Eva Jablonka
email: eva.jablonka@fu-berlin.de

A preliminary comparison of Chinese and German state mandated curricula for mathematics education (years 1 to 6)

Keywords: Mathematics curriculum, curriculum research, cross cultural studies, educational goals. China

The paper presents a preliminary analysis of official state-mandated curriculum documents for compulsory school mathematics education in China and Germany with a focus on years one to six. Both documents were issued in the context of relatively recent reform initiatives. In the context of these reforms some have observed an international harmonization of curriculum goals via the educational discourse disseminated by the OECD. This trend motivates the comparison. The preliminary comparison presented in this paper is part of a larger upcoming project that investigates how curriculum objectives are interpreted by actors at different levels of governance in systems with differing policies for controlling curriculum implementation. As a starting point for investigating the discretionary space granted to schools and teachers in their micro-curricular planning, the comparison presented here identifies the level of detail in the content standards.

In China, after an incipient period in the 1950s where Soviet mathematics education was investigated and "sinocised" through translating and adapting Soviet textbooks [START_REF] Ye | Developments and changes in the primary school mathematics curriculum and teaching material in China[END_REF], subsequent explorations of other systems appear much more diverse and critical. At a national forum convened by the Ministry of Education of the P.R.C. in 1959, investigations of Eastern European curriculum reforms were initiated; in the 1960s comparisons included the German Democratic Republic, the Soviet Union as well as the United States and Japan. In the late 1970s and the 1980s, the People's Education Press investigated textbooks from Germany (along with some from England, France, Japan and the United States) in order to look at the structuring of mathematical knowledge and arrangement of topics [START_REF] Xu | The development of school mathematics textbooks in China since 1950[END_REF]. Some dimensions of the recent Chinese curriculum reform were originally initiated by investigations of trends in mathematics education in a range of countries, including Germany, Hong Kong, the United States and Taiwan in the 2000s [START_REF] Lv | The evolution of mathematics curriculum and teaching materials in secondary schools in the twenty-first century[END_REF][START_REF] Ye | Developments and changes in the primary school mathematics curriculum and teaching material in China[END_REF].

In the context of incipient reforms in the 1950s, classroom pedagogy in some regions in northern China was found to draw on a five-step formal sequencing model for instruction (preparing; prompting; making comparisons; summarizing; applying) developed by the German philosopher Friedrich Herbart (who held Emmanuel Kant's chair in Königsberg) and his followers; the model was imported during the New Culture Movement. According to [START_REF] Ruan | High school Chinese language and literacy curriculum reforms[END_REF] this sequencing of instruction is still seen in many classrooms in China at the beginning of the 21st century.

Over the last decade the Chinese mathematics curriculum has been subjected to continuous reforms. Some shift in focus is according to Zhang (2018, p. 480) motivated by the needs created in an era of "knowledge-based economies and information-based societies, the fierce international competition and challenges", which require a capability for "knowledge innovation and scientific innovation, to be able to collect, sort and express information using mathematical methods, establish models, solve problems, and create value for society." The resulting revised compulsory mathematics curriculum standards (issued in 2011; implemented in autumn 2012) stress the importance of mathematical thinking and problem solving, including problems in real-world settings, in addition to the traditional emphasis on developing basic mathematical knowledge and skills [START_REF] Wang | Chinese mathematics curriculum reform in the twentyfirst century[END_REF]. Further, aspects of what often is associated with allegedly 'Western' pedagogies, including 'student-centered' activities such as experimenting and conjecturing, argumentation and discussion have found their way into official Chinese recommendations for curriculum implementation [START_REF] Li | Features and characteristics of Chinese new century mathematics textbooks[END_REF]. The changes in China also entailed a de-centralizing of educational administration and textbook production in order to allow for more diversity [START_REF] Cai | Problem solving in Chinese mathematics education: Research and practice[END_REF]. In comparison to the previous policy for controlling curriculum implementation via textbooks, this entails a shift towards proposing the use of a range of resources, only harmonized by the national standards, which consequently (compared with the old syllabus) expanded in scope and depth [START_REF] Xu | The development of school mathematics textbooks in China since 1950[END_REF][START_REF] Wang | Chinese mathematics curriculum reform in the twentyfirst century[END_REF].

In Germany, the unexpectedly low national performance in the OECD's PISA as well in the IEA's TIMSS-tests at the turn of the millennium stimulated a major debate on the quality of schooling and resulted in a new focus in educational policy. The Federal Ministry of Education and Research (BMBF) inter alia justified the need for developing outcome-oriented national educational standards by comparisons with some more successful education systems in the PISA-ranking. As a matter of fact, the selected comparator countries not only had adopted national curriculum standards, but also engaged in regular external evaluations of student achievement-outcomes. As [START_REF] Ertl | Educational standards and the changing discourse on education: the reception and consequences of the PISA Study in Germany[END_REF] points out, this new attempt of curriculum alignment across the Länder of the Federal Republic of Germany was conceptually different from earlier approaches of the Standing Conference of the Ministers of Education and Cultural Affairs (KMK) to reaching agreement on shared national standards since the late 1990s. The initiative not only aimed at modernizing and homogenizing local curricula, but also at alignment of educational aims and assessment. According to an expertise commissioned by the BMBF, the new national curriculum should describe competencies at a level of detail that not only affords conversion into learning tasks for students but in principle also into test items so that it becomes "possible to check whether the desired competencies have actually been acquired." And further, "This will determine the extent to which the education system has fulfilled its mandate (educational monitoring), and schools will receive feedback on the results of their work (school evaluation)" (Klieme et al., 2003, p. 10, transl. EJ). At the same time the expertise recommended that standards should determine competency outcomes (instead of listing topics to be covered) and so leave sufficient freedom for schools and teachers for micro-curricular planning.

The resulting national curriculum document bears some similarities with the US National Council of Teachers of Mathematics (NCTM, 2000) standards in its descriptions of general (topic-and context-independent) processes and the structuring of mathematical topics in broader areas. All Länder have agreed to use the national standards as a basis for developing new local curricula as well as in teacher education. In parallel they also founded institutes responsible for the control and the improvement of the quality of schooling. These institutes not only employ assessment for educational monitoring and school evaluation, but also aim at developing teachers' professional practices by exemplary operationalization of the curriculum standards in the form of assessment tasks (Blum et al., 2006).

The above sketch of diverse reform movements, in the context of which the Chinese and German mathematics curriculum documents have been produced, suggests that the texts under scrutiny represent hybrids of a range of pedagogic discourses from antecedent national curricula and other educational policy texts, international curricula as well as national recontextualisations of policy texts produced by supra-national institutions. The stated curriculum objectives in both systems reflect international mathematics education discourses that frame the outcomes as not only relevant for developing disciplinary knowledge, but stress applications ('mathematical literacy', 'practical mathematics', 'numeracy') [START_REF] Jablonka | The evolvement of numeracy and mathematical literacy curricula and the construction of hierarchies of numerate or mathematically literate subjects[END_REF]. Consequently, these curricula can be expected to share some objectives and so have sufficient in common to make an investigation of their differences meaningful. Further, while in both systems there are centralized education regulations, there are differences in modes of governance that shape the decisions of actors (schools, teachers, textbook writers etc.). The production of textbooks in the two systems certainly recontextualises official pedagogic discourse in different spheres, with more or less intersections with the economic field.

Goals and methodology

In addition to other curriculum control instruments, such as officially sanctioned textbooks, external assessments, school inspections and other measures of evaluation, an investigation of curriculum documents provides a starting point for investigating how much interpretive action is granted to teachers. Depending on the level of detail in prescriptions of curriculum standards, teachers might perceive greater or lesser freedom to decide upon what content to bring into focus and how to adapt their instructional strategies in line with their school ethos, their professional identities, and personal values. In turn, such curriculum adaptations might either mitigate or (unintentionally) reinforce unequal access to mathematical knowledge: on the one hand, access to the generative principles and styles of mathematical reasoning that underpin disciplinary knowledge, and on the other, to more skill-based forms (e.g., [START_REF] Dowling | The sociology of mathematics education: Mathematical myths/pedagogic texts[END_REF][START_REF] Jablonka | Potentials, pitfalls and discriminations: Curriculum conceptions revisited[END_REF]. This concern has been raised in the context of curriculum reforms in both systems: in China particularly with respect to increasing social inequality (Cai & Nie, 2011); in Germany with a focus on the challenges for teachers in finding ways to address the needs of heterogeneous groups of students [START_REF] Siller | Herausforderung Heterogenität. Grundvorstellungen als Basis und Bezugsnormdas Beispiel Terme[END_REF].

Based on the general research interest stated above, this preliminary comparison explores at which level of detail the prescriptions operate and which forms of mathematical knowledge and skills they privilege. It is part of a larger upcoming project that investigates how curriculum objectives are interpreted by actors at different levels of governance in systems with differing policies for controlling curriculum implementation. The larger project will investigate the articulation of curriculum objectives at three levels, which are commonly conceptualized as macro-, meso-and micro-levels, and will include case studies of classroom practice. While conceptualising the relation between these levels as "recontextualisation" of pedagogic discourses [START_REF] Bernstein | Pedagogy, symbolic control and identity: Theory, research, critique[END_REF], in the project this is not interpreted as constituting a hierarchical relation.

