside the box and that kind of stuff. And the other (less well-qualified) person might have a few more problems with that, maybe… and probably it's easier for the person that's really good at math to excel in whatever they want to do… which is harder for the other person Most Swedish interviews also contained some reference to the ways in which mathematics was seen as a service tool to other forms of activity. Typically, these focused on the natural sciences, as in the comments of Adam (V), who said, "you use math well in other subjects like physics", Mikaela (A), who believed that "I know … in chemistry or physics … I need maths. So I think to learn the physics or those things you need to learn, to know the maths first. So you can build on it" and Julio (V), who argued that "maths is the grounding for several subjects, so if you remove maths, it's not just maths skills that will be worse, but then it is physics, chemistry and yes science subjects just disappear, so maths is an important part". Others' comments were more general, as in Winston's comment that mathematics "works well with other subjects that also need the maths, like they complement each other", while Max commented that "sports is a subject where you use it (mathematics)".

Discussion

In this paper we set out to uncover what Norwegian and Swedish upper secondary students believed was the purpose of school mathematics, particularly as it has been an ever-present compulsory part of their school careers. The results seem much removed from mathematics as a problem posing and solving discipline [START_REF] Ernest | Mathematics and values[END_REF] for which teaching aims to "lead students to appreciate the power and beauty of mathematical thought (Dreyfus and Eisenberg, 1986, p.2). Indeed, the two dominant themes, mathematics as a support for functioning in the real world and mathematics as an entry into employment, are as utilitarian as it is possible to be, showing no connection to mathematics as a cultural artefact to be appreciated as would be art, music or literature. Moreover, the themes were not utilitarian in the sense that students saw themselves as having been educated for citizenship [START_REF] Noyes | Rethinking school mathematics[END_REF], made socially empowered [START_REF] Ernest | Mathematics and values[END_REF] or inducted into societal participation [START_REF] Watson | Red herrings: Post-14 'best' mathematics teaching and curricula[END_REF]. They were utilitarian solely from the perspective of personal advancement. There was no evidence that these students, many of whom were expecting to go to university and study mathematics-related subjects, saw mathematics as part of the "moral and ethical imperatives that operate in the human and in the more-than-human realms" (Davis, 2001, p. 21).

While such findings may be disappointing, we acknowledge that these students' perspectives have not emerged by chance but from individually unique experiences of mathematics. These ten-and eleven-year experiences, located in different countries and schools, will necessarily have influenced the formation of individuals' beliefs about the nature of mathematics and its purpose [START_REF] Cobb | Two children's anticipations, beliefs, and motivations[END_REF], and yet their collective voice was close on deafening. So, are there any explanations? Well, the literature is not particularly expansive on such matters, although a recent case study may offer some insight. In their study of largely disaffected Swedish upper secondary school students, [START_REF] Andersson | I am [not always] a maths hater": Shifting students' identity narratives in context[END_REF] motivated their students by means of tasks involving, inter alia, percentages related to personal economics. This approach, which was received positively by the students concerned, raises at least two important questions. The first is whether such an approach legitimates students' perception that school mathematics is about preparing them for the real world.

The second is whether students' positive reactions were a consequence of the task meeting their expectations of school mathematics. In short, which came first, the chicken or the egg?

  She said that You need it in daily life and… you´re in the shop, you want to buy something, and then you must add… sum the price of the things you buy and stuff… like if you work, and if you want to know how much you earn and you want to know your net income... if you know it yourself you can double check if the person who does it has done it right and stuff. While Ruben (V) added that "you can calculate numbers and stuff… it helps you on in life... if you work you can calculate your salary and stuff", before adding that "there is a lot of maths that you don´t need, but you need at least a part of it". Others offered less specific statements regarding

Proceedings of CERME10


text:
Cecilia Tuula Koljonen
email: tuula.koljonen@mdh.se.

