Coherence in a Range of Mathematics Curriculum Reforms

Abstract

This chapter considers the coherence – or lack of it – that is evident in mathematics curriculum reforms in a range of countries. Coherence is analysed both between and within components of curriculum, as is coherence of mathematics curriculum with the discipline of mathematics. Considering the level of coherence between the curriculum and the ‘curriculum system’ that provides the context for enactment of the intended curriculum identifies some important factors that can support or hinder the successful uptake of curriculum reforms.

This chapter addresses the series of key questions posed in the ICMI Study 24 discussion document for theme B:

What is the extent of coherence within and among different aspects of reformed curricula such as values, goals, content, pedagogy, assessment, and resources? How are curriculum ideas organised and sequenced for internal coherence in a curriculum reform? What are the effects of a lack of coherence? For example, regarding relations between high-stakes examinations and curriculum reforms? (2018, p. 580)

The main focus in this chapter is the documentation – written and otherwise – of the curriculum as it sits within the overall ‘curriculum system’ as outlined in the Introduction (Chap. 9) to this theme. Where possible and appropriate, there is reference to studies and other observations of the curriculum ‘in action’ as a means of assessing the extent of coherence, how it is achieved (or not) in practice, and its impact and effects.

In line with the definitions adopted for coherence and relevance in the Introduction to theme B (see Chap. 9), there will be an exclusive emphasis on coherence in relation to mathematics curriculum reforms. The response to the key questions posed in ICMI Study 24 – and other issues that emerged – draws on the analysis of a selection of mathematics curriculum reforms. These range in scale from reforms of mandated national curricula to reform initiatives at smaller scale related, for example, to particular aspects of the curriculum such as some particular mathematical content, mathematical process or pedagogical choice(s). The coherence between mathematics curriculum reforms and mathematics itself as the ‘parent discipline’ is also investigated, as are the interactions between the curriculum and the whole educational context for its implementation. A range of theoretical approaches and practical frameworks have been used in the analyses reported. These are discussed in detail in Chap. 13.

Our analysis here uses four possible ‘lenses’ to consider different forms of coherence of mathematics curriculum reforms:

• coherence between components of the curriculum;

• coherence within components of the curriculum;

• coherence with mathematics as the ‘parent discipline’;

• coherence of the curriculum with the wider curriculum system.

The first two of these are related directly to the six components of the framework for describing any mathematics curriculum proposed by Niss (2016). The ‘Niss framework’ has been outlined in the Introduction (Chap. 9, p. 121) to theme B and is drawn on in various ways in other chapters. The third lens for coherence reflects the need for a mathematics curriculum to facilitate learning of mathematics that is coherent with the logic and structures of the discipline. Whilst the fourth lens is external to the curriculum itself, the level of coherence of curriculum reforms with the existing curriculum system can have a significant impact on the enacted curriculum.

Coherence Between Components of the Niss Framework

This section contains analysis of several national mathematics curriculum reforms. We identify and discuss the coherence between components of those curricula. In each case, the Niss ‘goals’, taken as the statements of purposes and aspirations for the curriculum, are clearly identified. Coherence of the other curricular components with the goals is needed to support the realisation of those purposes and aspirations – strong alignment of all the components is evident in the success of the example from Portugal, a reform that can justifiably be described as ambitious in the context. The example from Brazil demonstrates some clear coherence between the content and goals, but that this is not uniform, while in Vietnam, although the goals and assessment seem to be well aligned, the current materials, student activity and support for teaching remain locked in the past and out of step with the reform goals.

Portugal

Since 2004, senior secondary education in Portugal has been structured around seven ‘tracks’, each targeted at a specific student cohort or trajectory. The tracks all contain a combination of compulsory and optional courses. At the commencement of these reforms it was decided to develop and offer a mathematics course (the MACS course) as an option in the social sciences track, in recognition that students on this track benefit from mathematical experiences and learning suited to their particular needs. The goal of the MACS course is, “significant mathematical experiences that allow [students] to appreciate adequately the importance of the mathematical approaches in their future activities” (Carvalho e Silva, 2003, as quoted in Carvalho e Silva, 2018, p. 310). Rather than focusing on specific concepts, MACS aims to “give students a new perspective on the real world with mathematics, and to change the students view of the importance that mathematical tools will have in their future life” (p. 310). Students engage with real situations, in order to “develop the skills to formulate and solve mathematically problems and develop the skill to communicate mathematical ideas (students should be able to write and read texts with mathematical content describing concrete situations)” (Carvalho e Silva et al., 2001, as quoted in Carvalho e Silva, 2018, p. 310). The approach is therefore interdisciplinary in nature, in order to be relevant for the particular cohort; Chap. 11 contains further discussion of this aspect of the MACS course.

Using the course “For all practical purposes” (COMAP, 2000) as the inspiration, the three topics for grade 10 are decision methods (election methods, apportionment, fair division); mathematical models (financial models, population models); statistics (regression, with graph models, probability models and inference) the areas covered in grade 11 (Carvalho e Silva, 2018, p. 310).

Implementation of MACS from 2004 faced a number of challenges:

• there was no tradition of such approaches to mathematics and, indeed, a previous initiative to implement a quantitative methods course that had similar intentions to those of MACS had failed during the 1990s;

• a lack of teacher knowledge in content areas such as graph theory and the mathematics of elections;

• few relevant teaching materials and no suitable textbooks;

• developing an examination as part of the secondary school diploma.

