Mathematical thinking and understanding in learning of mathematics

The concept “mathematical thinking” can be found in several studies of mathematics education, in national curricula or in media during the decades all over the world. We searched words “mathematical thinking” from the database of international scientific articles, and we found 456 707 mentions at first time. These are the main reasons why we have chosen “mathematical thinking” as the central concept of the Special Issue. The other interesting question from our point of view is how a student can express his/her mathematical thinking? By answering this question, we have made simple model for the teacher education purposes, and we call it “languaging” (of mathematical thinking). The articles o…

Opettajien matematiikkauskomukset ja matematiikan oppikirjan merkitys alkuopetuksessa

Matematiikan oppimisen iloa etsimässä? Tulkintoja opiskelijoiden kirjoittamien oppimimispäiväkirjojen pohjalta

Artikkelin tarkoituksena on tarkastella aikuisena luokanopettajiksi opiskelevien oppimisen ilon kokemuksia matematiikan peruskurssin aikana kirjoitettujen oppimispäiväkirjojen pohjalta. Fenomenologishermeneuttisen tutkimuksen aineistona on satunnaisotannalla valittujen 10 oppimispäiväkirjan oppimiskokemusten kuvaukset, jotka analysoitiin Atlas.ti -ohjelmalla. Oman matematiikkasuhteen merkityksen tiedostaminen, vertaistuki, yhteisöllinen ja salliva oppimisilmapiiri, onnistumisen kokemukset matematiikan tehtävissä, tavoitteiden asettaminen ja näkyväksi tekeminen sekä niihin pyrkiminen pitkäjänteisellä työllä nousivat esille merkityksellisinä oppimisen ilon kokemuksina. peerReviewed

A Case of Four Prospective Adult Class Teachers’ Mathematical Identity Work

Research on adult students’ the mathematical identity work has not been explored so much. In this study I try to understand four prospective adult class teachers’ identity works on the basis of their learning diaries during a mathematics education course (6 ECTS). These four prospective class teachers’ reported they had negative experiences of mathematics during their school years but later in their lives the views towards mathematics had changed to a positive direction. However, the identity work during the mathematics education course revealed that life experiences had not changed the most fundamental views towards mathematics, mathematics learning and teaching. During the mathematics edu…

"Sitten palikat ilmestyivät kahvikuppini eteen kangastuksenomaisesti. Heureka!! Oli hieno hetki." : Reflektiivinen oppimispäiväkirja aikuisopiskelijan matematiikka-identiteettityön todentajana

Early Signs of Mathematics Anxiety?

Abstract Mathematics anxiety refers to individual's negative affect when engaging in numerical and mathematical tasks. Researchers have recently connected high math anxiety to lower performance on math tasks, developmental dyscalculia, and lower self-efficacy towards math learning. Math anxiety scales have been made and validated mainly for secondary school and high school students. In our paper we are looking for early signs of math anxiety using a Pictorial Test (37 pictures) which we have developed for 6 to 8 year old children. Children were asked to concentrate on one picture at a time and to write down spontaneously their emotional and mathematical ideas. Most powerful sources of sadne…

NÄYTTEITÄ MURTOLUVUN KÄSITTEESTÄ ERI AIKAKAUSIEN OPPIKIRJOISSA

Tässä artikkelissa kartoitamme sitä, miten murtoluvun käsitettä on opetettu eräissä 1800–2000-lukujen alaluokkien oppikirjoissa, ja millaisia murtoluvun käsitekuvia tarkasteluun valituista oppikirjoista välittyy. Osoittautuu, että murtolukua on käsitelty monipuolisesti ja eri tavoin jo 1800-luvun oppikirjoissa ja osa näistä käsittelytavoista on edelleen käytössä. Uuden matematiikan kirjoissa korostuu jonkin verran konseptuaalinen ajattelu, muuten murtolukujen käsittelyä hallitsee laskemisen näkökulma. peerReviewed

How can i be a Better Teacher? Development of Finnish Adult Pre- service Teachers’ Pedagogical Thinking

Abstract Adult students with experiences of teaching have often certain views about good teaching and a teacher. One of the most important roles in teacher education is to support future teachers to develop their pedagogical thinking systematically. Teacher's pedagogical thinking comes up in her/his actions and reflections on the teaching and learning situations. This article is based on research project which aims at to find out pre-service teachers’ progress in pedagogical thinking during three different practicums in Adult Teacher Education Program. The article will describe this thinking process during the first practicum which is carried out after three months studying.

Pupils’ mathematical self-concept in the beginning of the sixth grade

The article shows the mathematical self-concept of the sixth grade pupils (n=57) in relation to their mathematical skills in school. Mathematical skills were measured with the help of a survey of the command of the decimal system and on the basis of the report grades in mathematics. The pupils’ mathematical self-concept was surveyed in the beginning of the sixth grade with the interviews and a questionnaire using the Likert-scale. The mathematical self-concept seemed to be more positive the better the mathematics skills pupils had. peerReviewed

Kielentämisen näkökulmia kuudennen luokan oppilaiden matematiikan sanallisten tehtävien ratkaisuihin

Artikkelissa tarkastellaan kuudennen luokan lopussa oppilaiden tuottamia ratkaisuja kahteen sanalliseen ongelmanratkaisutehtävään. Tutkimuksen toteutti matemaattiseen ajatteluun ja sen kielentämiseen perehtynyt opettaja. Tutkimukseen osallistui 35 kuudennen luokan oppilasta. Tutkimusaineistona olivat oppilaiden kirjalliset ratkaisut kahteen sanalliseen ongelmanratkaisutehtävään. Oppilaat kuvasivat kielentämisen keinoin omaa matemaattista ajatteluaan sekä tehtävien tuottamisessa että ratkaisuprossien avaamisessa. Tuloksista on nähtävissä, että oppilailla oli taitoja kielentää monipuolisesti matemaattista ajatteluaan. Matematiikan opetuksessa olisikin hyvä ohjata oppilaita ilmaisemaan matemaa…