Finnish teaching materials in the hands of a Swedish teacher: The telling case of

Keywords: Cultural scripts, educational context, primary school, teaching materials, telling case

A common perception in Sweden is that the best teachers do not rely on ready-made teaching materials. The position taken in this paper, building on socio-cultural theory, assumes that teacher materials can support teachers. Although there is an emerging body of research focusing on teachers' use of teaching materials, cross-cultural studies on this are scarce. The current study addresses this gap by offering unique insight into how a Swedish teacher makes use of teaching materials originally from Finland but slightly adapted to the Swedish context. Based on teacher interviews and classroom observations, I studied how the teacher planned for and enacted lessons. Findings indicate that she fits the material to her pre-existing practice and, thus, does not follow the material's original intentions. The results are compared with previous results on materials and their use, and finally some implications for Swedish mathematics education are presented.

Introduction

Finland is known as a country with good learning outcomes in mathematics (e.g. OECD, 2013). Furthermore, since the 1980s the country has had a tradition of producing exhaustive teacher guides (TGs) in collaboration with teachers, teacher educators and other experts [START_REF] Niemi | The societal factors contributing to education and schooling in Finland[END_REF]. These two factors have likely increased the interest in applying commercially produced Finnish teaching materials, such as textbooks and TGs, in Sweden as well as other countries, such as Italy. Applying new teaching material from Finland in Sweden could be achievable, as there are many similarities between the school systems in the two countriesfor instance, the inclusive nine-year compulsory basic education with no special tracking, and the national core curriculum that provides an overall outline for school education. In addition, primary school teachers in both countries often teach all subjects, and are free to choose which teaching materials to use. There are also differences between the countries' educational systems; e.g., Swedish teachers at primary school level are seldom subject specialists while Finnish teachers are well educated, but also the widespread negative talk in Sweden concerning the use of ready-made teaching materials [START_REF] Bergqvist | Matematikutbildningens mål och undervisningens ändamålsenlighet[END_REF], which does not occur in Finland. Swedish teachers' orientation toward ready-made teaching materials most certainly affects how they engage with and use them (cf. [START_REF] Stein | How curriculum influences students learning[END_REF]. Swedish teachers seldom use TGs in planning and enacting mathematical instruction [START_REF] Jablonka | Using texts and tasks: Swedish studies on mathematics textbooks[END_REF]. Instead, they rely mostly on the student textbook as their main source, a common feature of teachers in many parts of the world [START_REF] Remillard | Examining key concepts in research on teachers' use of mathematics curricula[END_REF][START_REF] Stein | How curriculum influences students learning[END_REF]. Also, Swedish compulsory school teachers have for the last two decades often organized individualized teaching, whereby students work individually in different areas and at their own pace [START_REF] Bergqvist | Matematikutbildningens mål och undervisningens ändamålsenlighet[END_REF][START_REF] Remillard | A cross-cultural analysis of the voice of six teacher's guides from three cultural contexts[END_REF]. The fact that students are taught largely according to the structure of their textbook has also resulted in less variation in teaching [START_REF] Jablonka | Using texts and tasks: Swedish studies on mathematics textbooks[END_REF]. However, Finnish teachers, especially primary school teachers in mathematics, trust and use commercially produced TGs extensively, and there are indications that Finnish teachers often organize whole-class teaching and use teaching methods other than individual seatwork. This has consequently led to a classroom practice that is different from the Swedish one [START_REF] Jablonka | Using texts and tasks: Swedish studies on mathematics textbooks[END_REF]Pehkonen, Ahtee & Lavonen, 2007). While there is growing interest in adopting and implementing mathematics materials in a new educational context, we know little about how imported mathematics materials are used or how they may influence classroom practice. Therefore, in this paper I aim to investigate the interplay between a Swedish teacher and the written curriculum as represented in the suggested lesson plans in a TG -Favorit Matematik (FM), originally from Finland. Moreover, I intend to show how this interplay may impact on enacted teaching. Teaching is viewed as a cultural activity, and cultural activities are represented in cultural scripts (cf. [START_REF] Stigler | The teaching gap: Best ideas from the world's teachers for improving education in the classroom[END_REF] and are consistent with the stable web of beliefs and assumptions within a cultural group. Scripts provide a background for interpreting behaviors; however, they do not describe, determine or predict the behavior of individual teachers [START_REF] Stigler | The teaching gap: Best ideas from the world's teachers for improving education in the classroom[END_REF]. Both teacher and teaching materials are significant participants and are situated in a socio-cultural context, a specific educational context. Through this, they both play a role in mediating that interplay, which is shaped by historical, social and cultural factors [START_REF] Brown | The teachertool relationship. Theorizing the design and use of curriculum materials[END_REF][START_REF] Remillard | Examining key concepts in research on teachers' use of mathematics curricula[END_REF]. Since cultural scripts are deeply rooted in practices and are hard to see from within a given culture, I opted for a case study approach, allowing for a deep analysis. I therefore anticipate that a study on the use of teaching material from one culture by a teacher from another culture will advance our understanding of the cultural scripts in both cultures, and of the participatory relationship between teacher and teaching material.

