Students' difficulties with proof, scholars' calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in integral calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students' exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish textbooks than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as directions for future research, are discussed.
Finland has recently reformed the national school curriculum, basing it on so-called twenty-first century skills. The curriculum prescribes that schools must organise at least one multidisciplinary learning module per year. In this case study, we investigate the implementation of such a module in secondary school (grade 9) from the perspective of the school headmaster and science and mathematics teachers. The data comprise interviews with the headmaster, teachers, and protocols from collegiate meetings. Despite some gains experienced by teachers concerning cooperation with colleagues, the results reveal challenges connected to diffusely defined learning goals, as well as to the vagueness of goals concerning the role of different subjects in the module. The study shows the importance of offering research-based support and time for shared sensemaking for stakeholders to avoid the problems identified when implementing multidisciplinary teaching.
In search of the reasons for Swedish students’ low achievement in algebra in inter-national and national evaluations, we investigate how the development of algebraic thinking is addressed in the Swedish national mathematics curriculum and two widely used mathematics textbook series for grades 1–6 in Sweden. The analytical tool used is based on the classification of ”big ideas” which research has shown as important for developing pupils’ algebraic understanding in early school grades. The results show that functional thinking, expressions, and equations are well represented topics both in the curriculum and the textbooks; however generalized arithmetic is a topic that is poorly developed in both the curriculum and the textbooks.
The aim of the study is to investigate and compare approaches to algebra in the Estonian, Finnish, and Swedish national core curricula (Grades 1–9). Despite the similarities in the school systems of these neighboring countries, the analysis reveals three quite different curricular approaches. The Estonian approach shows influences of the Russian Davydov School. The Finnish approach to some extent resembles the traditional one whereby algebra is addressed first at the lower secondary level and then in a formal manner. However, there are also characteristics typical of the functional view, which dominates the Swedish curriculum. Here, as opposed to the documents from the other two countries, a transition to more formal sophisticated methods at the secondary level is not visible at all. We discuss the results in relation to earlier research and students’ different learning outcomes in light of international evaluations.
The issue of students' misconceptions in mathematics and how to prevent and deal with them in teaching has been a major concern of mathematics educators since at least four decades. At the same time our knowledge about the processes of developing understanding and skills in proof and argumentation from early school years has increased. We argue, that there are connections between these two areas of studies important to make explicit for teachers. In this paper, we first elaborate the relation between the research on students' misconceptions and the ideas of developmental proof. Then we present the relevant results of an empirical study about how mathematics teachers in the field interpret this relation. Our conclusion is that there are important connections between these two research fields that are not always visible for teachers.
The most commonly used Finnish mathematics teacher guides (Grades 1-6) are investigated so as to determine what kind of resource they constitute for teachers in planning and enacting mathematics teaching and what kind of mathematics classroom they promote. The structure and the main contents of the guides were found to be quite homogeneous. The nature of communication was mostly descriptive, but the separate activities suggested for each lesson were quite explicitly described. Suggested activities, such as mental calculation tasks and homework assignment, were typically motivated by non-specific rationales, and many activities seemed to be taken for granted in the Finnish mathematics classroom culture. The results add both to knowledge about how to analyse teacher guides and to knowledge about Finnish educational features.
In the current paper, we present an analysis of a case study in which we have followed Swedish primary teachers who voluntarily began using translated Finnish curriculum materials, i.e. a textbook and teacher guide, in order to reform their mathematics teaching. The multifaceted data, consisting of questionnaires, interviews, protocols from collegial meetings and classroom observations, were gathered during the period 2010-2014. The analysis of the interplay within this cross-cultural setting reveals the special characteristics and the challenges existing in practice. Both the experienced and inexperienced teachers offloaded a great deal of their agency to the materials in order to become familiar with the ideas they mediated. Yet, the lack of a clear rationale behind the organization of the materials, as well as the suggested activities connected to taken-for-granted features of the Finnish teaching tradition, made fruitful interaction problematic. The changes teachers made in their classroom practice were tightly connected to the support offered in the materials, without which the teachers abandoned their new classroom patterns. Based on the results of this study, we suggest a number of general aspects that we regard as important to consider when implementing curriculum materials developed within another cultural-educational context.