The last of these was a “very controversial matter [for] the MACS course” (Carvalho e Silva, 2018, p. 311). In Portugal, students need to sit national examinations in four of the subjects in their ‘track’ in order to achieve their high school certificate. Since MACS is an option in the Social Sciences track, students must have the option of taking the final examination, which makes up 30% of their final grade for the course. However, the MACS syllabus encourages teachers to use a range of means for assessment designed to support students’ learning, but that are not generally seen as consistent with examinations. “Group work and individual work is recommended, [with assessment] assuming different forms: essays, personal notes, reports, presentations, debates” (p. 312).

The existence of a national examination was seen by the teachers’ association (Associação de Professores de Matemática) as “not compatible with the assessment suggested in the official syllabus (APM, 2007, as reported in Carvalho e Silva, 2018, p. 311). The association complains that teachers lose their freedom and try to ‘prepare’ students for the examination and this somehow “does not allow the innovation aspects of this program to pass fully into practice”. (p. 311) In other words, there was real concern about a lack of coherence between a number of the Niss components of the curriculum and the assessment component, given that at least part of the students’ assessment was to be carried out through a national examination.

After some exploration of models and means for achieving coherence between the intent of the MACS course and the external assessment, the current MACS examination “consists of several rather mostly open but simple questions, where some careful interpretation or model construction/analysis is required” (p. 312). Carvalho e Silva (2018, pp. 312–313) provided example items from the 2017 examination. The first asks students to use a graphic calculator to compare exponential and logarithmic models; the second sees them apply a voting method to a particular situation.

Evidence that the approach adopted has been successful in improving coherence of this aspect of the assessment with the intent and form of the enacted MACS curriculum – at least in the eyes of the student cohort – can be found in the current numbers of students opting to take the national examination in MACS. Carvalho e Silva reports:

As this course is accepted by very few Higher Education degrees, students that take this course can easily opt not to take the national examination. The number of students that take this examination is in fact very high. The total number of students taking exams is around 50, 000, and some 30, 000, take the main Mathematics A examination (p. 313)

In addition to achieving coherence between the goals and content and the assessment, the implementation of the MACS course has also achieved alignment of materials, forms of teaching and student activity through systematic and sustained effort over a number of years. These efforts are discussed later in this chapter as an example of achieving coherence between the curriculum and the curriculum system. The example of the MACS illustrates that an ambitious curriculum can achieve coherence between intentions and enactment.

Brazil

Established under a national legal framework in 2017 (for implementation from 2019), the National Curricular Common Base (BNCC) for elementary school¹ in Brazil is built around five thematic units that guide the formulation of skills to be developed in elementary school. Competence in the BNCC is defined as the “mobilization of knowledge (concepts and procedures), skills (practical, cognitive and social-emotional), attitudes and values to solve complex demands of everyday life, the full exercise of citizenship and the world of work” (Brasil, 2017, p. 8). As a result, “mathematical processes such as problem solving, research, project development and modeling can be cited as main forms of mathematical activity throughout this stage” (p. 8).

Dias and Cerqueira (2018) report on a qualitative and documentary analysis of the final version of Brazil’s National Curricular Common Base for the final years of elementary school finding on the one hand that, “The analysis of the BNCC for the Final Annals of Basic Education revealed the presence of social, symbolic and cultural components linked to the objects of knowledge and their respective [mathematical] abilities” (p. 227). These findings are evidence of broad coherence between the goals and content of the BNCC.

As reported by Dias and Cerqueira (2018, pp. 223–224), he developed a framework for analysing curriculum that combines Bishop’s (1988) identification that the mathematical formation of young people consists of three components (symbolic, social and cultural) with the framework for evaluating mathematics curricula provided by Silva (2009). The latter extends the four criteria proposed by Doll (1993) (wealth, reflection, reality and responsibility) also to include four more criteria (recursion, relationship, rigour and resignification). Dias found that in Brazil’s BNCC for elementary school mathematics the presence of the three components of Bishop and the eight criteria from Silva were not consistent. His analysis showed that while these characteristics are clearly identifiable for the curriculum at years seven and nine, they are not apparent for the content for years 6 and 8. In other words, the coherence was found not to be consistent (pp. 226–227). This type of analysis has the potential to provide curriculum developers with insights into the extent and consistency of a curriculum’s coherence with goals that characterise learning mathematics as a socio-cultural pursuit.

Vietnam

In Vietnam, the university entrance examination is high-stakes and the teaching tradition is one of close adherence to material presented in approved textbooks. There have been several curriculum changes in Vietnam over the last two decades, including in 2019–2020, bringing an intention to move towards a greater valuing of conceptual mastery and engagement with mathematical processes. As is the case elsewhere, these intentions for the goals and content of the curriculum bring with them challenges for teachers in terms of valid enactment of reforms.

Over time, the university entrance examination has become the de facto high school graduation examination. Trung and Phat (2018) provide specific examples of assessment items that, despite being constrained to be multiple choice in format, clearly require conceptual understanding rather than recall and reproduction of procedures. Figure 10.1 provides two examples of miultiple-choice assessment items in which identifying the correct response requires substantial conceptual understanding and mathematical reasoning. Hence, the conceptually-oriented questions common in the examination (assessment) since 2017 are coherent with the goals and content of the curriculum.

Fig. 10.1

From Trung and Phat (2018, p. 327)

On the other hand, the tradition of textbooks in Vietnam is highly procedural – it is the textbooks (i.e. the materials in the Niss framework) that are yet to be developed to be coherent with the goals. Chap. 12 contains further discussion of this matter which notes that the “cultural norm, and dominant approach, for mathematics teachers in Vietnam is a focus on procedure and memorization” (p. 181) and that this is strongly represented in the current textbooks.