The focus of the comparison are the state mandated curricula for mathematics education for years 1 to 6. For Germany, the recently issued common mathematics curriculum standards for Berlin and Brandenburg (Senatsverwaltung für […], 2015; henceforth the "G-BBM") has been chosen as an example. The document has been developed in order to align the local with the statewide standards. In the two Länder primary school comprises 6 grades (instead of the usual four). The Chinese national curriculum standards for compulsory mathematics education (Ministry of Education P. R. C., 2012; henceforth the "C-NCM") are grouped into three stages, grades 1-3, grades 4-6, and grades 7-9. Only the first two stages are of interest for the comparison. The G-BBM is for grades 1-10. It is structured in levels A-H within topics. For years 1-6, "as a rule" the levels A-D are to be reached and so these have been chosen for the comparison. One shared feature of the two documents is the coherent structuring of the mathematical topics beyond grade six, which affords useful comparison.

The most obvious challenge of this comparison is language. The Chinese document was subjected to a machine translation into English as well as into German. In some instances, these versions complemented each other and lead to improvements. The draft was further edited with the aid of an online German-Chinese dictionary. In many cases only knowledge of school mathematics helped to edit the translation. In addition, a semi-official English translation of an early trial-version has been consulted along with translations of key concepts in publications authored by insiders. Still, there remained quite some nonsensical portions of text.

Both the C-NCM and the G-BBM contain sections with mathematical topics (entitled "Content Standards" and "Topics and Content", respectively). These sections contain short statements with some thematic coherence under one or two levels of headings and subheadings with names for school mathematical areas and specific topics. In the texts, these statements are clearly separated by lay-out (paragraph spacing, arrangement in table cells, numbering). For the analysis, each visually separated portion of text has been counted as one statement. While simply counting statements admittedly does not provide deep insights about the authority relations established by the stateauthor with the teacher-reader, this simple comparison of content standards might reveal differences to be explored further.

A difference that emerged when reading the content statements concerns the "discursive saturation" of the suggested student activity; the notion here is used in the sense of [START_REF] Dowling | Sociology as method: Departures from the forensics of culture, text and knowledge[END_REF]. While interpreting the translated curriculum statements has been aided by an explanation of "action verbs" used in the curriculum document found in the Appendix of the C-NCM, an in-depth analysis of envisaged student learning activities clearly is not possible with the machine translation. However, [START_REF] Bernstein | Pedagogy, symbolic control and identity: Theory, research, critique[END_REF] notion of "classification" can be applied to the structuring of the content in the documents, as his notion does not differentiate between forms of expression and content.

Outcomes

The C-NCM has four main sections: "Section I: Preamble"; "Section II: Curriculum Objectives"; "Section III: Content Standards"; "Section IV: Recommendations" (including suggested pedagogic strategies, evaluation recommendations, guidance on textbook development, development and usage of resources). If from Section III only the text for grades 1-6 is taken (which is relevant here), the proportions of the sections (pages of the English version) are very roughly as follows: One fifth for sections I and II together; one fifth for section III; three fifths for section IV. Obviously, the section that describes content standards is comparatively short.

The German G-BBM has three main sections: "1. Development of Competencies in Mathematics", "2. Competencies and Standards", "3. Topics and contents". If from Section 3 only the text for levels A-D is taken, the proportions of the sections (pages) are very roughly as follows: Five eights for sections 1 and 2 together; three eights for section 3. Obviously, the section that describes "topics and content" is also comparatively short, but fills a larger proportion of the document than in the C-NCM.

Table 1 depicts the number of statements for each stage by topics and subtopics in "Section III: Content Standards" of the C-NCM. The abbreviations are explained under the table. Topics and sub-topics are enlisted in rows and columns in the order of appearance in the text. "Practical and integrated applications" (in some publications the English term is "Practice and Synthesis Application"), aims at providing opportunities for cross-disciplinary work as well as for innerdisciplinary connections, and so guide "students to focus on practice and application" and "making the abstract mathematical concepts deeply rooted in children's real lives" (Wang et al., 2018 p. 67). The standards for statistics and probability obviously are less detailed than for the other topics. Table 2 (next page) depicts the number of statements for the levels A-D by topics and subtopics in the section "3. Topics and contents" of the G-BBM.

A comparison of the two tables suggests that the German content standards (Berlin-Brandenburg) provide more detailed descriptions than the Chinese. There are three levels in structuring the content in G-BBM, while in the C-NCM there are only two. As to the content, the school mathematical area Quantities and Measuring of the G-BBM, a topic that only appears in the German document, is in the C-NCM integrated into Numbers and Algebra in the subtopic Common Quantities (NA 3 ), and into Graphics and Geometry in the subtopic Measurement (GG 2 ). Here the classification of the C-NCM is stronger as it reflects classificatory principles of academic mathematics. The other extra topic in the G-BBM, namely Equations and Functions, is in the C-NCM largely contained in the topic Numbers and Algebra.

Concluding comments

Contrary to what might have been expected, the German content standards (Berlin-Brandenburg) describe learning outcomes at a much higher level of detail than the Chinese. This conflicts with the official policy discourse which preceded their development. According to [START_REF] Klieme | Zur Entwicklung nationaler Bildungsstandards[END_REF], the new standards should describe broader competency outcomes. The similarities in the main structuring of the topics in both documents suggests some cross-system harmonization of objectives in the 2000s (see [START_REF] Lv | The evolution of mathematics curriculum and teaching materials in secondary schools in the twenty-first century[END_REF][START_REF] Ye | Developments and changes in the primary school mathematics curriculum and teaching material in China[END_REF]. The Chinese content standards, however, appears to leave more discretionary space to teachers and schools for making micro-curricular decisions. In both contexts, the role of accountability measures vis-à-vis the standardization of the curriculum needs to be investigated. Regarding the naming of the topics and sub-topics, the Chinese appears to be more specialized, but complemented by the topic Practical and Integrated Applications. If [START_REF] Bernstein | Pedagogy, symbolic control and identity: Theory, research, critique[END_REF] concept of "classification" is applied here, this interpretation suggests an intended move towards weakening (internal and external) classification by establishing more relations with other school subjects as well as with everyday practice. This might be one of the reasons for the opposition against the reform from mathematicians reported by [START_REF] Yin | Implementing the national curriculum reform in China: A review of the decade[END_REF].

One conspicuity that emerged in the course of the investigation concerns the use of verbs that refer to students' competencies in terms of "experiencing", "developing a feeling for" or "grasping". Examples of such statements are, "[Students are] able to feel the meaning of large numbers when associated with realistic material, as well as to make estimations." (First Stage, "Numbers and Algebra: Knowing Numbers"), "Experience that amongst all lines connecting two points the segment is the shortest." (Second Stage, "Graphics and Geometry: Knowing Figures"). The German document does not contain similar statements. A consistent investigation of this observation has been hampered by the low-quality machine translation. However, it can be taken as an indication for a greater focus on non-discursive dimensions of mathematical practice. A systematic investigation of differences in curriculum statements with regard to privileged forms of knowledge and skills will have to include an analysis of stated overall objectives as well as pedagogic recommendations or examples of student activities that are given to exemplify the envisaged classroom practice. An important follow up question then is whether the documents aim at providing insights into the principles on the basis of which the curriculum has been developed.

Table 1 : Number of statements in "Section II: Content" (First Stage and Second Stage) of the C-NCM.

 1 Abbreviations used for subtopics in Table 1 are as follow: NA 1 Knowing Numbers; NA 2 Number Operations; NA 3 Common Quantities; NA 4 Exploring Patterns; NA 5 Expressions and Equations; NA 6 Direct and Inverse Proportion; GG 1 Knowing Figures; GG 2 Measurement; GG 3 Figures and their Transformation; GG 4 Figures and their Positions; SP 1 Simple Statistical Data Processing; SP 2 Random Phenomena.

		First Stage			Second Stage
		(Grades 1-3)			(Grades 4-6)				Sum
	Numbers and Algebra	NA 1 NA 2 NA 3 NA 4 NA 1 NA 2 NA 5 NA 6 NA 4
		7	8	5	1	9	10	4	4	1	49
	Graphics and Geometry	GG 1 GG 2 GG 3 GG 4 GG 1 GG 2 GG 3 GG 4
		7	6	3	2	9	8	4	4		43
	Statistics and Probability	no sub-division		SP 1 SP 2
			3			6	2				11
	Practical and Integrated Applications no sub-division		no sub-division
			3					4			7
									Sum Total	110

Table 2 : Number of statements in section "3. Topics and Contents" (levels A to D) of the G-BBM.

 2 The German G-BBM does not include an extra cross-topic field (such as Practical and Integrated Applications in the C-NCM), but outlines six mathematical "process standards" (generic skills or key competencies), which are described in the section "1. Development of Competencies in Mathematics" and "2. Competencies and Standards". Na 2 Na 3 Nb 1 Nb 2 Na 1 Na 2 Na 3 Nb 1 Nb 2 Na 1 Na 2 Na 3 Nb 1 Nb 2 Ea 2 Eb 1 Eb 2 Eb 3 Ea 1 Ea 2 Eb 1 Eb 2 Eb 3 Ea 1 Ea 2 Eb 1 Eb 2 Eb 3 Da 2 Da 3 Db 1 Db 2 Da 1 Da 2 Da 3 Db 1 Db 2 Da 1 Da 2 Da 3 Db 1 Db 2 Abbreviations used for subtopics in Table 2 are as follow: Na Conceptions of Number: Na 1 Perceiving and representing numbers; Na 2 Ordering numbers; Na 3 Describing number relationships; Nb Conceptions of Operations and Calculation Strategies: Nb 1 Developing conceptions of operations; Nb 2 Using calculation procedures and strategies; Qa Conceptions of Quantities and Measuring: Qa 1 Utilizing conceptions of quantities and their units; Qa 2 Determining expressions for quantities; Qb Calculating with Quantities: Qb 1 Calculating quantities in contexts; Sa Geometrical Objects: Sa 1 Describing geometrical objects; Sa 2 Describing relationships between geometrical objects; Sa 3 Representing geometrical objects; Sb Geometrical Transformations: Sb 1 Utilizing geometrical transformations and their properties; Sb 2 Carrying out geometrical transformations; Ea Terms and Equations: Ea 1 Representing terms and equations; Ea 2 Solving equations; Eb Mappings and Functions: Eb 1 Exploring mappings and functions; Eb 2 Representing mappings and functions; Eb 3 Utilizing properties of functional relationships; Da Data: Da 1 Collecting data; Da 2 Representing data; Da 3 Evaluating statistical investigations; Db Counting strategies and probabilities: Db 1 Using counting strategies; Db 2 Determining probabilities of events.

