Finnish Mathematics Research Articles

핀란드 1~2학년 초등 수학 교과서에 제시된 프로그래밍 교육을 위한 수학적 요소 분석

Since 2016, Finland has been implementing programming education in elementary school mathematics. This study analyzed Finnish first and second grade elementary mathematics textbooks and tutorials that reflected programming education. Previous studies on programming were analyzed to extract mathematical elements for programming instruction, and based on the extracted sequence, rules, selection, variables, possibilities, logic elements, functions, and algorithm, the contents of elementary mathematics textbooks in Finland were analyzed. As a result of the analysis, in Finnish mathematics textbooks, activities suitable for programming learning were evenly presented by utilizing all content elements as a whole, and a separate lesson for programming was also organized. In addition, the basic content of programming suitable for the student level was presented mathematically, and the level was changed and presented as the grade level went up. Looking at each element, activities for variables and logic elements accounted for a large part, and students were provided with opportunities to learn various mathematical elements related to programming during the first and second grades of elementary school. Based on these results, this study is expected to provide significant resources for applying programming to mathematics education in Korea.

Pedagogical rationales of flipped learning in the accounts of Finnish mathematics teachers

ABSTRACT The focus of this study is on the pedagogy of flipped learning (FL) in mathematics teaching. There has been extensive research into FL, but less research on teachers’ pedagogical rationales when adopting this pedagogy. The present study addresses this research gap by examining interviews with mathematics teachers in Finland. These teachers identified themselves as FL advocates. A thorough analysis of the teacher interview data inspired by a grounded theory approach revealed three main pedagogical rationales for FL in the teachers’ accounts, namely, Individualising Learning, Fostering Self-regulated Learning, and Fostering Engagement. Individualising Learning emphasises attempts to differentiate and humanise learning mathematics in heterogeneous student groups. Fostering Self-regulated Learning highlights the teachers’ emphasis on students’ responsibility in goal-oriented activity that is supported by self-paced learning. Fostering Engagement is related to the teachers’ attempts to create a personally motivating learning environment for students. The results of this study contribute to the research into FL in two ways. First, the teachers of FL view self-regulation as an objective of education, and not just as a means of education. Second, the teachers underscore general learning skills over disciplinary learning in mathematics.

Prospective mathematics teachers’ self-referential metaphors as indicators of the emerging professional identity

Ideals play a key role in a student teachers’ identity work. They form targets to strive for and a mirror for reflection. In this paper, we examine Finnish mathematics student teachers’ metaphors for the teacher’s role (N= 188). We classified the metaphors according to a model that identified teachers as subject matter experts, didactical experts, and pedagogical experts, with the addition of another two categories, self-referential and contextual. For the exploration of emerging professional identities, we studied the self-referential metaphors, which formed the most common category in the data. We observed that every third metaphor described either student teachers’ personalities or their incompleteness as teachers, or new beginnings or eras. Although these aspects were expected, they also inform us as teacher educators of the values and ideals that student teachers have in terms of teaching and being a teacher. The metaphors that mathematics student teachers produced illustrated their identity processes and their emerging identity as a mathematics teacher.

Compute mindlessly. Not! Map consciously

This paper utilizes concept mapping as a tool for conscious and deliberate knowledge building in mathematics and its extension to algorithms. Currently, alleged defects in mathematics education are obvious: instead of conceptual elaboration, everyday praxis relies on routine computations that are likely to lead into alienated concepts with weak connections to prior knowledge. A concept map visualizes the existing conceptual structure, and whenever new information is brought in, it will be placed in the map by clearly explicating its linkage to the previous concepts. In the Finnish mathematics education, such new knowledge is programming content that is integrated into elementary school mathematics in 2014 Finnish National Curriculum. This content is crystallized as the requirements of computational and algorithmic thinking, the utilization of respective data structures, and adequate amount of hands-on practice to internalize good coding conventions. This study examines secondary (N = 19) and higher education students (N = 10) and their conceptual knowledge of mathematics concentrating on the domain of algorithms in particular. The concept maps drawn by the students are evaluated using the SOLO taxonomy. To conclude, a consensus map of algorithms is represented and linked to the elementary mathematics syllabus.

