Kirsti Hemmi

Analysing proof-related competences in Estonian, Finnish and Swedish mathematics curricula—towards a framework of developmental proof

Many countries are revising their mathematics curriculum in order to elevate the role of proof and argumentation at all school levels and for all student groups. Yet, we have very little research on how proof-related competences are aimed to be developed in the mathematics curricula of different countries in Grades 1 to 12. This article contributes to filling this gap by analysing and comparing three countries’ curricula from the perspective of developmental proof. For this purpose, we created an analytical frame of proof-related competences that could be connected to the development of students’ understanding and skills concerning argumentation and mathematical proof. The analysis reveals three quite different trajectories with specific characteristics, shortcomings and strengths.

The Culture of the Mathematics Classroom During the First School Years in Finland and Sweden

This chapter elaborates findings from a longitudinal ongoing cross-cultural study comparing the teacher education and classroom practices in Finland and Sweden. The focus is on the cultural scripts of mathematics instruction during the first school years (ages 6–8). Firstly, we present a description of the contexts of each country concerning primary teacher education and the transition from preschool to school. We then characterise the dominating conceptualisations of the mathematics classroom practices for the early years in both countries, building on several analyses of different data sources. We focus especially on the intricate balance between flexibly building mathematics on pupils’ ideas of familiar everyday phenomena within a thematic teaching style on the one hand, and on the other, the organisation of learning environments strictly based on a predetermined hypothetical learning trajectory. Finally, we discuss our findings in light of the international literature on early mathematics education and transition from preschool to school.

Investigating Finnish Teacher Guides as a Resource for Mathematics Teaching

ABSTRACT 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.

The State of Proof in Finnish and Swedish Mathematics Textbooks—Capturing Differences in Approaches to Upper-Secondary Integral Calculus

ABSTRACT 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.

Challenging traditional classroom practices: Swedish teachers’ interplay with Finnish curriculum materials

ABSTRACT 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.

Support or Restriction: Swedish Primary School Teachers’ Views on Mathematics Curriculum Reform

This paper reports the results of a quantitative study on primary teachers’ (n=253) views on the introduction of steering documents and national examinations for Grade 3. While the majority of the teachers experience the reform as empowering, some teachers feel the new curriculum and the national examination restrict their teacher professionalism. We found differences in how teachers viewed the reform depending on whether they had graduated before or after the reform in 1994. The differing views can be connected to teachers’ beliefs about teacher professionalism and the relation between teaching, learning and maturing. We discuss our findings also in the light of curriculum development during the last four decades.