		Level B				Level C				Level D
		(End of grade 2)			(End of grade 4)			(End of grade 6)			Sum
			Na		Nb			Na		Nb			Na		Nb
	Numbers and Operations	Na 1 8	5	5	4	6	4	4	4	4	7	5	5	6	5	7	79
		Qa		Qb		Qa		Qb		Qa		Qb
	Quantities and Measuring	Qa 1	Qa 2		Qb 1		Qa 1	Qa 2		Qb 1		Qa 1	Qa 2		Qb 1
		7	5		3		7	4		4		8	8		6		52
			Sa		Sb			Sa		Sb			Sa		Sb
	Space and Shape	Sa 1	Sa 2 Sa 3 Sb 1 Sb 2 Sa 1 Sa 2	Sa 3 Sb 1 Sb 2 Sa 1 Sa 2 Sa 3 Sb 1 Sb 2
		6	3	9	4	4	4	5	7	3	4	3	5	4	2	2	65
		Ea			Eb		Ea			Eb		Ea			Eb
	Equations and Functions	Ea 1 3	6	3	2	1	4	5	2	2	2	3	3	2	2 1	41
	Data and		Da		Db			Da		Db			Da		Db
	Chance	Da 1 2	3	3	3	3	2	2	3	2	5	1	2	1	1	2	35
															Sum Total	272


Kristina Palm Kaplan
email: kristina.palm.kaplan@edu.uu.se

Reform on a shaky ground? A comparison of algebra tasks from TIMSS and Swedish textbooks

Keywords: Algebra, curriculum reform, large-scale assessment, textbooks

Background

In the Swedish curriculum reform of 2011, TIMSS and PISA were explicitly mentioned as a point of departure [START_REF] Skolverket | Kommentarmaterial till kursplanen i matematik[END_REF]. More precisely, algebra was singled out as a topic where students needed to improve their knowledge. Given the large influence of international large-scale assessments, ILSAs, the items of these tests should be further investigated [START_REF] Säljö | Promoting teaching, learning and informed decision-making through the lenses of international large-scale assessment: looking beyond numbers[END_REF].

Comparative studies using data from ILSAs have concentrated on their results, not their tasks [START_REF] Jablonka | Comparative studies in Mathematics Education[END_REF]. Moreover, while textbook tasks are common in comparative studies, these often focus on how the learning of a specific topic is presented in textbooks from different countries [START_REF] Jablonka | Comparative studies in Mathematics Education[END_REF]. They are not considered in relation to curriculum reforms.

The aim of this study is to problematize the role of ILSAs in national curriculum reforms. It builds on results from a previous study of textbook tasks by Palm Kaplan and Prytz (submitted) and is a part of a thesis project which problematizes the role of ILSAs and textbooks in curricular reform, by looking at the algebra in TIMSS 2003-2015 and algebra chapters in Swedish textbooks. This study focuses the case of Swedish school algebra and the 2011 curriculum reform through looking at algebra items from TIMSS 2003, compared to changes in year 8 algebra tasks in three Swedish textbook series. The research question is  To what extent is the algebra in TIMSS consistent or in conflict with algebra chapters in Swedish textbooks?

Theory and method

The algebra items in TIMSS 2003 are studied to understand what the student is expected to engage with, when solving the items. To get a detailed picture of the items, discourse analysis is combined with an analysis of the algebra offered. These two aspects are discussed in terms of school algebra discourses and algebraic activities respectively. The school algebra discourses are constructions in linguistic resources [START_REF] Luke | Text and Discourse in Education: An Introduction to Critical Discourse Analysis[END_REF], steering what choices are possible to express different ideas while the algebraic activities mainly builds on Blanton et al.'s [START_REF] Blanton | The Development of Children's Algebraic Thinking: The Impact of a Comprehensive Early Algebra Intervention in Third Grade[END_REF] 'big ideas' in algebra.

One of these is functional thinking, which is the activity to investigate, represent and generalize relationships between co-varying entities [START_REF] Blanton | The Development of Children's Algebraic Thinking: The Impact of a Comprehensive Early Algebra Intervention in Third Grade[END_REF], e. g. number patterns. The analytic frameworks are further developed on the poster and in Palm Kaplan and Prytz (submitted).

The analysis is conducted in several steps. The two aspects mentioned above are analyzed in the items to obtain the distribution of the algebraic activities in the school algebra discourses. The distribution is then compared to the results of the Palm Kaplan and Prytz (submitted) textbook analysis of 1557 algebra tasks from algebra chapters in six Swedish textbooks. These books come from three textbook series which were on the market before and after the 2011 curriculum reform. TIMSS 2003 entail 45 valid algebra items out of 47. Two trend items are left out since in TIMSS 2007, they are coded by TIMSS as data and chance instead of algebra. Items including more than one activity or more than one discourse are also left aside in this study, for reasons concerning statistical analysis in the thesis. All items are used with permission from the Swedish National Agency of Education. They are chosen as a starting point for further analysis, which will also include the algebra items from TIMSS 2007, 2011 and 2015 in order to understand the development of TIMSS' algebra.

Results

The distributions of the algebraic activities in the school algebra discourses are reasonably corresponding when TIMSS-items are compared to Swedish algebra chapters. However, there are differences between the proportions in the distributions: Swedish algebra chapters concentrate on mainly two algebraic activities while TIMSS 2003 entail other algebraic activities to a larger extent. For example, functional thinking is more stressed in TIMSS. The proportions of the school algebra discourses also differ in the TIMSS-items from the Swedish algebra chapters, and a whole new discourse, the Relational discourse, was identified in the TIMSS-items. This discourse constructs algebra as relationships between variables in graphs, tables and symbols; whereas the other discourses, identified in the algebra chapters, do not use graphs as a resource and only occasionally use tables. In Swedish textbooks though, chapters on functions and relations are often saved for year 9, which is the year after the TIMSS-test is taken. Though it is difficult to assess since there are few algebra items in one TIMSS-test compared to in textbooks, these results make one of the points of departure for the Swedish curriculum reform in 2011 questionable.


Jöran Petersson
email: joran.petersson@mnd.su.se

Judy Sayers

Paul Andrews

Time series analysis: Moving averages as an approach to analysing textbooks

Keywords: comparative textbook analysis, foundational number sense, moving average, time series

, namely simple addition and subtraction, is distributed and sequenced across three English, year one, textbooks. The analyses are compared empirically with four other methods found in the literature to show how time series analysis using moving averages helps address the shortcomings of these different approaches.

Introduction

For many teachers of mathematics, irrespective of where they work, the textbook they use is not only the major resource for lesson planning and the provision of tasks for students but also the means by which the curriculum within which they work is realised [START_REF] Stein | The role of mathematics curriculum materials in large-scale urban reform[END_REF]Tarr, Cháves, Reys & Reys, 2006;[START_REF] Stacey | Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks[END_REF]. However, textbook analysis is in some way probabilistic in the sense that teachers make decisions as to how they use any book, leaving the analytical question "what would students learn if their mathematics classes were to cover all the textbook sections in the order given? What would students learn if they had to solve all the exercises in the textbook?" (Mesa, 2004, pp. 255-256). That is, researchers are typically interested in understanding the "likely impact of the text on the users (teachers and students)" (Stacey & Vincent, 2009, p.276). Moreover, in those cultures in which textbooks are unregulated, often leading to a plethora of choice for teachers, students may receive very different opportunities to learn [START_REF] Charalambous | A comparative analysis of the addition and subtraction of fractions in textbooks from three countries[END_REF][START_REF] Huntley | One-step and multi-step linear equations: A content analysis of five textbook series[END_REF][START_REF] Tarr | From the written to the enacted curricula: The intermediary role of middle school mathematics teachers in shaping students' opportunity to learn[END_REF].