This article explores effective mathematics teaching as constructed in Finnish and Swedish teacher educators’ discourses. Based on interview data from teacher educators as well as data from feedback discussions between teacher educators and prospective teachers in Sweden and Finland, the analysis shows that several aspects of the recent international reform movements are visible in the discourses in both countries. However, the Swedish teacher educators tend to conceptualize effective teaching as interactions with individual children, building on students’ ideas and emanating mathematics from everyday situations, while the Finnish teacher educators stress the importance of a clear presentation of mathematics, routines and homework as well as specific goals for every lesson. The results of this cross-cultural study cannot be generalized to the two countries but rather show interesting conceptualizations of effective teaching, adding to international theory building.
This article explores effective mathematics teaching as constructed in Finnish and Swedish teacher educators’ discourses. Based on interview data from teacher educators as well as data from feedback discussions between teacher educators and prospective teachers in Sweden and Finland, the analysis shows that several aspects of the recent international reform movements are visible in the discourses in both countries. However, the Swedish teacher educators tend to conceptualize effective teaching as interactions with individual children, building on students’ ideas and emanating mathematics from everyday situations, while the Finnish teacher educators stress the importance of a clear presentation of mathematics, routines and homework as well as specific goals for every lesson. The results of this cross-cultural study cannot be generalized to the two countries but rather show interesting conceptualizations of effective teaching, adding to international theory-building.
This study explores prospective primary school teachers' misconceptions concerning basic concepts in chemistry. The data was gathered during the years 2012-2016 among first-year primary school teachers in Finland (N = 389). The results show that every year, as many as 40%-80% of teachers share the same kinds of misconceptions as children ages 5-12 (depending on the investigated concepts). The results further show that the prospective primary teachers do not develop a solid foundation in chemistry during their school years, indicating that this area should be addressed more effectively in Finnish elementary schools. To raise the quality of school instruction, it is crucial that teacher education curricula consider this when methods of instruction in science courses are developed for prospective primary teachers.
Given that curriculum materials serve as cultural artefacts, this study addresses the need for more research on curriculum materials in different contexts. Most studies concerning curriculum materials have been conducted in US and, therefore little is known about the nature of materials in other cultural-educational contexts. The aim of this paper is to identify the underlying cultural norms of potentially constructed classrooms, by analysing recurrent activities in the most commonly used Finnish teacher guides at primary-school level. We identified three norms embedded in them: (1) creating opportunities for learning through a variety of activities and communication; (2) keeping the class gathered around a specific mathematical topic; and (3) concurrent active involvement of teachers and students. The results add to knowledge about both teacher guides and the Finnish educational context. Moreover, it adds to the growing body of methodologies, as our analytical approach is novel in the context of textual analysis.
Understanding of equality and solving equations are some of the big ideas in algebra. They have been in focus in early algebra research for some decades and in many countries it is now usual to work with equalities and solving equations using informal methods from early school years. However, it is not clear how the transition to formal methods of equation solving could be conducted in order to maintain students' interest and enhance their algebraic understanding. We shed light on the issue by reporting on what happens when three teachers introduce equation solving with formal methods in Grade 6 (age of 12) in Finland. We especially consider how the introduction could support students' development of an algebraic understanding of equality and their engagement in more formal mathematics.
Theories for conceptualizing educational policies aimed at improving classroom instruction at scale are under development in the educational sciences. In using such theories, it is essential to note the specific educational context. In this article, we conceptualize the role of contextual factors when operationalizing Cobb and Jackson (2012) [Journal of the Learning Sciences] in the Swedish context. Drawing on data and results from a large-scale project carried out during 2012-2017 and studies of Swedish educational contexts, we conceptualize contextual factors for large-scale projects. Besides rather obvious explicit contextual factors such as ongoing policies and practices, we elaborate on how the underlying, more implicit contextual factors of (1) the positioning of teachers within the educational system, (2) the positioning of teachers within the classroom, and (3) traditions of visible and invisible pedagogy affected the establishment of policy in the Swedish context. Insight into these factors deepens earlier frameworks of context, and helps not only to operationalize the policy within the context but also to make explicit hidden features of a cultural context that are important to influence if the aim of the educational policy is to reorganize school practices.