Coherence Within Components of the Niss Framework

The reforms discussed in the previous section are drawn from planning and change initiated at the national level; this section deals with particular aspects of reforms that, whilst they may be situated in a national effort, are considered in terms of a more narrowly defined focus for reform. A number of the examples of mathematics curriculum reform also demonstrate coherence – or lack of it – among the Niss components.

Recent curriculum reforms in both Costa Rica and Vietnam have had an emphasis on problem solving and mathematical modelling as core themes. In both cases, the intention is for student activities to incorporate mathematical modelling; that is, for students to develop, use and refine mathematical models as a means for solving problems and gaining insights that relate to the ‘real world’. Japan provides a third example that strives for coherence in the teaching of proof across the curriculum.

Costa Rica

The reform of mathematics education in Costa Rica has involved a significant reorganisation and renewal of many aspects of the curriculum as an “explicit bid to develop Costa Rican society in its full breadth and complexity” through a focus on competence.

The functional focus of the mathematics curriculum advocates knowledge that focuses on development of one’s own cognitive strategies, stressing the use of different forms of representation, argumentation abilities, and modelling techniques to pose and solve problems in context. In sum, its purpose is to develop schoolchildren’s mathematical competence by improving their thinking and giving them certain autonomy. (Lupiáñez & Ruiz-Hidalgo, 2018, p. 263).

The reform also highlights mathematical processes that are applied across content areas:

• reasoning and argumentation;

• posing and solving problems;

• connecting;

• communicating;

• representing.

Of special importance for the reform are also “disciplinary core ideas” (MEP, 2012) that indicate priorities and that permeate all components of the curriculum including content and topics, advice, suggestions and instructions for teachers.

‘Active contextualisation’, one of the core ideas for the Costa Rican mathematics curriculum, recognises the importance of posing problems in authentic situations in order to mathematically model situations. The real contexts can have varying origins including the popular media, school and community. Active contextualisation in the Costa Rican mathematics curriculum sees these realistic contexts being explored through questions that are either interesting, authentic, or didactically relevant (after Maaß, 2006, as cited in Lupiáñez & Ruiz-Hidalgo, 2018, p. 265).

A range of initiatives were established to support the implementation of the emphasis on active contextualisation, particularly in relation to building the knowledge and capacities of teachers. Lupiáñez and Ruiz-Hidalgo (2018) report that the teachers involved in one professional development program “showed considerable advance in conceptual clarification of various central notions of the reform” (p. 266). The results were mixed, however:

[In relation to the core idea of] “active contextualization,” although the teachers recognize the role and importance of its application, they have considerable difficulty proposing contextualized tasks […] express[ing] regret and worry that they cannot find phenomena and fields of problems that enable them to propose relevant tasks and authentic questions. (p. 266)

In other words, despite focused professional development on the topic, at the time of reporting many teachers’ approaches, and the student activities they motivate, lack coherence with active contextualisation as a major goal and focus of the curriculum reforms in Costa Rica.

Vietnam

From 2015, education in Vietnam has embarked on a major transformation designed to create and develop students’ subject-specific core competences, as well as:

Common competences for all subjects and educational activities that contribute to the formation and development: self-control and self-learning competence, communication and cooperation competence, problem solving and creativity competence. […] (Nguyen, 2018, p. 285)

The 2018 draft curriculum highlights the process of mathematical modelling as a focus. Students are required to develop and use mathematical models to solve problems and describe situations, to relate solutions to the real context being modelled, and to modify models as necessary.

Interdisciplinary application of mathematics is also an emphasis in the curriculum. Comparing the content on trigonometric functions in the mathematics and physics curricula for years 11 and 12, Nguyen (2011):

find(s) a reasonable arrangement between the contents of the two disciplines. Specifically, circular motions are associated with the trigonometric circle is mentioned in grade 10. Next, the trigonometric function is studied in Mathematics in grade 11 and its applications in Physics like waves, sound, harmonic oscillation, [...] present in grade 12. (p. 289)

Hence the goal of developing mathematical competence in mathematical models is supported by coherence within and between the mathematics and physics curricula.

However, the approach to modelling in mathematics textbooks in Vietnam is not coherent with the emphases intended in the curriculum. Nguyen (2011, as cited in Nguyen, 2018) reports that there are only:

traces of modeling in the application of mathematical knowledge to some of the problems that arise from reality. In high school mathematics textbooks, these exercises are very rare and are often placed in the readings section or at the beginning of some chapters. (p. 288)

In the case of trigonometric functions, not only do mathematics textbooks contain few real-life examples, but even when these are present, the students are given the model and merely asked to work with it to solve given problems. Whilst this can be seen as experience with applications of mathematics, the students are not engaging with mathematical modelling – certainly not in the spirit of the intended curriculum.

Nguyen (2011, as cited in Nguyen, 2018, p. 289) provided an example (Fig. 10.2).

Fig. 10.2

A ‘modelling’ exercise in the algebra and analysis textbook, grade 11

This exercise requires students to undertake a range of substitutions and computations in a procedural manner. The astronomical context is rich and a different approach would provide students with opportunities to engage with, build and appreciate harmonic models of natural phenomena in ways that are much more coherent with the goal of the mathematics curriculum to promote and provide experience with the process of mathematical modelling and associated processes (reasoning, problem solving, communication, etc.).

In addition to this lack of coherence of the materials for teaching with the goals and content of the reformed curriculum, Vietnam faces issues in ensuring that assessment is coherent with the intentions of the curriculum in terms of evaluating mathematical modelling. As outlined earlier in this chapter, Trung and Phat (2018) note a move to more conceptually-based assessments in the examination system in Vietnam that has significant coherence with the goals of the curriculum. The particular case of mathematical modelling presents additional challenges for assessment. Key among these is to develop assessment strategies for the examinations that test students’ capacity to develop and use mathematical models effectively. It is also necessary to provide teachers with means to measure student performance in mathematical modelling if they are to assist their students to develop.