Much of the world's textbook-related research has employed some form of qualitative description, typically aligned with an analytical framework related to the topic under scrutiny, supplemented by frequency analyses. That is, researchers often count and compare the number of occurrences of particular forms of task in different textbooks. Such studies usually focus on either mathematical content knowledge or mathematical processes, although there are exceptions such as [START_REF] Borba | Analysis of the role of the calculator in Brazilian textbooks[END_REF] study of calculator use in Brazilian primary school textbooks. Research focused on mathematical content knowledge has included, for example, studies of primary school textbooks' treatment of fractions in Kuwait, Japan, and the USA [START_REF] Alajmi | How do elementary textbooks address fractions? A review of mathematics textbooks in the USA, Japan, and Kuwait[END_REF], Cyprus, Ireland, and Taiwan [START_REF] Charalambous | A comparative analysis of the addition and subtraction of fractions in textbooks from three countries[END_REF], division of fractions in grades six and seven in Chinese, Japanese and US textbooks [START_REF] Li | Conceptualizing and organizing content for teaching and learning in selected Chinese, Japanese and US mathematics textbooks: The case of fraction division[END_REF], multiplication and division of fractions in grades five and six in the textbooks of the USA and Korea [START_REF] Son | How reform curricula in the USA and Korea present multiplication and division of fractions[END_REF], inverse relations in US and Chinese primary textbooks [START_REF] Ding | Opportunities to learn: Inverse relations in U.S. and Chinese textbooks[END_REF], functions in the middle school textbooks of 15 countries [START_REF] Mesa | Characterizing practices associated with functions in middle school textbooks: An empirical approach[END_REF]; geometry in English and Japanese grade eight textbooks [START_REF] Jones | Interpretations of national curricula: The case of geometry in textbooks from England and Japan[END_REF], number sense in year one textbooks in England and Sweden (Löwenhielm, Marshall, Sayers & Andrews, 2017) the distributive property [START_REF] Ding | A comparative analysis of the distributive property in U.S. and Chinese elementary mathematics textbooks[END_REF] and so on.

Textbook-related studies on mathematical processes has included, for example, examinations of mathematical problem solving (Brehmer, Ryve & Van Stenbrugge, 2016;[START_REF] Fan | Representation of problem-solving procedures: A comparative look at China, Singapore, and US mathematics textbooks[END_REF], mathematical problem posing [START_REF] Cai | An analysis of problem-posing tasks in Chinese and US elementary mathematics textbooks[END_REF], general perspectives on reasoning and proof in Australia [START_REF] Stacey | Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks[END_REF] and the USA [START_REF] Stylianides | Reasoning-and-proving in school mathematics textbooks[END_REF], geometry-related proof for students in the grades 6-9 in the USA [START_REF] Otten | The mathematical nature of reasoning-andproving opportunities in geometry textbooks[END_REF] and France and Japan [START_REF] Miyakawa | Comparative analysis on the nature of proof to be taught in geometry: The cases of French and Japanese lower secondary schools[END_REF], calculus-related reasoning and proof for upper secondary students in Finland and Sweden [START_REF] Bergwall | The state of proof in Finnish and Swedish mathematics Textbooks-Capturing differences in approaches to upper-secondary integral calculus[END_REF].

Other studies have attempted to identify and demonstrate the different emphases within a textbook by going beyond simple quantification. For example, [START_REF] Tarr | From the written to the enacted curricula: The intermediary role of middle school mathematics teachers in shaping students' opportunity to learn[END_REF], in their analysis of American teachers' use of different types of textbooks, introduced what they called the emphasis index, which was defined as the number of lessons taught by a teacher on a topic divided by the number of lessons in the textbook on the same topic. Interestingly, a similar procedure could be applied uniquely to a textbook, where the emphasis index would be the number of lessons on a particular topic in the book divided by the total number of lessons. However, while frequency analyses have the potential to offer insight into the topics privileged by a textbook's authors they typically offer little with respect to how mathematical ideas are sequenced. Indeed, even though some scholars have conceded that they "examined only proportions of various types of opportunities and did not attend to the sequencing of activities, which may be important with regard to students' learning" (Otten et al., 2014, p.75), few have examined the distribution of codes within a text. Of those studies that have attempted explicitly to examine the distribution of codes, four methods stand out. Firstly, method 1, researchers have identified, in the sequence of all tasks, the position of the first task of interest [START_REF] Fujita | The order of theorems in the teaching of Euclidean geometry: Learning from developments in textbooks in the early 20th century[END_REF]. Secondly, method 2, researchers have analysed the proportion of a textbook covered before an occurrence of interest [START_REF] Alajmi | How do elementary textbooks address fractions? A review of mathematics textbooks in the USA, Japan, and Kuwait[END_REF][START_REF] Li | Conceptualizing and organizing content for teaching and learning in selected Chinese, Japanese and US mathematics textbooks: The case of fraction division[END_REF]. Thirdly, method 3, studies, particularly those analysing series of textbooks that cross several grade boundaries, have displayed relative percentages across grades [START_REF] Borba | Analysis of the role of the calculator in Brazilian textbooks[END_REF][START_REF] Ding | Opportunities to learn: Inverse relations in U.S. and Chinese textbooks[END_REF]. Moreover, Ding displayed results in a line diagram, where the distribution of a code per grade level in one book series could be compared with the distribution of the same code per grade level in another book series. This takes us to our final method and the inspiration for our contribution. Fourthly, method 4, studies have exploited timeline analyses, whereby coded observations were presented as dots on a horizontal axis representing the sequence of all tasks through the textbooks. With such an approach, "large swaths of these textbooks were not coded", allowing the reader to see, at a glance, how topics under scrutiny are both located and emphasised (Huntley & Terrell, 2014, p. 758). One advantage of such diagrams is that they give a picture that is fine-grained down to occurrence of individual codes. However, this is also its draw-back. For larger data sets, the graphic in the time-line diagram might get too jammed making it difficult to discern any useful information in the diagram. In this paper we present a comparison of several of these methods before offering a novel methodological perspective on the analysis of textbooks that goes beyond what others have achieved. In so doing, we aim to make a meaningful contribution to the methods of comparative textbook analysis.

The context of this study

This paper is a preliminary account of textbook analyses conducted as part of the Foundational Number Sense (FoNS) project, which is an ongoing Swedish Research Council-funded study of how FoNS, a set of eight core number-related competences, is acquired by year one children in England and Sweden. These core competences, which research has shown to be essential for later mathematical success, derived from a constant comparison analysis of literature from mathematics education, psychology, special educational needs and generic education [START_REF] Andrews | Identifying opportunities for grade one children to acquire foundational number sense: Developing a framework for cross cultural classroom analyses[END_REF]. While it is important to understand how textbooks structure opportunities for learning, England and Sweden are of particular interest as both systems, due to perceptions of systemic failure on international tests of achievement like TIMSS and PISA, have begun importing and adapting textbooks used in high-achieving countries. In Sweden, textbooks from Finland and Singapore have been imported, while in England, textbooks from Singapore and Shanghai.

In the study presented here, which is principally a methodological contribution, we examine aspects of three textbooks used with English grade 1 classes, Abacus, Inspire Maths (hereafter, Inspire), and Maths -No Problem (hereafter, MNP). These were chosen because Abacus is a long-standing series written by a well-known primary mathematics education researcher. The other two, drawing on societal perceptions that English mathematics teaching would be improved by the adoption of Singaporean practices, were English-authored adaptations of Singapore textbook series.

Comparing different display methods on the same data

Many educational research analyses yield codes of zeros and ones for the absence or presence of specific phenomena, particularly analyses of the tasks found in textbooks. Thus, depending on whether analysed properties are present in each task, one outcome of this process might be a sequence {0, 0, 1, 0, 1, 1,...} of ones and zeros. The manner in which the results from such analyses are displayed necessarily depends on the research question and could be the presentation of frequencies or the distribution of such codes throughout the data. When comparing textbooks, differences in the frequencies of particular codes can be compared by means of, say, χ 2 tests. However, when the research goal is to focus on the sequential distribution of codes, the focus of this paper, other methods are needed. In this paper, as a response to this, we offer time series analysis as a novel method for comparing the sequential distribution of particular task properties in textbooks. In so doing, we compare this novel approach with others found in the literature.

In the following, as indicated above, we draw on FoNS-related analyses of three English year one textbook series, in which every task that explicitly expected a student response was coded for its FoNS-related opportunities. Thus, every task could be represented as set of eight ones and zeros ranging from (0, 0, 0, 0, 0, 0, 0, 0,), in which no FoNS-related codes were identified, to (1, 1, 1, 1, 1, 1, 1, 1), in which all FoNS codes were identified [START_REF] Löwenhielm | Opportunities to acquire foundational number sense: A quantitative comparison of popular English and Swedish textbooks[END_REF]. However, with the objective of comparing methodologies, space prevents an analysis of all eight FoNS categories, so we restrict this presentation to just one, in order to exemplify the advantages and disadvantages of different analytical approaches. Thus, the following focuses solely on FoNS category seven, simple addition and subtraction within the number range 0 -20 due, in part, to the significance of such competence in later mathematical learning.

The figures of table 1 show the results of the two methods employed by [START_REF] Fujita | The order of theorems in the teaching of Euclidean geometry: Learning from developments in textbooks in the early 20th century[END_REF] and [START_REF] Alajmi | How do elementary textbooks address fractions? A review of mathematics textbooks in the USA, Japan, and Kuwait[END_REF] respectively. The first, based on the position of the first occurrence of the code in question [START_REF] Fujita | The order of theorems in the teaching of Euclidean geometry: Learning from developments in textbooks in the early 20th century[END_REF], shows substantial differences in the number of tasks completed before simple arithmetical operations are introduced. However, due to differences in the total number of tasks in each series, it is not necessarily straightforward, without additional calculation, to discern whether the positions of the first occurrence are comparable across the three books. In this respect, the second method, based on the percentage of all tasks presented before the first occurrence 2012), gives a better indication of how deep into the book the first occurrence of each code occurs and, of course, their relative positions. Thus, it can be seen that the introduction of simple arithmetic in the two textbooks based on the Singaporean tradition, Inspire and MNP, occurs not only substantially later than in the English textbook, Abacus, but also in different positions relative to each other.