フィンランドの算数科・数学科カリキュラムに関する考察―汎用的コンピテンスに焦点を当てて―

フィンランドの算数科・数学科カリキュラムに関する考察―汎用的コンピテンスに焦点を当てて―

Canadian and Finnish upper-secondary school mathematics teachers’ perceptions of autonomy

Autonomy is a site of political struggle; it is essential for teachers, but challenged in relation to global policies and standardised testing. In this qualitative study, the accounts of Canadian and Finnish mathematics upper-secondary school teachers are examined in relation to their perceptions of autonomy. Teachers report on similar autonomous classroom operations, and similar hindrances to autonomy in school-wide operations. However, contrasting outlooks on autonomy were found among the Finnish and Canadian teachers. This contrast informs us about the role culture plays in the different ways autonomy is internalised and perceived by teachers. The findings deepen the understanding of autonomy for teachers, as well as its implications, in different contexts.

Mathematics Student Teachers’ Epistemological Beliefs about the Nature of Mathematics and the Goals of Mathematics Teaching and Learning in the Beginning of Their Studies

This article examines Finnish mathematics student teachers’ epistemological beliefs concerning the nature of mathematics and the goals of mathematics teaching and learning solely in the beginning of their studies at university. A total of 18 students participated in a study consisting of a short questionnaire and interviews. The data was analyzed by categorizing the views presented in terms of four mathematics-related orientations. According to the results, student teachers often regarded mathematics mainly as a static system, but their beliefs about the goals of mathematics teaching and learning consisted of features that derive from all of the four orientations. Future teachers’ beliefs can be affected by teacher education and hence the results are important for teacher educators to be recognized.

Student teachers’ types of probing questions in inquiry-based mathematics teaching with and without GeoGebra

ABSTRACTPrevious studies have produced several typologies of teacher questions in mathematics. Probing questions that ask students to explain are often included in the types of questions. However, only rare studies have created subtypes for probing questions or investigated how questioning differs depending on whether technology is used or not. The aims of this study are to elaborate on different ways of asking students to give explanations in inquiry-based mathematics teaching and to investigate whether questioning in GeoGebra lessons differs from questioning in other lessons. Data was collected by video recording 29 Finnish mathematics student teachers’ lessons in secondary and upper secondary schools. The lesson videos were coded for the student teachers’ probing questions. After this, categories for the types of probing questions were created, which is elaborated in this paper. It was found that the student teachers who used GeoGebra emphasized conceptual probing questions during the explore phase of a lesson slightly more than the other student teachers.

Investigating Finnish Teacher Guides as a Resource for Mathematics Teaching

ABSTRACTThe 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

ABSTRACTStudents’ 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.

Emotion work and affective stance in the mathematics classroom: the case of IRE sequences in Finnish classroom interaction

Although according to the Finnish curriculum the learning environment in mathematics lessons should promote supportive interaction, Finnish pupils’ attitudes toward and self-beliefs regarding mathematics deteriorate during basic education. This article investigates emotion work in teacher–student interaction in Finnish mathematics classrooms; the aim is to identify and suggest situational sources for the negative affective stances that seem to prevail among Finnish pupils. For the specific topic of analysis, we chose IRE sequences, especially teachers’ responses to pupils’ incorrect answers. Our data consist of ten sixth-grade mathematics lessons. Using conversation analysis as our method, we analyze examples that highlight the participants’ emotion work displayed in verbal and non-verbal practices. We suggest that it is pedagogically important that the teacher practices communicating the message that incorrect answers are appropriate student contributions and that in analyzing interaction in mathematics education it is important to pay attention to the emotional aspects of the interaction.

Two finnish girls and mathematics: Similar achievement level, same core curriculum, different competences

Mathematical thinking and problem solving are essential parts of learning mathematics described in the Finnish National Core Curriculum for Basic Education. Evaluations on both have been done at national and international level. However, in a request for deeper understanding of pupils’ mathematical thinking we need to move beyond paper tests. This paper is a first look into the mathematical thinking of two Finnish girls, Emma and Nora, in their final year of Finnish comprehensive school. After solving a real-life situated problem in a classroom, the girls talk about mathematics and problem solving in an interview. The focus of the analysis is on the learning objectives, core content and final-assessment criteria related to thinking skills and methods in the Finnish curriculum. Also some results on metacognition and affect will be reported. The results suggest that while both pupils have similar achievement level in mathematics, their competences are different: Emma is more competent in problem solving and Nora is more self-confident and self-guided in learning mathematics and can more easily recognize mathematics outside school.