Japan

The 2017 national curriculum in Japan has at its core a general process for ‘finding out’ and solving problems in mathematics in which phenomena from both the ‘real world’ and the ‘mathematical world’ are dealt with through a three-part, common approach:

• problems are represented mathematically;

• problems are ‘focused on’ (i.e. problem solving);

• results are reflected on and interpreted in terms of the context.

Proof and proving play an important role in this process of ‘doing’ mathematics in the Japanese curriculum. However, just like their peers in many other countries, it is reported that Japanese students in grades 7–9 (and beyond) experience difficulties in the areas of proof and proving in mathematics. Miyazaki & Fujita (2015, as cited in Miyazaki et al., 2018, p. 275) observed that the course of study that is the statement of the Japanese curriculum only requires proof and proving, but does not propose the way to realise it in the curriculum. Moreover, the course of study requires that students learn various properties of plane and three dimensional figures mainly based on congruency and similarity, and also the meaning of proofs, and how to prove formally.

Although it encourages the gradual introduction of formal proofs until the end of grade 8, the curriculum does not offer a clear plan on how to gradually implement the learning processes of planning and constructing a proof. In other words, students are exposed to proofs, but not to the processes involved in proving, even though the curriculum emphasises active problem solving, in which proving plays a critical role. These authors’ response to this lack of coherence has been to propose and develop frameworks for ‘exploratory proving’ for grades 7–9.

Miyazaki and Fujita “define explorative proving as having the following three components: producing propositions, producing proofs (planning and construction), and looking back (examining, improving, and advancing)” (2015, p. 1399). They subsequently add, “Careful mapping of the transition between these components as part of a teaching and learning sequence allows for systematic development of students’ knowledge, understanding and capacity with proofs and proving” (p. 1399). This approach is an example of a way of explicitly teaching students one of the key processes of mathematics – being able to prove a mathematical result, and how to assess the logic of a proof they encounter, “that reflects the nature of mathematics, but also cultivating generic competencies of authentic explorative thinking” (p. 1402).

Miyazaki et al. (2018) provide examples of the framework and transitions of exploratory proving in Geometry, the domain which has traditionally included some treatment of proof, as well as in algebra, functions and data handling – domains in which attention to proof has largely been absent in the Japanese curriculum and, arguably, many other countries. Their analysis shows that the transitions between the components of exploratory proving are different in the different domains. The approach, once translated into actual learning materials by “combining local transitions of our frameworks with units in the Course of Study, the developed curriculum can provide teachers with a realizable plan on how to gradually implement the learning processes, and evaluate students’ ability” (p. 275). The approach will enable explicit attention to proving to occur in domain-specific ways. As a result, it is anticipated that students will develop a richer and more robust capacity in proof and proving in mathematics; there will be greater coherence between the goals and the mathematics being taught and learnt through the forms of teaching and student activities that materials motivate, and much more:

proving activities are flexible, dynamic and productive in nature, and various aspects of proving activities are interrelated and resonant with each other. We can see that proving activities ‘breathe life’ into mathematics teaching and learning and are intellectually stimulating in numerous ways, for example: producing inductively/deductively/analogically propositions, planning and constructing proofs for these produced propositions, and reflecting on and looking back at producing propositions, including planning and constructing proofs to overcome local and global counter examples and difficulties and then refining propositions and proofs. Mathematics as an activity is continuously developed by these processes which work dynamically together as ‘intellectual gears’, as if small paddle wheels (various aspects of proving) give power to propel a big paddle steamer (mathematics). (Miyazaki & Fujita, 2015, p. 1397)

Hence, their work can be seen as a means for creating greater coherence between goals and student activities through detailed attention to a key element of the goals (in this case, ‘proving’). It is an example that illustrates the challenges in aligning these aspects of mathematics curricula and the significant effort required to achieve coherence.

Coherence with Mathematics as the ‘Parent Discipline’

The Niss framework allows analysis of the coherence between and of the six components of a mathematics curriculum it identifies. Another aspect of coherence that is important in mathematics curricula is internal to the mathematics itself. This section considers recent and ongoing reforms to high school geometry curriculum on Israel in some detail as a means for exemplifying the complexity in achieving this form of coherence, followed by shorter comments on some other contemporary reform proposals.

Schmidt et al. (2005, as quoted in McCallum, 2018, p. 4) identify the importance of ordered, logical progression of the mathematics concepts and content:

We define content standards […] to be coherent if they are articulated over time as a sequence of topics and performances consistent with the logical and, if appropriate, hierarchical nature of the disciplinary content from which the subject-matter derives. (2005, p. 528)

That is not to say that there is only one possible sequence or hierarchy of topics and associated student learning. There is no ordained reason, for example, to introduce fractions before decimals, or negative numbers before elementary algebra – it is possible to create a logical and internally coherent sequence whatever order is chosen for these topics.

As an example of the ‘hierarchical nature’ of the discipline, the equation of a straight line is learned at school both as a part of analytic geometry as an analytic representation of straight line as a geometric object, and as a graph of a linear function. As a graph of a linear function, it is learned earlier (typically around eighth grade), and generally appears spirally in higher grades, in relevant analytic contexts. The slope as tangent of an angle can only be defined when the trigonometric functions are defined, and at the first stage only for acute angles. Hence this extension of the concept of slope must be delayed until the trigonometric ratios for right angled triangles have been developed.