Book

FoNS The third method, represented in Table 2, shows the relative percentages across the various workbooks in each series [START_REF] Borba | Analysis of the role of the calculator in Brazilian textbooks[END_REF]. Here, each series is partitioned into a different number of booklets; Abacus is in three booklets, Inspire in four and MNP in two. What is clear, interestingly, is the variation in the distribution of tasks focused on code 7, simple addition or subtraction, across the three sets of workbooks; Abacus incorporates substantial proportions of such material across all three booklets, while both Inspire and MNP incorporate such tasks only in the first half of their respective series. Of course, presenting the same data as line diagrams in the manner of [START_REF] Ding | Opportunities to learn: Inverse relations in U.S. and Chinese textbooks[END_REF] may have offered a clearer picture, but little by way of additional insight into the actual sequencing of the tasks.

The fourth method, Figure 1, shows a timeline with the position of each task coded for simple addition or subtraction, code 7, shown as a dot [START_REF] Huntley | One-step and multi-step linear equations: A content analysis of five textbook series[END_REF]. As with method 3, this highlights well differences between Abacus and the two Singapore-based books, Inspire and MNP.

The former shows simple arithmetic tasks distributed throughout the workbooks, while the latter confirms that such tasks only occur within the first half of the series. They also show a not dissimilar pattern within the two Singapore-based books, with periods of extended opportunity, followed by a gap with occasional revisitations before a second extended opportunity. That being said, such graphs offer insight only into the sequencing of activities within a textbook and not necessarily an indication how children's day-to-day learning may be structured.

Abacus

Moving averages

The fifth method, which is our contribution to the debate, involves moving averages as an approach to time series analysis in which data are logged at equally spaced points in time. Typically, these are undertaken to "understand the underlying dynamics, forecast future events, and control future events via intervention" of stochastic processes (Fan & Yao, 2003, p. 9). Now, while a school mathematics textbook is not stochastic, the use of time series, whereby data are successive tasks, should offer a clear indication of a textbook's sequential emphases over time. However, the graphs yielded by time series analyses are typically extremely noisy, as with daily records of air temperature [START_REF] Wakaura | A time series analysis on the seasonality of air temperature anomalies[END_REF], a problem that can be overcome by means of moving averages. Here, rather than report single data points, moving averages are based on a sequence of overlapping sets of data or moving windows. This process smoothes out short-term fluctuations in time series so that longer-term patterns become more visible and the influence of outliers is eliminated. Mathematically, a moving average means substituting a single data point with , where is the arithmetic mean of its neighbouring data points as in equation 1.

(Equation 1)

Of interest here, is the size of the divisor, 2n + 1, which represents the total number of data points included in the calculation and is dependent on the time period chosen for the calculation. That is, refers to the original point, , and its neighbouring data points, n before and n after. In the context of a mathematics textbook, the width of this window could be the number of tasks that an average student is expected to cover each day, or each week or each month and this choice depends on the research question. Thus, one choice of the width of the moving average window could be , roughly corresponding to a single week's workload across the year. This means that wherever the moving average diagram shows 'above zero', then the pupil would have met that coded property during that week. Figure 2 shows the results, for code 7, simple addition or subtraction, for different time periods. The vertical axis refers to the average code score, taking values from 0 to 1, for any given time interval, and the horizontal axis the year for which the textbook is intended for use. The solid line refers to MNP, the dotted line to Inspire and the dashed line to Abacus. As can be seen, analysing the daily opportunities for code 7 produces a noisy diagram that is difficult to interpret, particularly with the three textbooks are presented together. However, as seen in the weekly graphs, a longer time period begins to smooth out much of this short-term variation and expose patterns of emphasis. Therefore, in such analyses it is important to select a window that is sensitive to the context of the data source in to obtain a graph that shows neither too little nor too much of the data's details. In the context of this study, the monthly graph is probably the most illuminating. The two Singapore books are virtually indistinguishable but Abacus, shows continued exposure, albeit at different levels of intensity, to tasks focused on simple addition and subtraction, opportunities, we know from several of the analyses above, missing in the second half of the two Singapore-inspired series.

In conclusion, we believe that graphs presenting moving averages offer a novel but methodologically important tool in the analysis of textbooks. In fact, it can be argued that it represents a generalisation of the four methods compared above, albeit dependent on the choice of time interval. For example, where the moving average curve in figure 2 rises above zero, it shows where the first occurrence of a code occurs, which is the information provided by methods 1 and 2 in table 1. Moreover, it shows where in a series of books a code occurs, as yielded by method 3 in table 2, corresponding to a non-overlapping (disjoint) average, while moving averages do overlap. Finally, moving averages offer a direct generalisation of strategy 4, since a moving average with a window of length one would reproduce the diagram in Figure 1.

Figure 2 .

 2 Figure 2. Method 5 (moving average) for code 7

Table 1 . Method 1 (1st occurrence) and method 2 (% tasks before first occurrence)

 1

		category	Position of first occurrence	Percentage of tasks before first occurrence
	Abacus	7	22nd of 1522	1%
	Inspire	7	168th of 907	18%
	MNP	7	175th of 1955	9%

Table 2 . Method 3 (% occurrences per booklet) for code 7 Figure 1. Method 4 (timeline dot plot) for code 7

 2

Acknowledgment

The authors gratefully acknowledge the support of the Swedish Research Council (Vetenskapsrådet), project grant 2015-01066, without which the work reported here would not have been possible.


Hendrik Van Steenbrugge
email: hendrik.van.steenbrugge@mdh.se

Heidi Krzywacki
email: heidi.krzywacki@helsinki.fi

Janine Remillard
email: janiner@upenn.edu

Tuula Koljonen
email: tuula.koljonen@mdh.se

Rowan Machalow
email: machalow@gse.upenn.edu

Kirsti Hemmi
email: kirsti.hemmi@edu.uu.se

A cross-cultural study of curriculum systems: mathematics curriculum reform in the U

Keywords: Cross-cultural study, Curriculum reform, Mathematics education

This paper relates to the mathematics curriculum systems of the United States, Finland, Sweden, and Flanders (Belgium). These four regions are in the midst of curriculum reform, which provides interesting grounds for cross-cultural comparison. Our analysis builds on a framework that focuses on curriculum policy, design and enactment in each of these regions and draws on interview data with teachers in all four regions, sample cases of curriculum use, context descriptions, and available descriptions of mathematics education in these four regions. This leads to a more nuanced understanding of the particular curriculum systems through which reform manifests, and sheds light on a challenging balance concerning a curriculum reform that is both coherent across a region and supported by teachers.

Mathematics curriculum reform: a delicate process

Curriculum reform is a delicate process because multiple factors influence implementation, and, ultimately, student performance. If a curriculum is to promote region-wide reform, it should be coherent across that region. Further, there is evidence that the teacher has a crucial role, in that teachers should embrace the underlying vision (e.g., [START_REF] Tarr | The impact of middle-grades mathematics curricula and the classroom learning environment on student achievement[END_REF]. Also crucial for educational change is to understand the educational system to which the reform applies [START_REF] Andrews | Negotiating meaning in cross-national studies of mathematics teaching: Kissing frogs to find princes[END_REF][START_REF] Miyakawa | Paradidactic infrastructure for sharing and documenting mathematics teacher knowledge: a case study of "practice research[END_REF][START_REF] Stigler | The teaching gap: Best ideas from the world's teachers for improving education in the classroom[END_REF]. This paper aims to add to a better understanding of the mathematics curriculum systems in the U.S., Finland, Sweden, and Flanders (Belgium). All four regions have recently undergone mathematics reform, or are in the midst of reform, which makes them interesting sites for comparison of curriculum systems. The paper's central goal is to describe the curriculum systems of these four regions, and to consider consequences for teacher involvement in, and region-wide coherence of the region.

Curriculum policy, design, and enactment framework

Because we understand teachers' use of resources to be situated in a broader school system, we draw on the curriculum enactment process as conceptualized in [START_REF] Remillard | Conceptualizing the curriculum enactment process in mathematics education[END_REF] (See Figure 1). Remillard and Heck differentiate between an official and operational curriculum. The official curriculum, authorized by governing agencies includes curricular aims and objectives; assessments; and the designated curriculuma set of instructional plans specified by a governing agency. The operational curriculum captures the enactment process. It acknowledges the central role that teachers have in interpreting and mobilizing curriculum resources and differentiates between a teacher-intended and enacted curriculum and student outcomes. The location of instructional resources outside of the official and operational curriculum allows to fit both (centralized) systems in which the instructional resources are part of the official curriculum, and other systems in which they are not.

The framework assumes a definition of curriculum, which we also subscribe to: "a plan for the experiences that learners will encounter, as well as the actual experiences they do encounter, that are designed to help them reach specified mathematics objectives" (Remillard & Heck, 2014, p. 707). We use the term instructional resources to refer to the resources used to support curriculum enactment. These resources include curriculum resources that sequence a particular content such as student textbooks and teacher's guides, but also other resources such as digital (online) applications.