I have previously analyzed TGs from four Finnish textbook series in mathematics, and found that their structure, form and content were relatively homogeneous [START_REF] Koljonen | Finnish teacher guides in mathematics -Resources for primary school teachers in designing teaching[END_REF]. In Koljonen, Ryve and Hemmi (under review), we captured what kind of mathematics classroom the Finnish guides promote. Recurrent cultural scripts of the classroom practices were found, comprising: keeping the class around a specific topic; keeping the teachers and students active; clear lesson goals are vital features; different recurrent activities; concrete material; and embedded differentiation. Due to these findings and the different classroom practices in Sweden and Finland, it is of interest to investigate a Swedish teacher's interplay with Finnish teaching material as a way to compare the written and the enacted curricula grounded in two different cultural platforms. The research questions guiding this study are: 1) How does a Swedish primary school teacher, locally regarded as competent, interact with a Finnish teacher guide while planning and implementing teaching? 2) How does this interaction influence the classroom practice?

Methodology

This study is part of a larger cross-cultural project examining the interplay between Swedish and Finnish teachers using the same mathematics teaching materials. The data for this project are comprised of semi-structured interviews with four primary mathematics teachers from each country. The interview questions cover seven themes: teacher's education; teacher's experience; school settings; classroom culture; beliefs about mathematics and its teaching; TGs; and planning of lessons. Additionally, three consecutive mathematics lessons per teacher were videotaped. When videotaping during the lessons, I used two cameras: one teacher camera that captured the teacher's actions and talk, and one whole-class camera focused on the students' actions and talk. I conducted and transcribed the audio-recorded interviews (50-110 minutes) and the videotaped lessons (40-60 minutes). FM [START_REF] Asikainen | Favorit matematik 3A Lärarhandledning[END_REF] includes references to the Swedish national core curriculum (Lgr 11), but does not describe how the lesson goals actually serve to prepare students to meet the curriculum goals. Earlier studies [START_REF] Koljonen | Finnish teacher guides in mathematics -Resources for primary school teachers in designing teaching[END_REF] revealed that FM lacks educative support (cf. [START_REF] Brown | The teachertool relationship. Theorizing the design and use of curriculum materials[END_REF] for teachers as well. For example, the rationales behind its suggested lesson activities are rarely explicitly discussed, which is a critical component in teacher learning. Each lesson (4 pages) in FM has a similar structure, both visually and content-related; for instance, clear recurrent headings located in the same place on every page, and a variety of optional activities presented for each lesson. The activities are all linked to the central content and the lesson objectives, from which the teacher is to choose appropriate activities for their practice.

As a starting point in the larger cross-cultural project, I selected one of the Swedish teachers, Cecilia (fictitious name), to exemplify a single case as this approach offers possibilities for deeper theoretical insights that would otherwise go unseen (Andrews, In press). Cecilia graduated in 2010 as a compulsory school teacher (F-6), and was prepared to teach all other subjects besides mathematics as well. Thus, she is not a mathematical subject specialist. However, one of the criteria for selecting the teachers was that they were regarded as locally competent (cf. [START_REF] Clarke | The LPS research design[END_REF]. Among the other teachers, Cecilia is recognized and esteemed for her locally defined 'teaching competence' and has been nominated by the school's principal and the municipality and is thus regarded as a local subject specialist. At the time this study was conducted, Cecilia was in her third year of teaching with FM. She teaches children in Grade 3; her 24 students come from a constrained socio-economic area, with mostly non-native speaking families. Cecilia volunteered to participate, knowing the study was on FM and its use.