However, internal mathematical coherence depends not only coherence of ‘topics’ – what would be traditionally seen as the content – but also on coherence in relation to the ‘performances’ of doing mathematics with and through that content that are aspects of the student activity. These performances or processes include substantial mathematical capabilities such as reasoning, proving, communicating and formulating and solving problems.

Israel

One feature of reforms of the Israeli mathematics curriculum that commenced implementation in 2014–2015 was a reshaping of the levels of courses to match the abilities and needs of different student cohorts. At the high school level (years 10–12), the intermediate curriculum (called the ‘four point’ curriculum) – which is intended for the middle 50% or so of the students seeking to matriculate – provides students with a basis for study and career trajectories in the life sciences, economics and the social sciences more generally.

The geometry component of the new intermediate (or four point) mathematics curriculum demonstrates the issues and complexities inherent in achieving internal mathematical coherence. Barabash (2018) notes that:

The guidelines […] include […] integration of analytic geometry, trigonometry, and synthetic geometry; linking mathematical rigour to development of intuition and visualisation-based valid reasoning; the Ministry’s policy (particularly the intended students’ characteristics), technological innovations, possibilities created by [dynamic geometry software], and experimental mathematical ideas that support systematic inductive reasoning. (p. 183)

Based on these guidelines the curriculum is elaborated according to three principles about integration of aspects of geometry, the form of problems that are to be posed to students (number of steps; use of numerical data only) and the use of digital platforms “to enhance inductive conjecturing followed by deductive testing (proof or refutation) of hypotheses thus formulated” (p. 183). The goals of the geometry curriculum emphasise different aspects and forms of reasoning, visualisation and representation, applications and critical evaluation of results. There is a heavy emphasis in the documentation of the curriculum on providing advice on teaching, including teaching sequences and sample problems, such that the list of topics is a relatively small component of the document itself.

The development and implementation of the new four-point geometry curriculum, and especially the extensive advice being given, provided opportunities in relation to ‘internal coherence’ within the geometry curriculum itself. This coherence is between its goals and principles; between the characteristics and possible academic or professional trajectories of the students for whom the curriculum is intended, and the content, the level, the complexity of tasks, etc. (student activity). In addition, there is the matter of coherence between this specific component of the whole high school curriculum and its other parts, i.e. analysis, statistics, algebra – the geometry curriculum’s ‘external coherence’.

The high school geometry curriculum in Israel connects synthetic geometry, analytic geometry and trigonometry in the course of teaching and learning. As noted above, this is explicitly stated as one of the leading features of the geometry part of the curriculum. These connections are pursued consistently through the three-year high school teaching plan. As examples of what can be termed ‘inner geometric interdisciplinarity’ Barabash identifies that:

In addition to the list of topics, clarifications are added to enhance the spirit of the document, such as: “equation of a straight line by slope and a point on it, as an analytic implementation of the axiom of parallels”; “equation of a straight line by two points on it, as an analytic implementation of the axiom claiming the existence and uniqueness of a straight line passing through two given points”; (p. 184)

Thus, the connections between the different ways of thinking geometrically are made explicit and clear.

Continuing this theme of the straight line illustrates some ‘inner-mathematical interdisciplinary connections’ that are central to the coherence of the geometry curriculum with the mathematics curriculum. The straight line, primarily a synthetic-geometric object, appears to have various meanings; in particular, it is of paramount importance as the graph of a linear function. Its central property of constant slope is geometric in nature. The constant slope property is expressed and worked-with using the techniques of analytic geometry and geometric theorems (triangle similarity, angle formed by parallel lines and a secant, etc.). The relationship between the slopes of two perpendicular straight lines also has a geometric basis. Therefore, the straight line in a coordinate system embodies the mathematical inner interdisciplinarity in the sense that it cannot be uniquely attributed to any one mathematical field. This is similarly also the case for points, segments, and many other geometric objects.

Barabash analyses a learning sequence that draws on and ties together analytic geometry, trigonometry and synthetic geometry. The first exercise uses a triangle on the co-ordinate plane to focus on:

critical testing a wrong supposition that inexperienced students might find correct. … [Subsequent exercises] use another triangle located correspondingly in the coordinate plane for a similarly guided exercise leading to the conclusion that tan (a – b) ≠ tan a – tan b. The recursive appearance of such questions guides a student toward the habit of doubting and testing “self-evident” beliefs. (pp. 185–186)

Hence, the high school geometry curriculum in Israel is an example of student activities that are coherent with the goals and content and which provide for achieving the goals and students learning the content. The coherence achieved reflects careful attention to the interplay between synthetic geometry, analytic geometry and trigonometry as the mathematical bases of the curriculum and the development of detailed student materials and guidance for teachers.

Japan

Similarly, the approach to proof and proving from Japan as outlined earlier in this chapter is also an example of building components of a curriculum that provide coherence with the mathematical processes of proof and proving as they apply in algebra, geometry, function and data handling (i.e. across the discipline). This is achieved by applying a common theoretical developmental framework that can lead to students having an appreciation of what it means to prove in mathematics in general, as well as the similarities and differences in proof and proving between the domains of the discipline.

An Alternative

The examples above – and indeed all the mathematics curriculum reforms outlined in this chapter – use an orthodox view of the discipline of mathematics, as envisaged by Schmidt and colleagues (above). As a group the examples highlight both the importance and the feasibility of developing curricula that are in some respects coherent with the parent discipline. This is seen to be important if students are to be engaged and inducted into what it means to work in the discipline.