Context and method of study

This study is part of a larger cross-cultural study on elementary school teachers' use of printed and digital instructional resources in the U.S., Finland, Sweden, and Flanders (the northern part of Belgium, which has its own educational system). Although largely an opportunity sample, the selection of these four regions addresses both constants and contexts [START_REF] Osborn | New methodologies for comparative research? Establishing 'constants' and 'contexts' in educational experience[END_REF]) comprising a sound rationale for comparison. Talking to the constants, all four regions value local educational authority, emphasize similar aspects as to the mathematical curriculum, and teachers rely on (printed) curriculum resources when teaching mathematics. Our previous analyses of printed curriculum resources also shed light on differences in provided teacher support, surfacing contextspecific assumptions of teaching and learning mathematics (e.g., [START_REF] Remillard | A cross-cultural analysis of the voice of six teacher's guides from three cultural contexts[END_REF].

When designing and analyzing interviews on resource use in the four regions, we were faced with challenges of equivalence, validity, and comparability [START_REF] Clarke | The validity-comparability compromise in crosscultural studies in mathematics education[END_REF][START_REF] Osborn | New methodologies for comparative research? Establishing 'constants' and 'contexts' in educational experience[END_REF], and with challenges related to the undertake of such a study in a cross-cultural team of researchers. To develop the team's prerequisite intersubjectivity [START_REF] Andrews | Negotiating meaning in cross-national studies of mathematics teaching: Kissing frogs to find princes[END_REF] needed to fully understand the completed interviews as situated in their specific context, we developed case descriptions illustrating curriculum use for one teacher per context, and context descriptions. This paper draws primarily on these four case and context descriptions, but also on interview data specifically relating to the selection of instructional resources and additional readings on mathematics curricula in these four regions (i.e., [START_REF] Hemmi | Mathematics Curriculum: The Case of Finland[END_REF][START_REF] Remillard | Mathematics Curriculum in the United States: New Challenges and Opportunities[END_REF][START_REF] Van Steenbrugge | Developing a reform mathematics curriculum program in Sweden: relating international research and the local context[END_REF][START_REF] Verschaffel | All you wanted to know about mathematics education in Flanders, but were afraid to ask[END_REF].

Ten teachers in Finland, the U.S., Flanders, and Sweden were interviewed in fall 2017 and again in spring 2018 on their use of resources when planning and teaching mathematics (Note: In Sweden and the U.S., one teacher was unavailable for the second interview; in Finland, nine instead of ten teachers have been interviewed so far). The first interview was more general and addressed teacher and school backgrounds, what resources teachers used, teachers' views on the curriculum resources being used, and teachers' general beliefs on teaching and learning mathematics. The second interview focused in more detail on teachers' actual use of both print and digital resources, centered around a walk-through of planning, decisions, and enactment of a lesson that the teacher taught recently. Input for Interview 1 initially came from team members' previous related research on curriculum use and was modified during subsequent team meetings. Interview 2 was also developed collaboratively, based on findings and experiences from Interview 1 and our knowledge of each of the contexts.

Each case description was prepared by a team member who is a cultural insider, written in English for shared use. We first applied low-inference codes to the interviews to index excerpts of the interviews. These codes identified, for instance, teachers' descriptions of resources, how they were used, reasons for use, background information on the teacher and school, and teacher beliefs on curriculum use and teaching and learning mathematics. The process of coding was tried out individually, discussed in, and refined by the team. Having coded two interviews for one teacher per region, we gathered similarly-coded statements and applied the following structure to the cases: a) teacher education and teaching background, b) information about school and class, c) selection process of the resources, d) use of resources and purposes for use, e) teacher beliefs and conceptions, f) changes in resource use.

The process of writing and reading cases made us aware that significant insider knowledge was necessary to make sense of them, which is why we also developed context descriptions. Context descriptions are organized according to the following structure: a) school system-structure, b) pathways into teaching elementary mathematics, c) school environment, d) financial resources for organizing education, e) decision-making mechanisms in schools in relation to mathematics education (including the selection of instructional resources), f) student assessment, and g) monitoring and quality assurance of education.

An important step in the process of developing the cases and context descriptions was full-team review and discussion of them. In fact, we arrived at a common structure and approach through incremental development, review, and discussion.

Building on the curriculum policy, design, and enactment framework [START_REF] Remillard | Conceptualizing the curriculum enactment process in mathematics education[END_REF], we came to the following analytical structure to compare the four educational systems, based on case and context descriptions, interview data, and the abovementioned additional readings:

-Educational jurisdiction and school funding; -Most recent central mathematics curriculum, including name and launching date, initiators, structure, novel aspects, requirement of adoption; -Role of assessments; -Curriculum specification in addition to central curricular aims and objectives; -Instructional resources and influential factors, including resource market, designers of curriculum resources, embedment of a digital platform, selection of resources, acceptance criteria.

Curriculum systems in the U.S., Finland, Sweden, and Flanders Regulations and incentives to steer local authority are present to different extents. The curriculum systems of the U.S. and Flanders have the most explicit mechanisms to steer local authority. In the U.S., states possess authority in relation to educational policy and, sometimes, delegate policy to school districts, but the Government by means of applying specific funding mechanisms influences policy and curriculum use at state, district, and school level. In Flanders, schools are in principle free to determine how to work toward the attainment targets, but the Government, through regulations such as school inspectorates and the requirement to adopt a learning plan, and through targeted funding, sets the framework of the curriculum system and influences curriculum policy and use at the local school level. In Sweden, the Government also sets the framework of the curriculum system, but influences curriculum use at a more implicit level, through rolling out a nation-wide professional development program following the curriculum reform. From our study, it appears that central regulation is the least well manifested in Finland. The Finnish National Board of Education commissions on regular interval-base an expert group to develop a new curriculum. Schools and teachers are provided with guiding documents and regulations, but are not checked upon application of the guidelines and regulations.

Across the four regions, curriculum resources served as interpreters of the official curriculum, hereby serving as mediators between the intended curriculum and the classroom [START_REF] Valverde | According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks[END_REF]. Additionally, and talking to the systems of Finland and Flanders, curriculum resources can also potentially influence curriculum making. In Finland, teachers at times rely on the learning sequence in commercial curriculum resources to design their crucial school-level curriculum. Currently in Flanders, new curriculum resources, often complemented with digital applications, are published before the actual launch of the new attainment targets, hereby possibly influencing the novel aspects of the mathematics reform related to digital competence.

Following the curriculum policy, design, and enactment framework [START_REF] Remillard | Conceptualizing the curriculum enactment process in mathematics education[END_REF], we allocate teachers to have a central role in interpreting and mobilizing the curriculum. Indeed, we find related evidence in our interview data. Our comparative analysis also reveals differing levels where significant curriculum interpretation happens to reside. In Sweden, the bulk of interpretation happens at the individual teacher level. In Finland, significant interpretation is applied to compose a school-level curriculum, whereas in Flanders, major interpretation of the attainment targets is located above the school-level, by the umbrella organizations issuing learning plans. In the U.S., significant curriculum interpretation resides in the assessments.

Curriculum reform: a delicate balance between region-wide coherence and teacher approval

Our study of the mathematics curriculum systems of the U.S., Finland, Sweden, and Flanders, suggests that a curriculum reform that is both region-wide and supported by teachers, is a challenging balance. Both Flanders and the U.S., through their layered curriculum infrastructure, succeed most toward a region-wide curriculum coherence, but this goes at the cost of teacher involvement in the reform process. In Finland, teachers are most involved in reform through the design of a school-level curriculum, but this goes at the cost of a nation-wide curriculum-coherence. Sweden stands out in that teachers were asking for reform and that the Government answered the call by means of rolling out a nationwide professional development program. It still has to be seen to what extent that results in curriculum coherence. In all four regions, commercially published curriculum resources are a central aspect in a region-wide curriculum reform. Given this significant position, it is remarkable that only in the U.S., sometimes quality criteria are issued that curriculum resources have to pass.

Figure 1 .

 1 Figure 1. Visual model of the curriculum policy, design, and enactment system (Source: Remillard & Heck, 2014, p. 709)

Table 1 . The curriculum systems of the U.S., Finland, Sweden, and Flanders

 1 Table1includes our descriptions of the curriculum systems of the U.S., Finland, Sweden, and Flanders. Looking across the table helps to attain a more nuanced understanding of these curriculum systems, which, from the outset share similarities such as local authority, use of a primary (usually printed) curriculum resource available from a commercial publishing market, and the raise of digital resources. Surfaced similarities and differences relate to a) regulations and incentives to steer local authority, b) role of curriculum resources in curriculum reform, and c) curriculum interpretation. We discuss these aspects below and relate them in a final section to two crucial aspects of curriculum reform: coherence and embracement of the reform by teachers.

		U.S.	Finland	Sweden	Flanders
	Educational jurisdiction and school funding
	Jurisdiction	School districts	Finnish	Swedish	Flemish
		with oversite	government	government	government
		from states
	School	State and local	National and	Government to	Government; also
	funding	taxes	local funding	municipals to	targeted funding
				schools
	Most recent central mathematics curriculum
	Name &	Common Core	National Core	Läroplan (LGR	Attainment
	launching date	State Standards	Curriculum	11), 2011 -	targets;
		(CCSS), 2010	(NCC), 2016	revised 2018	1998/under
					development
	Initiative	State governors	Finnish National	Swedish	Flemish
		& educational	Board of	government	government
		leaders, private	Education	commissioned	commissioned an
		foundations	commissioned	the National	entity
			expert group	Agency of
			(about every 10

This study is funded by the Swedish Research Council (2016-04616).