Data analysis

Teaching is viewed as a cultural activity [START_REF] Stigler | The teaching gap: Best ideas from the world's teachers for improving education in the classroom[END_REF]) and a design activity, whereby teachers craft instruction, and do so with different degrees of artifact appropriation: offloading, adapting and improvising [START_REF] Brown | The teachertool relationship. Theorizing the design and use of curriculum materials[END_REF]). Cecilia's interaction with TGs is characterized through these three analytical constructs. Offloading emerges when a teacher follows material and assigns a great degree of authority to the teaching material. That is, the agency for the delivery of content lies in the material. Adapting, on the other hand, occurs when a teacher reflects when elaborating with the material. Here the agency is embedded in both the material and the teacher. Improvising, finally, relates to when a teacher does not closely follow the material. That is, the agency lies with the teacher as she relies on her own strategies for instruction, with minimal reliance on the material. The relationship is further characterized as participatory or non-participatory (cf. [START_REF] Remillard | Examining key concepts in research on teachers' use of mathematics curricula[END_REF]. When the teacher regularly and deliberately uses the material, and also looks at it critically, this provides an intimacy between teacher and material and is thus categorized as a participatory relationship. Meanwhile, if the teacher's use of the material is more tacit and sporadic, it will lack intimacy and is thus categorized as a non-participatory relationship. My intention is to characterize Cecilia's interaction with the material in use (FM) and to compare the written curriculum in FM and Cecilia's enacted curriculum. I do not intend to evaluate which degree of interaction (offloading, adapting or improvising) or the relationship (participatory or not) is better than the other. However, I acknowledge that comparison and evaluation are intertwined (cf. [START_REF] Jablonka | Why look into mathematics classrooms? Rationales for comparative classroom studies in mathematics education[END_REF]. Below, I present Cecilia as the telling case through some merged snapshots of from both interview and classroom recordings.

The telling case of Cecilia

During her interview Cecilia said that it is a waste of time making a written plan, because "if a lot of the students don't understand today's lesson, we would have to repeat it tomorrow and then my intended plan would crash if I'd written it down" (Cecilia, 9 Nov 2015). On the one hand, such comments indicate that Cecilia, as a locally competent teacher, trusts her ability to deliver the mathematical content with appropriate strategies for instruction. On the other hand, it can also be due to a lack of time that she does not write her plans, as she stressed that the ongoing national professional development program, Matematiklyftet, takes time away from all the things she has to do. Cecilia stated that she starts her planning for the introduction phase not by using the TG but the student textbook:

I turn to the current page in the textbook and see that the next passage is about multiplication by 9. Immediately, I have an idea about which strategies I want the students to know, and notice that the book is using the same strategy as me... but I prefer to create my own [instructions] using my own language. (Cecilia, 9 Nov 2015)

Cecilia's prospective mental plan is partially consistent with the textbook's plan. However, here the agency stays with Cecilia, since she claims to have her own mental plan. Her use of and interaction with the textbook could be understood through the Swedish teachers' context, in which they are not accustomed to using TGs in planning and teaching. In addition, the minimal support provided for how TGs may be used may compel Swedish teachers to turn to the textbooks instead. This and several other similar excerpts led me to infer that Cecilia is influenced by Swedish culture, as she states that she "prefers to create her own instructions", reflecting a generally held perception of Swedish teacher competence. During the interview, Cecilia mentioned, due to lack of time, that she occasionally glances at the TG to get a skeleton plan for the lesson. She then looks at its "introduction box", which suggests how to introduce the lesson's topic on the board. Hence, from the interview I infer that she perceives the material as worth having in the classroom but not necessary for planning. I infer that she improvises when planning, and that the agency lies with her. I further infer that she has a more tacit than close relationship with the material, especially since she seldom uses the TG and hardly ever reflects on the material or its impact on the context.