On the other hand, Tarp (2018) proposes a radically different mathematics curriculum that takes as its base a different conception of the discipline that is not obviously coherent with mainstream views of the discipline. His “question guided re-enchantment curriculum in counting [that] could be named ‘Mastering Many by counting, recounting and double-counting’” (p. 320) is a creative alternative. A case is made for the internal coherence of the curriculum (pp. 321–324), and this raises the question as to whether basing a mathematics curriculum on an alternative view of the discipline can be sufficiently rigorous, and what might be the challenges and advantages of doing so.

Tarp’s proposal brings into focus two issues for mathematics curricula. The first is whether at any particular moment in history there is a tacit assumption that there is a single conception of the discipline, or even one “mainstream” conception of it. Such conceptions evolve with time and, therefore, depends on the educational system considered. Moreover, the ways curricula are formulated and enacted contribute tend to modify or sustain this conception and reject alternative conceptions.

The second issue is that a proposal for a curriculum that is radically different (such as Tarp’s) tend to be critiqued in terms of rigour, benefits and the challenges, if the proposal were to be adopted. These factors should also be addressed as carefully in reviews of mathematics curricula that reflect mainstream views of the discipline.

Coherence of the Curriculum with the Curriculum System

The curriculum system is an articulation of many of the factors that are in place in a particular educational setting, including wider resources and constraints, such as policy settings of governments; responses to globalisation; leadership structures; all aspects of teachers’ capacity; societal values and so on. There can be many factors that motivate a government or curriculum authority to embark on a reform of the mathematics curriculum. Whether reforms are a response to the society expressing a desire to better equip young people for citizenship; for more young people to undertake STEM careers; findings from national or international assessment programs that aspects of student performance need to be improved; or from other research, reforms both reflect and ultimately have impacts on the curriculum system.

Hence, by definition, mathematics curriculum reforms are designed and have the intention of, to some extent, disrupting the current state of the curriculum system, for example in areas such as the organisation of schooling, teacher beliefs, availability and appropriate use of learning technologies, textbooks, examinations and other assessment structures. This interdependence of curriculum reforms on the one hand, and the curriculum system is therefore an important fourth dimension of coherence of mathematics curriculum reforms. In this section we use examples to explore coherence between the curriculum and the ‘curriculum system’. An initial question is what is the impetus for reform? In particular how research on student outcomes, as a measure of the success or otherwise of the existing curriculum and curriculum system, can inform the need for reform of a mathematics curriculum, and, potentially, the directions and emphases of those reforms.

This section begins with a substantial example of research informing reforms in China. We then turn our attention to the issues of coherence between a curriculum and the prevailing curriculum system during implementation. By far the most important issues for the effectiveness of uptake of the reforms outlined in this chapter are those that relate to teacher capacity; some other examples suggest means for building coherence between the curriculum and the curriculum system. This discussion and analysis begins with a reasonably detailed consideration of recent reforms in Mexico. This is followed by observations about the coherence of the curriculum with the curriculum system apparent in examples presented earlier in the chapter.

China

The Mathematics Basic Activity is a key element of the new mathematics curriculum in China designed to change “the Chinese conceptualization of mathematical basics” (Guo & Silver, 2018, p. 245). A new curriculum for grades 1–9 was introduced from 2011; for grades 10–12 the new curriculum commenced in 2017. The focus of this aspect of the new curriculum (i.e. the Mathematics Basic Activity) is experiential learning, rather than learning through instruction.

A key driver for this reform was recognition that whilst Chinese students excel at numerical and algebraic computation, spatial reasoning, and logical reasoning, they do less well with non-routine problems that involve “increased attention to mathematical processes associated with problem solving, invention and creativity” (p. 247) as is seen in the mathematics curricula of other countries such as USA and Japan.

Of the “two main forms of mathematics basic activity experience: practical experience in mathematics and thinking experience in mathematics” (p. 246), Guo & Silver report an investigation of the second of these about thinking. The “new aspect of the Chinese mathematics curriculum is that it is the student’s way of mathematics thinking accumulated from experiencing and understanding the processes of mathematics inductive reasoning [initially] and mathematics deductive reasoning [later, in order to verify and prove results arrived at by inductive reasoning]” (p. 246).

They analyse findings from a study by Guo and Shi (2013) of the performance of students from seven middle schools in different parts of China on a set of six problems (with sub-problems) that were “drawn from a variety of sources […] intended to assess students’ proficiency in generating a general rule or conclusion through a process that starts from a specific and simple problem” (p. 248). Only about 1% of the students produced responses in the “highest category that involved evidence of proficiency with mathematical reasoning” (p. 250), with the rest of the students either able only to imitate procedures with little or no mathematical reasoning (80%), or showing some capacity with mathematical reasoning.

Given that processes such as mathematical reasoning are highly valued as goals for twenty-first-century mathematics education, these results emphasise the need for the reform embodied in the Mathematics Basic Activity as it is now incorporated in the new Chinese curriculum for grades 1–9. However, much more needs to be done to create coherence between the intended and enacted curricula. The 2013 study shows that the, “instructional practices and curriculum emphases in these Chinese classrooms and schools have not been sufficient to support the majority of students to obtain the kinds of experience envisioned by the curricular reform” (p. 251).

Greater coherence between the curriculum, with its emphasis on the Mathematics Basic Activity and the curriculum system will be an important goal that will be facilitated by developing and making available materials (textbooks, etc.) that are coherent with the goals of the reformed curriculum.

Mexico

The curricular reform of 2011 introduced a competences approach to the Mexican educational system for the first time.

The Integral Reform for Basic Education is a public policy that promotes the comprehensive training of all preschool, primary and secondary students with the aim of favouring the development of life competencies and the achievement of a certain profile at the end of the basic education, [all of this is] based on expected learning and the establishment of Curricular, Teaching Performance and Management Standards. (SEP, 2011, p. 17).