Only the raisins or all the pie? The implementation of the Varga-Neményi method in Finnish schools

At the beginning of a course held at Helsinki University in 2000, Julianna Szendrei asked us, Finnish teachers the question: Who do we want to get to the finish line with – the few best pupils, „the average” or every single pupil, even if there will be some who will have to wear roller skates to get there. According to Eszter C. Neményi, who introduced all her courses with this, the teachers are driven by two kinds of love in their work: the love of children and the love of mathematics. Now, these thoughts are important and familiar to us, Finnish teachers as well. It is exactly the similar values, the similar view on teaching that aroused our interest: How do you teach here in Hungary? What novelties could we learn from you? This was the situation in 2000. Now we are here to tell you about Finnish mathematics teaching and about what we learnt from you, apart from the fact that our school culture is somewhat different from yours, how we are proceeding in the direction you set for us, and how we have formed the seven principles of Varga-Neményi to „our image”.

PISA, TIMSS and Finnish mathematics teaching: an enigma in search of an explanation

Finnish students’ success on all three content domains of each of the four cycles of the OECD’s Programme for International Student Assessment (PISA) has created much international interest. It has also prompted Finnish academics to offer systemic explanations typically linked to the structural qualities of Finnish schooling and teacher education. Less well-known has been the modest mathematics performance of Finnish grade 8 students on the two Trends in International Mathematics and Science Study (TIMSS) in which Finland has participated, which, when compared with its PISA successes, has created something of an enigma. In this paper, we attempt to shed light on this enigma through analyses of Finnish mathematics classroom practice that draw on two extant data sets—interviews with Finnish teacher educators and video-recordings of sequences of lessons taught on standard topics. Due to the international interest in Finnish PISA success, the analyses focus primarily on the resonance between classroom practice and the mathematical literacy component of the PISA assessment framework. The analyses indicate that Finnish mathematics didactics are more likely to explain the modest TIMSS achievements than PISA successes and allude to several factors thought to be unique to the Finns, which, unrelated to mathematics teaching practices, may be contributory to the repeated Finnish PISA successes. Some implications for policy-borrowing are discussed.

Finnish Mathematics Teachers Talk about Restructuring Practice

The case study presented in this article addresses how five Finnish mathematics teachers, colleagues in a Swedish speaking teacher training school, talked about on-going restructuring of their teaching and learning practices in terms of experiences and expectations five months into an action research process that they initiated in the early 1990s. In this action research the teachers were the agents of change (Leder, 1989); I joined the process as an outsider researcher.

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.

Finnish Mathematics Teaching from a Reform Perspective: A Video-Based Case-Study Analysis

This article offers a qualitative analysis of videotaped mathematics lessons taught by four teachers in a provincial university city in Finland. My study is framed not only by Finnish success on Programme for International Student Assessment (PISA) but also by the objectives of current mathematics education reform, which are consistent with PISA’s goals of measuring mathematical literacy. The analyses indicated that conceptual understanding and procedural fluency were addressed by all four teachers. However, adaptive reasoning, strategic competence, and the development of a productive disposition appear rarely. I observed few occasions where students were invited to solve authentic problems.

Comparison of the Classroom Practices of Finnish and Icelandic Mathematics Teachers

Mathematics teachers in Finland and Iceland are on different tracks. Based on a recent video study, the classroom practices of Finnish mathematics teachers seem to be rather traditional. This is in contrast to the Icelandic teachers many of whom use progressive-minded, learner-based instructional strategies. The classroom practices in Finland include substantial whole-class interaction, while many students in Iceland are getting used to learning independently, without significant collaboration with others. This paper discusses three pedagogical dimensions on which mathematics teaching in Finland and Iceland differ: individualization, learner control, and content-related discourse.

Levels of students' understanding on infinity

Here we report some results of a two-year study for grades 5-6 and 7-8 (during the academic years 2001-03). The study included a quantitative survey for approximately 150 Finnish mathematics classes out of which 10 classes were selected to a longitudinal part of the study. Additionally, 40 students from these classes participated also a qualitative study. This paper will focus on students' understanding of infinity and the development of that understanding. The results show that most of the students did not have a proper view of infinity but that the share of able students grew, as the students got older.

Instructional practices and teachers' beliefs in finnish mathematics education

Instructional practices and teachers' beliefs in finnish mathematics education