The video data reveal that, while mobilizing the teaching, Cecilia sequenced her lessons into four distinct episodes. The allocated lesson time of her lessons consists of: the introduction phase, taking approximately 22% (~10 min) of the lesson time; what to work on in the student textbook; the students' individual seatwork, taking approximately 56% (~25 min) of the lesson time; and the closure of the lesson. Cecilia always starts her lessons by showing a strategy or method in the introduction phase that is applicable to that day's pages in the student textbook, and by referring to textbook: "Hey, listen! Last Thursday we went over page 90 in the textbook, as we used these hands [pointing to the cut-out hands in fabric on the blackboard] as one strategy for multiplying by 9. Today we'll revise it" (Cecilia's L1, 9 Nov 2015). Here, Cecilia is simultaneously showing the textbook pages they have been working with. This revision is not included in the TG's suggested lesson plan, and no elaboration or reflection is revealed. But this could also be due to the evaluation at the last lesson closure. Hence, I infer that Cecilia is improvising. Nevertheless, the video data frequently show that the delivery of the content is based on the material, as she offloads the agency to the textbook as she follows the textbook pages, lesson by lesson. Cecilia is very firm during the interview that the cut-out fingers she refers to are not from the TG but were instead an idea that simply came to her. During Lesson 1, she first shows two examples of the old strategy before introducing a new strategy for multiplying by 9: "Hey guys, listen! At the top of page 91, it says 'Multiply and write in the table'… This is a different strategy… So, let's try this too! Ehh, they want us to think like this... Can you give me a multiplication from the 9 table, Ali?" (Cecilia's L1, 9 Nov 2015). This extract illustrates Cecilia turning to a rather close offloading to the student textbookespecially when she says they want us to think like this. The textbook lacks a description of how to deal with this task, and the fact that Cecilia does not explain to the students how "to think" indicates that she has not elaborated on this task beforehand. I infer that she shows this task since it is included on that day's lesson pages in order to prepare the students for their individual seatwork. However, the TG offers some explanation, and recommends that they fill in this table together in the whole-class setting, which Cecilia has missed since she does not read the TG carefully or regularly. This displays that her relationship with the TG is rather tacit. The video data further reveal that after the introduction Cecilia always tells the students which pages to work with during the individual seatwork. She does this through the material's website and the SmartBoard, where she shows the students the pages. She also writes the pages on the whiteboard. This procedure is not stated in the material, which confirms that Cecilia is improvising and maintaining the agency. The following is an example of how she transitions the students into their individual textbook work: "I think most of you managed to do both pages 90 and 91, and possibly also 92 or 93. On page 94 it says 'We rehearse', and these two pages are the goal of today's lesson" (Cecilia's L1, 9 Nov 2015). This extract additionally confirms that Cecilia is closely offloading to the textbook, as she assigns a great degree of authority to it. At no time does she present the lesson objectives, which are clearly visible in the TG. Instead, she mentions that the lesson goal is to do pages 94 and 95. Neither does she use the different recurrent activities or concrete materials included in the TG during any of these lessons.

The video data further reveal that, during the individual seatwork, some students are working on other pages than the ones Cecilia had announced before they started working individually, and some are even working in a textbook for Grade 2. This is not in line with the material's intention, as it offers embedded differentiation instead. As shown in the video data, Cecilia always closes the lessons with a blind evaluation to determine whether she can move on or if rehearsal is necessary. Now, close your eyes and answer YES to my question by raising your hand. If your answer to my question is NO … leave your hand on the table […] 'I feel confident about the strategy of using my fingers to multiply by 9' Okay, those of you who have your hands up can put them down. 'I still think it feels a little hard to use this strategy, using my fingers to multiply by 9' Thanks! ... 'I feel pleased with what I did during the mathematics lesson today ' Great! (Cecilia's L1, 9 Nov 2015) This extract confirms that Cecilia does not just say she wants all her students to be on track but that she also checks this before ending the lesson. In so doing, she is checking their understanding of the "old" strategy for multiplying by 9, but not whether they understand the "new" strategy she has introduced, or the objectives displayed in the guide. Two of the questions are related to the mathematics, whereas the last is connected to students' individual seatwork. There is no support in the material for how to end the lessons, so Cecilia trusts to experience and improvises the evaluation. Hence, from the video data I deem that she uses the textbook for support for the students' individual seatwork but not for her actions or events when mobilizing the teaching. The video data show that Cecilia improvises, but does not critically reflect on the material or its impact on the educational context, or make any changes in relation to the material. In addition, at several points, the interview and video data show collectively that Cecilia largely offloads the agency to the textbook and uses it on an ad hoc basis. Cecilia's use of the TG is minimal. Thus, this settles her weak interaction with the material; i.e., having a non-participatory relationship with it.