This policy required substantive changes in the approach, goals, and content of the 2011 reformed curriculum:

The Articulation of Basic Education is the beginning of a transformation that will generate a school focused on educational achievement by addressing specific learning needs for each of its students, so that they acquire the competencies that allow their personal development. (p. 18)

In particular, in contrast with the 1993 curriculum that had separate curricula for elementary and middle school levels and goals that were mostly functional, the 2011 curriculum expected elementary and middle school students to develop the following mathematical competencies:

• solve problems autonomously;

• communicate mathematical information;

• validate procedures and results;

• use techniques efficiently.

These were carried out through content arranged in “three thematic axes” (numerical sense and algebraic thinking; form, space and measure; information handling), rather than the six directly content-based organisers in the previous curriculum.

Importantly, in contrast with the 1993 curriculum which did not deal with assessment, the reforms of 2009–2011 address issues of assessment through guidance about the nature and focus of assessment. Importantly, there is an intention that, “students should be evaluated on their know-how and on the application of the mathematical contents” (SEP, 2011, as quoted in Hoyos et al. 2018, p. 255), as a means for generating coherence between the assessment and the goals and content of the curriculum.

Further questions about coherence between the goals and content of the recent reforms in Mexico and the materials and student activities, as presented in the curriculum documentation and official textbook, are considered in Chap. 12 (materials and technologies), with the overall finding that these components are little changed from the previous curriculum, thus leaving teachers ill-supported in the face of the shifts in the goals and emphases of the curriculum.

The magnitude of the changes has resulted in many challenges to the curriculum system, in particular the teachers’ capacities and beliefs. The importance of teacher support – through opportunities for in-service professional development – to help them develop the capacity needed to enact curriculum reforms is well demonstrated in the findings about the effectiveness of successive reforms of the mathematics curriculum in Mexico. Whereas the initial reform of 1993 included coherent classroom and teacher materials in the form of an ‘educative’ teachers’ guide, along with associated in-service professional development, the follow-up reform has been characterised by less coherent materials for teachers and students and no teacher development. In addition, it is not clear that there has been support for coherence of Assessment, beyond the requirements in the curriculum documentation.

As an indication of the potential for these inconsistencies and lack of support for teaching and learning in mathematics classrooms that is coherent with the intentions of the 2011 curriculum, PISA assessments of learner cohorts show that the percentage of Mexican students that were below level 2 (i.e. attaining the level 1 or zero) in PISA 2009 was 51%, with this figure rising to 57% in PISA 2015 for students who have been substantially taught under the 2011 curriculum, indicating that there may be issues for the attained curriculum, with a greater proportion of Mexican students in the poor levels of performance (Hoyos et al. 2018).

Portugal

Issues of teacher preparedness, curriculum materials and support for effective teaching in the implementation of the MACS course in Portugal that was designed to meet the needs of students who are taking a social sciences study and career trajectory – as outlined above (see p. xx) – were substantial. They have largely been addressed through “a carefully designed plan [that has] allowed today’s situation where thousands of students opt for this course” (Carvalho e Silva, 2018, p. 314), that was put in place over a decade or more, beginning in 2001.

The approach has had several elements. To support the use of effective teaching strategies and associated student activities, Written teaching materials have been produced by the authors of the MACS program and others. The Ministry of Education edited and made available translations of relevant COMAP publications, and new textbooks have been produced. From 2001, a cadre of specialist MACS teachers were selected and prepared for a role in leading and supporting others through in-service professional development programs that helped them develop the knowledge and skills in teaching the MACS course. Several universities have included content relevant to MACS such as Election Theory, Apportionment and Graph Theory. in their courses for pre-service teachers of mathematics.

An important feature in these and other programs of support has been ongoing engagement with the teaching profession. The Teacher Association APM, in its 2007 report, as quoted in Carvalho e Silva, 2018, p. 315, said that, “APM participated actively with proposals, teacher preparation, discussions, preparation of materials, etc. [… and] the process […] has been exemplary”. The authors of the MACS course had a permanent “contact with teachers in the field, asked for contributions from all the teachers, mathematicians and other specialists, integrated in a very satisfactory manner the several suggestions sent to them, and the authors also organized meetings to discuss the work being done in a very open way” (p. 315).

Whilst the extended time provided to generate coherence between the MACS curriculum and the Portuguese curriculum system was necessary, the success of the enterprise is the result of the multi-faceted, systematic and inclusive nature of the initiatives taken. Both elements – sufficient time and targeted initiatives – are necessary; neither is sufficient by itself.

England

Nor are such situations necessarily either static or convergent. Golding (2018), in an extensive suite of longitudinal curriculum enactment studies that focused on learners aged from 5 to 18, shows how recent aspirational curriculum reforms in England, targeting a renewed emphasis on deep conceptual fluency, mathematical problem-solving and reasoning, were initially well-supported in many schools by espoused teacher beliefs, teacher-educative materials (Davis & Krajcik, 2005) and early assessments that were all coherent with curriculum intentions. Over time and given teacher commitment to professional development, good progress was often (though by no means uniformly or universally) made towards classroom practice well aligned with the goals of the curriculum. This underlines that the development and embedding of teacher change is a complex process, so that reasonable stability of curriculum is also valuable – and that there might well be advantages to evolution, rather than revolution, of intended curriculum.

However, England operates with a marketized assessment regime, and teachers commonly talked about choosing assessment providers which are perceived to offer the most accessible routes to good grades in high-stakes assessments, rather than those whose assessments are most coherent with curriculum intentions. Over time, then, and in the context competition between providers, assessments were seen to progressively dilute central intentions. Further, as teachers became more familiar and confident with emerging assessments, they frequently developed alternative classroom practices, often involving mathematically incoherent subsets of the curriculum, which they taught to key groups of students. While curriculum systemic coherence is challenging to establish, then, it would appear even more challenging to sustain, and in this case proved fragile in several respects.