Kahden alkuopetuksen matematiikan oppikirja -sarjan didaktinen analyysi

Matkalla kohti ymmärrystä : matematiikan peruskurssin opetuksen periaatteiden tarkastelua

Artikkelin tarkoituksena on avata tuleville luokanopettajille suunnatun matematiikan peruskurssin kulmakiviä aikuiskoulutuksessa. Nämä kulmakivet ovat rakentuneet omien opettajakokemusteni luokanopettajakoulutuksessa sekä opettaja työnsä tutkijana pohjalta. Tutkimusteni ja saatujen kokemusteni pohjalta ovat löytyneet seuraavat opetukseni rakennusaineet: positiivisen matematiikkakuvan rakentaminen ja tukeminen, ymmärryksen rakentaminen koulumatematiikasta opiskelijan omien oivallusten kautta kielentämisen ja konkreettisten välineiden avulla sekä yhteinen tiedonrakentelu yhdessä opiskelijaparin ja koko opiskeluryhmän kanssa. Kaikkia edellä mainittuja rakennusaineita yhdistää avoin ja luottamu…

Sustainability Development in Mathematics Education : A Case Study of What Kind of Meanings Do Prospective Class Teachers Find for the Mathematical Symbol “2/3”?

In this article, our focus is on sustainable development in mathematics education from the point of view of teacher training. The aim was to develop prospective teachers’ content knowledge and pedagogical content knowledge of school mathematics. As a case study, we chose the mathematical symbol “a/b”, and examined how prospective class teachers in Finland connect it to the concepts of fraction, ratio, division, rational number or probability. Mathematics textbooks often have a central role in lessons, and they affect strongly how pupils understand concepts and the relationships between them. We chose languaging as a multi-semiotic approach to interpreting what kind of meanings the prospecti…

Adult Pre-Service Teachers Applying 21st Century Skills in the Practice

In this research we present how a group of Finnish adult pre - service teachers (n=21) incorporated 21 st CS through their methods of working instructional methods in the final practicum in autumn 2013. The research material consists of pre - service teacher’s practicum plans, taped lessons and written practicum synthesis. The nature of teaching has changed significantly during the past decades, as a result of which teaching has never been more difficult or more important. This is a challenge also for teacher education programs. The adult teacher education program at Kokkola University Consortium Chydenius focuses on developing 21 st century skills (21 st CS) to enhance the use of technolog…

The Potentials of Tangible Technologies for Learning Linear Equations

Tangible technologies provide interactive links between the physical and digital worlds, thereby merging the benefits of physical and virtual manipulatives. To explore the potentials of tangible technologies for learning linear equations, a tangible manipulative (TM) was designed and developed. A prototype of the initial TM was implemented and evaluated using mixed methods (i.e., classroom interventions, paper-based tests, thinking aloud sessions, questionnaires, and interviews) in real classroom settings. Six teachers, 24 primary school students, and 65 lower secondary school students participated in the exploratory study. The quantitative and qualitative analysis revealed that the initial…

Murtoluku vai suhde?

Artikkelin tarkoitus on tutkia miten luokanopettajaopiskelijat tulkitsevat koulumatematiikan käsitteitä murtoluku ja suhde heille annetussa tehtävässä, jossa on matematiikan symbolikielen merkintöjä ja niihin liittyviä kuvia. Tutkimukseen osallistui yhteensä 92 opiskelijaa kahdesta yliopistosta. Tuloksista näkyi useimpien opiskelijoiden proseduraalinen lähestymistapa annetun ratkaisun tulkinnassa käsitteellisen näkökulman jäädessä vain harvojen opiskelijoiden valinnaksi. Tulos on opetussuunnitelman ja oppimateriaalien näkökulmasta odotettu. Murtolukujen monipuolinen ymmärtäminen on haasteellista. Tämä tuo haasteita opettajankoulutuksen ja täydennyskoulutuksen kehitta…

Multimodal Communication and Peer Interaction during Equation-Solving Sessions with and without Tangible Technologies

Despite the increasing use of technologies in the classroom, there are concerns that technology-enhanced learning environments may hinder students’ communication and interaction. In this study, we investigated how tangible technologies can enhance students’ multimodal communication and interaction during equation-solving pair work compared to working without such technologies. A tangible app for learning equation solving was developed and tested in fourth and fifth-grade classrooms with two class teachers and 24 students. Video data of the interventions were analysed using deductive and inductive content analysis. Coded data were also quantified for quantitative analysis. Additionally, teac…

Academic Literacy Supporting Sustainability for Mathematics Education : A Case: Collaborative Working as a Meaning Making for “2/3”?

In this article, we focused on sustainable development in mathematics education from the point of view of academic literacy in mathematics (ALM). ALM was understood here through three integrated components: mathematical proficiency, mathematical practices, and mathematical discourse (languaging). ALM skills support 21st Century competences which are important for citizen skills. Both ALM skills and 21st Century competences support lifelong learning and sustainable education. Citizens of future society need both ALM and 21st Century competences to model and solve the issues of sustainable development. We want to develop prospective teachers’ content knowledge and pedagogical content knowledg…

Children's Mathematical and Emotional Expressions Inspired by Pictures

Children's talk about mathematics and mathematical talk