Constantinos Xenofontos
email: constantinos.xenofontos@stir.ac.uk

The use of problem in upper-primary and lower-secondary textbooks of the Republic of Cyprus

Keywords: Transitions, mathematical problem, textbooks, primary, secondary

 framework about the structural characteristics of problems was utilised. Findings conclude that coherence between the two textbook series is limited, as in the primary series the term 'problem' is extensively used in a particular way, while in the secondary series, a significant decline of the number of tasks labelled as problems is observed.

Transitioning from primary to secondary school

Many pupils preparing to transition from primary to secondary school appear to have predetermined ideas about the challenges and difficulties of mathematics at the next school level [START_REF] Attard | Students' experiences of mathematics during the transition from primary to secondary school[END_REF]. As a consequence, there is a general decline of pupils' engagement with mathematics as they move from primary to secondary education [START_REF] Martin | Exploring the ups and downs of mathematics engagement in the middle years of school[END_REF], a decline of their self-efficacy beliefs, motivation, and performance [START_REF] Bouffard | Students' transition from elementary to high school and changes of the relationship between motivation and academic performance[END_REF], and the reinforcement of stereotypes regarding gender and mathematics performance [START_REF] Denner | Latino children's math confidence: The role of mothers' gender stereotypes and involvement across the transition to middle school[END_REF]. While research evidence suggests that the observed differences could partly be attributed to teachers' self-efficacy beliefs [START_REF] Midgley | Change in teacher efficacy and student self-and task-related beliefs in mathematics during the transition to junior high school[END_REF], as well as to teachers' and parents' emphases on goal [START_REF] Friedel | Changes in efficacy beliefs in mathematics across the transition to middle school: Examining the effects of perceived teacher and parent goal emphases[END_REF] across the two school levels, little is known about the impact of the use of instructional materials by primary and secondary mathematics teachers [START_REF] Howard | Mathematics and manipulatives: Comparing primary and secondary teachers' views[END_REF], and more specifically, how similarly or differently textbooks are used at these two school levels [START_REF] Fan | Textbook research in mathematics education: development status and directions[END_REF].

Here, I focus on the national textbooks of a highly centralised educational system, that of the Republic of Cyprus. The mathematics curriculum of Cyprus (as with all school subjects) is prescribed centrally by the Ministry of Education and Culture -MoEC (Mullis, Martin, Goh, & Cotter, 2016). In order to bridge the gap between the previous primary and secondary mathematics curriculum, the curriculum in effect, introduced in 2010, is built around the same five general topics, from pre-primary education up to the last year of upper-secondary education (MoEC, 2010). These five topics are numbers, algebra, geometry, measurement, and statistics-probability. Furthermore, this unified mathematics curriculum claims to be built on four principles, with principle 2 stating that the curriculum places emphasis on problem solving (MoEC, 2010) Both in primary and secondary education, the vast majority of mathematics teachers' instruction methodology depends on the respective national textbooks, prepared by the MoEC [START_REF] Xenofontos | The cultural dimensions of prospective mathematics teachers' beliefs: Insights from Cyprus and England[END_REF][START_REF] Xenofontos | Opportunities of learning through the history of mathematics: the example of national textbooks in Cyprus and Greece[END_REF]. Although I do not, in this study, examine textbook use by teachers, I do explore how the concept of problem is presented in the upper-primary (grade 6) and lower-secondary (grade 7) textbooks, which were introduced after the initiation of the reform in 2010. Such an approach sees textbooks as a potentially implemented curriculum [START_REF] Schmidt | A splintered vision: An investigation of U.S. science and mathematics education[END_REF], or, as Mesa (2004, p. 255-256) puts it, "a hypothetical enterprise: What would students learn if their mathematics classes were to cover all the textbook sections in the order given? What would students learn if they had to solve all the exercises in the textbook?"

Why problem?

In the mathematics education literature, the term problem is one of the most widely used; yet, there does not seem to be an agreement as to what it means. In general, my views coincide with Schoenfeld's (1985, p. 74) statement that "being a 'problem' is not a property inherent in a mathematical task. It is a particular relationship between the individual and the task that makes it a problem for that person". For the purposes of this study, however, I do not want to use any particular definition of the term, as my goal is to examine how problem is conceptualised and formed in the national textbooks of Cyprus. Such a deliberate choice may serve as a proxy to how problems and problem solving are conceptualised at the level of policy-making, that is, the intended curriculum (Mullis et al., 2016;[START_REF] Schmidt | A splintered vision: An investigation of U.S. science and mathematics education[END_REF]. Also, I acknowledge that a number of comparative studies have concluded that problem and problem-solving are perceived differently across educational systems [START_REF] Cai | Cognitive analysis of U.S. and Chinese students' mathematical performance on tasks involving computation, simple problem solving, and complex problem solving[END_REF][START_REF] Xenofontos | Defining mathematical problems and problem solving: Prospective primary teachers' beliefs in Cyprus and England[END_REF].

In attempts to analyse the types of problems found in mathematics textbooks, various colleagues have used well-defined frameworks. For instance, in their work with two widely used textbook series in Singapore, [START_REF] Fan | Problem solving in Singaporean secondary school mathematics textbooks[END_REF] distinguish between routine and non-routine problems, with various other categories falling under the latter (namely problem-posing problem, puzzle problem, project, journal task). In subsequent work, [START_REF] Fan | Representation of problem-solving procedures: A comparative look at China, Singapore, and US mathematics textbooks[END_REF] identified various similarities and differences in the promotion of problem-solving strategies in Chinese, Singaporean, and US textbooks, again, using a very structured, predetermined framework related to how the researchers understood the concepts of problem and problem-solving. In the same spirit, [START_REF] Xin | Word problem solving tasks in textbooks and their relation to student performance[END_REF] examined the distribution of seven types of word problems (namely: multiplicative comparisoncompared, multiplicative comparison-referent, multiplicative comparison-scalar, rate times a quantity, fair share or measurement division, and proportion problem type) in Chinese and US middle school textbooks, noting a more balanced distribution in the former than the latter. In turn, [START_REF] Son | Teachers' selection and enactment of mathematical problems from textbooks[END_REF] investigated how teachers select problems from textbooks and present them in class; however, the researchers "use problems and tasks interchangeably" (p. 493), meaning that they did not utilise a specific definition of what a problem might be.

Methodology

This study is based on the following research question:

How similar/different are the characteristics of the tasks explicitly labelled as problems in the upper-primary and the lower-secondary national textbooks of Cyprus?

This question is of particular significance, mainly because in its rhetoric the MoEC declares that the new unified curriculum aspires to promote a smooth transition from one school level to another (MoEC, 2010).

Instead of utilising a predetermined definition of the term problem, I chose to focus only on those tasks explicitly labelled as problems in the two textbook series. By explicitly, I refer to a presentation of the term 'problem' in the instructions provided, i.e. "solve the following problems". Nonetheless, to enable comparisons to be made, I used an adapted framework based on [START_REF] Borasi | On the nature of problems Educational Studies in Mathematics[END_REF] ideas about the structural characteristics of problems. In her work, Borasi acknowledges the difficulties in deciding whether a specific task is a problem or not, the same way [START_REF] Schoenfeld | Mathematical problem solving[END_REF] recognises the role of the individual solver in the labeling of a task as a problem. Borasi, therefore, proposes four structural characteristics of tasks, which are independent of the solver. The first is the formulation, and refers to how instructions about what needs to be done are presented. These can take three forms: a question, a statement, or no presentation of instructions at all (open for the solver to decide). The second is the context in which the task is presented, and can take two forms: purely mathematical or applied [START_REF] Blum | Applied mathematical problem solving, modeling, applications, and links to other subjects: state, trends and issues in mathematics education[END_REF]. The third is the set of the acceptable solutions. There may be no acceptable solution to a problem, one and only solution, or more than one. Finally, the fourth characteristic has to do with the methods of approach. In respect to this, [START_REF] Polya | Mathematical discovery: On understanding, learning and teaching problem solving[END_REF] classified problems in four categories: one rule under your nose (when the problem can be solved by simply applying an algorithm just presented), application with some choice (when the suitable algorithm must be selected among others previously studied), choice of a combination (when in order to reach the solution some of the algorithms previously learnt must be suitably combined), and approaching research level (when the elaboration of a new algorithm is required or when the task cannot be solved algorithmically). The term 'approaching research level' is used by Polya himself to refer to tasks that are genuinely problematic, and perhaps is closer to [START_REF] Schoenfeld | Mathematical problem solving[END_REF] views about what a problem is.

As explained earlier, this paper focuses on the tasks explicitly labelled as problems in the upperprimary (grade 6) and lower-secondary (grade 7) national mathematics textbooks of Cyprus. Each task was described in terms of the four structural characteristics (formulation, context, set of acceptable solutions, methods of approach). For grade 6, the textbooks are organised in six parts, while for grade 7 they are organised in two parts.

Findings

In grade 6 textbooks, 228 tasks were identified as being explicitly labeled as problems, while in grade 7 textbooks, only two tasks were identified. Although the total number of tasks (labelled as problems or not) in the two series was not counted, a quick scan through the textbooks shows that in each grade (6 and 7) there are more than 1000 tasks.

Below, figure 1 presents an example from grade 6, and how it was coded with the use of [START_REF] Borasi | On the nature of problems Educational Studies in Mathematics[END_REF] four structural characteristics. Figure 2 demonstrates another example from the textbooks of grade 6, which, contrary to the algorithmic task of figure 1, was identified as approaching research level, as there is no standard algorithm that could be applied for its resolution. In figure 3, one of the two tasks identified in grade 7 textbooks is presented. In fact, both tasks were coded under "approaching research level", due to their high complexity and lack of any particular algorithm to be applied.