Other Examples

The design of curricular reforms in Costa Rica has the ambitious goal of:

reorganization of the weight of the main dimensions and elements of the curriculum to give them greater cohesion and depth. As a whole, the changes represent an explicit bid to develop Costa Rican society in its full breadth and complexity. (Lupiáñez & Ruiz-Hidalgo, 2018, pp. 261–262)

However, enactment of that curriculum is compromised by a lack of coherence between its intentions and methods and a key element of the curriculum system – the preparation, training and support of teachers. The authors report on a study that found that many teachers seem to have limited capacity to faithfully enact that curriculum, particularly in relation to “active contextualization of […] one of the disciplinary core ideas” (p. 261) in that curriculum.

Barabash (2018) outlines the challenges to the coherent enactment of reforms of high school mathematics in Israel that are created by the curriculum system in that country. These include factors such as ministerial policy, budget and logistic considerations that have an impact on time allocations, and the fact that many teachers’ beliefs and traditional teaching approaches have become entrenched during several decades of exam-oriented teaching. Work to address these issues and create greater alignment between the curriculum system and the curriculum began in 2014 and is continuing.

Some Possible Responses to Achieve This Coherence

Olsher and Yerulshamy (2018) describe a ‘bottom-up’ process² being used in Israel that brings together mixed groups of teachers, administrators, consultants, researchers, textbook authors and others to curate existing curriculum materials by creating and sharing digital tags that relate materials to aspects of the curriculum. By working together to ‘tag’ curriculum materials, they build common understandings of what it means for the whole curriculum system to enact the mathematics curriculum in ways that are faithful to the goals and processes of that curriculum. These common understandings have the potential to build coherence in the enactment of the curriculum and the curriculum system through the roles the various educators involved play in that system.

In Vietnam, addressing coherence for the assessment component in relation to mathematical modelling is only part of what is needed for greater coherence within the high school curriculum. Nguyen (2017) has also found that teachers lack the knowledge and skills to effectively implement a modelling approach in their teaching of mathematics and recommends some means for helping teachers build their capacity including:

• training teachers about interdisciplinary teaching and how to incorporate mathematical modelling;

• preparing materials and setting up teaching situations with interdisciplinary themes associated with modelling as a source for teachers to refer and use;

• organising lesson study for mathematics and other subject teachers as ongoing professional development.

It is likely that variations on these types of approaches will be applicable in many settings where, in order for the implementation of reformed curricula to be faithful to the goals, greater alignment between the curriculum and the curriculum system is required.

The examples above provide means for implementation that is faithful to the goals. This is not to say that the goals themselves may not be the source of problems. The goals may have been developed without due regards for the capacity and orientation of the education system that is expected to implement the curriculum. Critical analysis of the goals is also necessary.

Conclusion and Key Messages

In a study such as this, it is only possible to present a snapshot of the field that draws on and analyses some examples of mathematics curriculum reform, in this case in terms of their coherence. We have used four different lenses on coherence and identified ways in which reforms are seen to have coherence when viewed from these different perspectives.

Using the Niss framework to analyse the coherence of the range of curriculum reforms presented in this chapter has highlighted a number of key messages.

• Overall coherence of a reformed mathematics curriculum requires careful and consistent attention to the coherence between all the components of the curriculum.

• The fact that mathematics curricula encompass not just mathematical content, but also mathematical processes and thinking, often with social and cultural overlays, means that it is necessary to give attention to coherence in relation to all these aspects of the curriculum.

• Research and development is needed to support and inform coherence in mathematics curricula. The design of such programs needs to investigate coherence at specific interfaces between Niss’ components (e.g. between goals and content or between student activity and assessment, etc.) and provide practical advice that helps promote coherence between components.

• Given that high stakes assessment (examinations) are important in many countries, lack of coherence between assessment and goals of the curriculum can have a significant impact on the enacted curriculum. Many teachers resist change to their existing practice, and textbook writers feel no real urgency to align with the goals of the curriculum when they perceive that examinations have not changed. Burkhardt (1987) coined the term WYTIWYG (What You Test Is What You Get) to make the point that, in mathematics, assessment tends to drive both what is taught and how it is taught. This seems to be as true now as it was more than 30 years ago.

Whilst coherence between and within the components of the curriculum, and between the curriculum and the discipline of mathematics, are all necessary, such coherences are not sufficient for effective enactment of reformed mathematics curricula. It is critical that the curriculum system is also aligned with and supportive of the reforms intended. In example after example the lack of effort to support teachers to develop knowledge and skills that give them the capacity to work with their students (and colleagues) in the spirit of the curriculum has been cited as resulting in inconsistent and inadequate implementation. The best examples of alignment between reformed mathematics curricula and the local curriculum system seem to have three characteristics in common.

• Alignment is achieved over an extended period of time during which the curriculum itself remains a constant.

• There is a comprehensive and targeted program designed to achieve the best possible alignment between the elements of the curriculum system (teacher capacity, values, societal expectations, structure of schooling, place of student voice, etc.) and the curriculum itself.

• The program for alignment is characterised by respect for, and engagement of the wide range of ‘players’ in the curriculum system (teachers, students, school administrators, officers of education systems, teacher educators, researchers and textbook authors, etc.).

Coherence is likely to be a universal aspiration for mathematics curriculum reforms – truly achieving it presents many challenges and requires commitment to persistent, collaborative work on many fronts.