Discussion and conclusion

In this paper, I present the telling case of Cecilia, a Swedish teacher, locally regarded as competent. The aim is to reveal her interaction with an imported TG from Finland when placed in her specific context. Thus, the material is sited in a new educational context. In the analysis I compare the written Finnish TG with Cecilia's actual classroom practice. The analysis is therefore combined with in-depth descriptions and snapshots of events, and is thus in line with the telling case (cf. Andrews, In press) as an attempt to make visible how she interacts with the Finnish material and how this interaction may have affected her classroom practice.

First, how does Cecilia interact with the Finnish material? My analysis revealed that Cecilia uses the student textbook when teaching, and that she offloads agency to the textbook. This interaction is categorized as non-participatory since it lacks intimacy. Her interaction with the TG is even weaker, and more sporadic and tacit than with the textbook, and is thus also non-participatory. When she trusts in her own knowledge and experience, improvising occurs, especially in regard to the repetition at the beginning of the introduction phase and the closure of the lesson with the blind evaluation. No adaptation was observed, since no equally embedded agency was found. Cecilia says she creates her lesson plans mentally. However, her focus is not on the entire lesson, since she only prepares the introduction phase. Even though she has chosen FM due to her judgment of its overall good quality, she does not seek support for teacher learning or to enhance the variety in her lessons through its recurrent activities. Second, is Cecilia's classroom practice affected by her interaction with the Finnish material? My analysis revealed that Cecilia's enacted classroom practice mirrors the "typical Swedish" practice, with short introductions and then individual seatwork most of the time (cf., e.g., [START_REF] Remillard | A cross-cultural analysis of the voice of six teacher's guides from three cultural contexts[END_REF]. Cecilia does not keep the students together around a specific mathematical topic by using the embedded differentiation, and no concrete materials are used during these three lessons. No objectives are stated, either. These are all important parts of the cultural scripts found in Finnish TGs (Koljonen et al., under review). Thus, Cecilia's classroom practice is in contrast to those promoted by the Finnish TG. I deem that Cecilia's practice is marginally affected by her relationship with the material. This may be because it does not offer enough support for how to use it, or explain its intentions, therefore forcing Cecilia to follow the common norms of Swedish classroom practice; as well as the fact that it is challenging for teachers to change their teaching [START_REF] Stein | How curriculum influences students learning[END_REF][START_REF] Stigler | The teaching gap: Best ideas from the world's teachers for improving education in the classroom[END_REF]. Further studies are needed to capture the essence of the Swedish classroom practice when using imported material.

My conclusion is that the use of the originally Finnish material has not had the intended impact on the practices as promoted by the guides. Instead, Cecilia uses and confirms her preexisting culture rather than the intended one as in the Finnish TG (cf. [START_REF] Davis | Teachers and science curriculum materials: Where we are and where we need to go[END_REF][START_REF] Stein | How curriculum influences students learning[END_REF][START_REF] Stigler | The teaching gap: Best ideas from the world's teachers for improving education in the classroom[END_REF]. One possible implication of this is that it may be hard to implement material from other educational contexts, even if it is quite similar and is assumed to change or even improve the quality of teaching. Yet without targeted support for how to use new material it is hard, even if a teacher is regarded as competent, to independently conduct changed or improved teaching and simultaneously maintain or gain pedagogical autonomy. This is especially important since the Finnish material lacks educative support and, thus, is not regarded as educative material [START_REF] Hemmi | Investigating Finnish teacher guides as a resource for mathematics teaching[END_REF][START_REF] Koljonen | Finnish teacher guides in mathematics -Resources for primary school teachers in designing teaching[END_REF]. I argue that this requires that teacher materials be included in professional development programs, as previously argued for by [START_REF] Ball | Reform by the book: What is-or might be-the role of curriculum materials in teacher learning and instructional reform[END_REF], in order to proficiently convey and highlight the principles of the materials and adjust them to the new context that is underpinned by the social and cultural practice. It remains to be seen whether subsequent case studies of the other teachers in the larger project reveal whether the above-mentioned tentative conclusions hold for the larger data set.

Proceedings of CERME10