Table 1 illustrates the distribution of these tasks for each form of a structural characteristic. In both grades' textbooks, certain tasks were identified, in which the term problem appeared in the instructions. However, these were classified as problem-posing tasks, as they invited pupils to write a problem that would meet particular criteria (i.e. 'translate' a symbolic expression/representation into a real world situation). In the textbooks of grade 6, eleven such tasks were identified, while in the textbooks of grade 7, there were four tasks of this type. Below, figures 4 and 5 demonstrate an example from the primary and secondary textbooks, respectively. More than one 0 1

Methods of approach

One rule under your nose 110 0

Application with some choice 37 0

Choice of a combination 73 0

Approaching research level 8 2

Table 1: The identified tasks and their structural characteristics

Discussion

When approached from different perspectives, the findings of this study offer insight into at least two matters. Firstly, regarding transition from primary to secondary education, this paper suggests that pupils in grade 6 (upper primary) are significantly exposed to the term problem when interacting with their mathematics textbooks, compared to grade 7 (lower secondary). There appears to be a change of discourse, as we move from grade 6 to grade 7. This does not necessarily mean that children in grade 7 are exposed to less tasks that, according to [START_REF] Polya | Mathematical discovery: On understanding, learning and teaching problem solving[END_REF], approach research level. Nonetheless, an abrupt disappearance of the term problem as children move from primary to secondary education is apparent. Although the latest educational reform intends to bridge the gap between the mathematics curricula at various transition points (MoEC, 2010), this does not seem to be happening in an effective manner. In fact, questions are raised about the extent to which the intended "cohesive and coherent curriculum from pre-primary to upper secondary education" (MoEC, 2010, p. 15) is, in reality, cohesive and coherent. Secondly, when these findings are taken as a whole, and especially when the problem-posing tasks in the two series are added to the equation, the study echoes my previous work examining pre-service teachers' problem-solving related beliefs [START_REF] Xenofontos | The cultural dimensions of prospective mathematics teachers' beliefs: Insights from Cyprus and England[END_REF][START_REF] Xenofontos | Defining mathematical problems and problem solving: Prospective primary teachers' beliefs in Cyprus and England[END_REF]. In the Republic of Cyprus, there seems to be a very particular, perhaps culturally specific, understanding of what a mathematical problem is:

A 'problem', for the educational system of Cyprus, seems to be a real-world task, in which the instructions regarding what needs to be done are mainly presented in the form of question, or, sometimes, with a statement. There is always one (and only) acceptable solution to this task, which can either be solved by applying a straight-forward method/algorithm or a combination of methods/algorithms, which in any case, are already known.

Such a cultural perception of mathematical problems is, I think, extremely problematic, especially when the new curricula claim, in their rhetoric, to be placing emphasis on problem solving (MoEC, 2010). Nevertheless, I do acknowledge that this study, as all studies, carries a set of limitations that has the potential to introduce new research avenues. As explained, in this paper I only examine those tasks explicitly labelled as problems. There appear to be other tasks in the textbooks that are not labelled as problems, the methods of approach of which, in [START_REF] Polya | Mathematical discovery: On understanding, learning and teaching problem solving[END_REF] terminology, approach research level. Furthermore, even though teachers in Cyprus seem to rely heavily on textbooks [START_REF] Xenofontos | The cultural dimensions of prospective mathematics teachers' beliefs: Insights from Cyprus and England[END_REF][START_REF] Xenofontos | Opportunities of learning through the history of mathematics: the example of national textbooks in Cyprus and Greece[END_REF], what we don't know is how they use textbooks and other instructional materials in the classroom. Finally, future research could examine and compare primary and secondary in-service teachers' beliefs and practices in relation to problem-solving, so that we can have a complete picture of how to make the transition, as smoothly as possible, from one school level to another.

Figure 1 :Figure 2 :

 12 Figure 1: An example of a task (grade 6, part B, page 89)

Figure

  Figure 3: An example of a task (grade 7, part A, page 94)

Figure

  Figure 4: A problem-posing task (grade 6, part D, page 21)

  . Yet, no further clarifications are provided as to what a problem is, what problem solving means, and what kind of related skills are desirable.


Renaud Chorlay
email: renaud.chorlay@espe-paris.fr

Aline Bernardes
email: aline.bernardes@uniriotec.br

Tanja Hamann
email: hamann@imai.uni-hildesheim.de

Antonio Oler-Marcén

Introduction to the papers of TWG12: History in Mathematics Education

Introduction

History of mathematics in mathematics education, and history of mathematics education continue to receive much attention. However, empirical research and coherent theoretical/conceptual frameworks within this area have emerged relatively recently. The purpose of this CERME TWG is to provide a forum to approach mathematics education in connection with history and epistemology.

TWG12 welcomes both empirical and theoretical research papers, and poster proposals related to one or more of the following issues:

1. Design and/or assessment of teaching/learning materials using the history of mathematics, preferably with conclusions based on empirical data; all levels can be considered, from early-age mathematics to tertiary education and teacher training. 2. Surveys on the existing uses of history or epistemology in curricula, textbooks, and/or classrooms in primary, secondary or tertiary levels, and in teacher training; 3. History of mathematics education; 4. Relationships between, on the one hand frameworks for and empirical studies on history in mathematics education and, on the other hand, theories, frameworks and studies in other parts of mathematics education research.

Even though the inception of this TWG is fairly recentit started in CERME6 ( 2009)it has deeper institutional roots within the maths education research community. Indeed, the HPM studygroup (History and Pedagogy of Mathematics) was created at the 1972 ICME conference; it has been organizing satellite conferences to the ICME meetings since 1984, and has several active regional branches (HPM-Americas, European Summer Universities). In CERME11, 13 papers and 3 posters were presented in TWG12, with participants coming from a large range of European countries and beyond (Brazil, Tunisia). The topics covered by TWG12 were also reflected in Kathy Clark's plenary address on History and pedagogy of mathematics in mathematics education: History of the field, the potential of current examples, and directions for the future.

Before going into any details, it should be stressed that this TWG has four general but distinctive features which give these meetings their specific flavour. Firstly, its topic lies at the intersection of different fields of researchmaths education research and history of mathematicswhich requires versatility and methodological vigilance [START_REF] Fried | Can Mathematics Education and History of Mathematics Coexist?[END_REF][START_REF] Chorlay | History of Science, Epistemology and Mathematics Education Research[END_REF]. Secondly, the strength of the historical and the HPM community varies greatly among countries, and these meetings play a crucial role for researchers working in relative isolation, and with difficult access to resources in the field. Thirdly, the scope of TWG covers both history in mathematics education and history of mathematics education, which are two significantly different research topics (TSG 24 and 25 in ICME13); connecting the two lines of investigations is a constant challenge. Fourthly, since the topic of TWG12 is neither specific to one level of the educational system (from primary education to teacher-training) nor to any single mathematical topic (be it fraction concepts, algebra, proof, etc.), the work in TWG12 intersects that of most other TWGs. It should be noted that, for this edition, there was little intersection with what was covered in TWG8 (Affects and the teaching and learning of mathematics), TWG10 (Diversity and maths education), in spite of the fact that it is not uncommon for outsiders of the HPM research communityamong which most policy-makers and curriculum-designersto ascribe such goals to the historical perspective in teaching.

These four features made this meeting not only useful but also challenging and exciting. As the final discussion made clear, the general feeling among the participants was that one of the main outcomes of this meeting is that we actually learned a lot from the one another, both from their papers and from the lively discussions.

Some significant features of the 2019 conference

A large share of the papers bore on teacher training, either for prospective or in-service maths teachers. Two case studies, one presented by Barreras and Oller-Marcén, and one by Bernardes and Bruna-Correa, showed how confronting teachers with mathematical documents from the past can be a tool both for research into teachers' content knowledge, and for the professional development of teachers. From another perspective, van den Bogaart and Schorcht focused on teachers' beliefs, either when studying some history of maths as part as their per-service training, or when assessing the didactical potential of classroom activities with some historical content.

By contrast, few papers were dedicated to what has long been a central line of research in the HPM community, namely the use of historical documents in the classroom. However, these few papers point to fresh research directions. For instance, the poster of Olsen and Thomsen aimed to connect the studies on history of maths in the classroom with those on the use of ICT. Chorlay's paper bore on primary school mathematics and argumentation, thus illustrating the new approach in the French IREM community, with its greater emphasis on the actual teaching effects of experimental sessions and its rather new involvement with pre-secondary mathematics. This new approach led to the publication of a textbook (Moyon & Tournès, 2018), which was awarded the 2019 prize for best science teaching resource by the Paris Academy of science.

On a par with teacher-training, a significant share of the papers bore on the history of mathematical education; a fact which bears witness to the vitality of this research field at the international level. We are pleased that TWG12 of CERME provides opportunities for researchers working in different fieldsmathematics education research, history of mathematics, and history of mathematics educationto interact fruitfully. The papers on topics from history of maths education presented in this CERME conference gave food for thought for mathematics education researchers working on textbooks (Hatami and Pejlare), resources for teaching and the professional development of teachers

  (Krüger), testing and assessing(Smestad & Fossum). Several papers (Zwaneveld & De Bock, Weiss & Känders) bore on fairly recent historical developments, thus shedding a new (and occasionally critical) light on current trends. History of education was complemented with inputs from sociology in Hamann's work, in order to sketch a general model for education reform.