Search

In this plenary, I reflect on the MathTrack reseach project that examines the role of visual attention in the multimodal social interaction in the classroom contexts of collaborative non-routine problem solving. The project is using multiple mobile eye-tracking devices to record teacher and student visual attention when students work in groups solving a non-routine geometry problem. Project outcomes include methodological innovations for working with eye movement data, findings about joint representational attention, and the importance of eye contact in teacher-student interaction. Our experience suggests that eye movement research in classrooms should focus on analysing visual processes and making within-person analyses.

Tutkimus perustuu Opetushallituksen ja Kansallisen koulutuksen arviointikeskuksen keräämään pitkittäisaineistoon. Samaan ikäluokkaan kuuluvat oppilaat ovat osallistuneet kansallisiin matematiikan kokeisiin ja matematiikka-asenteita kartoittaviin kyselyihin vuosien 2005–2015 aikana neljällä eri mittauskerralla perusopetuksen kolmannelta vuosiluokalta toisen asteen loppuun. Tutkimusaineiston kokonaisotos käsittää yhteensä 3896 oppilasta. Tutkimuksessa keskitytään tarkastelemaan matematiikassa parhaiten menestyneitä opiskelijoita. Matematiikan parhaiksi osaajiksi määritetään kansalliseen matematiikan kokeeseen osallistuneet lukiolaiset, jotka saivat pitkän matematiikan ylioppilaskokeesta arvosanan laudatur tai eximia cum laude approbatur (n = 146). Ensin tutkimuksessa selvitetään, miten parhaiden osaajien matematiikka-asenteet muuttuivat perusopetuksesta lukion loppuun ja toiseksi, miten opetuksen pedagogiset ratkaisut yläkoulussa ja lukiossa selittävät osaamiseltaan parhaiden tyttöjen ja poikien asenteissa tapahtuneita muutoksia. Selittävien tekijöiden analyysissa käytetään päätöspuuanalyysia (DTA) ja lineaarista regressioanalyysia. Matematiikan parhaiden osaajien matematiikasta pitäminen kasvoi lukio-opintojen aikana, mutta minäkäsitys ja kokemus matematiikan hyödyllisyydestä laskivat. Matematiikassa parhaiten menestyneiden tyttöjen asenteissa tapahtuneet muutokset poikkesivat asenteiden yleisestä muutossuunnasta. Parhaiden tyttöjen minäkäsitys kasvoi yläkoulun ja lukion aikana lähes parhaiten menestyneiden poikien tasolle ja tytöt pitivät matematiikasta lukion lopussa poikia enemmän. Matematiikassa parhaiten menestyneiden tyttöjen ja poikien asenteiden kehittymistä selittivät erilaiset opetuksen pedagogiset ratkaisut. Molemmilla myönteisiä asenteita vahvistivat yleisesti oppilaskeskeisyyteen, yhteistoiminnallisuuteen ja oppijoiden tarpeiden huomioimiseen liittyvät pedagogiset ratkaisut. In English The study is part of a longitudinal research. Students belonging to the same age group were followed from the third grade of primary education to the end of upper secondary level. The data was collected by EDUFI and FINEEC during 2005–2015. The data consists of 3896 students. The target group consists of mathematically high-achieving students at the end of their studies in upper secondary school. The definition of high-achieving students based on success in matriculation examination of advanced math (n = 146). First we examine how high-achieving students’ attitudes towards mathematics change from primary education to the end of upper secondary level and then how pedagogical solutions in grades 7–9 and in upper secondary level explain boys’ and girls’ changes in attitudes. Decision tree analysis (DTA) and regression analysis were used to analyse the data. The results indicated that high-achieving students liked more mathematics during upper secondary school but self-concept and experience of usefulness of mathematics decreased. The changes of high-achieving girls’ attitudes differed from the general trend of change. High-achieving girls’ self-concept increased almost to the same level than high-achieving boys’ during grades 7–9 and in upper secondary school. Additionally, girls liked mathematics more than boys at the end of upper secondary level. The development of high-achieving girls’ and boys’ attitudes was explained by different pedagogical solutions. Both girls’ and boys’ positive attitudes were generally reinforced by pedagogical solutions related to student-centeredness, cooperative learning and paying attention to students’ needs. Fulltext in Finnish.

This paper presents a case study of following visual attention during collaborative geometry problem solving. We first analyse the emergence and spread of the incorrect idea of curved lines as the optimal shortest solution. Then, we examine the different visual representations made for solution ideas and how these are observed by the collaborating peers.

This paper presents a systematic review of university students’ emotions in connection with virtual learning based on 91 articles published between 2002 and 2017 in four international journals that focus on virtual learning and educational technology or on learning in higher education. These journals were considered potential channels for research on emotions in virtual learning and higher education. The objective was to analyse the articles for concepts and theoretical background related to virtual learning and emotions, contextual focus, methodological choice, and/or results. The review showed that the most common emotion‐related concept was “satisfaction.” The most common context for the articles was a complete non‐physical learning environment (e.g. Second Life). Approximately 60% of the articles used quantitative methods. The most common design for studying emotions was an explanatory design. Students’ emotions were mainly studied through concepts related to emotion (e.g. “satisfaction”). Yet only a few of the studies focused on the fluctuation of emotions in the course of events, relying instead on post hoc data that treat students’ emotions as traits rather than states.

This LUMAT special issue is a collection of selected papers from the 24th international conference of mathematical views that was held on August 20–22, 2018 in Helsinki, Finland. The conference was a wonderful opportunity to elaborate issues related to mathematics-related affect among colleagues interested in this area of research. The keynote at the conference was given by Reinhard Pekrun with a title: “Achievement emotions in mathematics”. Out of the 25 conference presentations, 12 were submitted as a manuscript for peer review. We had one reviewer selected among MAVI 24 participants, and another reviewer was invited among mathematics affect researchers who were not at the conference. After the review and revisions, we ended up with eight articles that you can read in this special issue. Non

D. B. McLeod's framework that identified three main topics of research in mathematics related to affect: emotions, attitudes, and beliefs. The main focus of such studies over the years concerned: the structure of affect and the relation between affective variables and achievement, the role of affect in mathematical problem solving and problem posing, and change in students' affect. One important area of research on mathematics-related affect is the role of different affective constructs in the process of mathematical thinking with problem solving. Despite the improved understanding of mathematics-related affect, the general trend still is that enjoyment of mathematics decreases over the school years. The reflections about methodology developed over the years in Congress of European Research in Mathematics Education (CERME) have had a great impact on the development of several critical issues in the field of affect. Comparative research on mathematics-related affect has confirmed that while some research findings about affect are universal, some other findings are contextual.

Tässä luvussa oppilaan matematiikkakuvalla tarkoitetaan oppilaan tunnesuhtautumista matematiikkaan sekä matematiikkaan liittyvää motivaatiota ja uskomuksia. Uskomukset voivat koskea matematiikkaa, sen oppimista sekä oppilasta itseään matematiikan oppijana. Matematiikkakuva on olennainen osa matematiikan oppimistuloksia, sillä se vaikuttaa oppilaan tulevaan oppimiseen ja elämänuran kannalta olennaisiin koulutusvalintoihin. Oppilaan matematiikkakuvan muodostumiseen ja muuttumiseen vaikuttavat hänen matematiikasta saamansa kokemukset. Oppilaan matematiikkakuva ei kuitenkaan ole pelkästään yksilötason kysymys, vaan luokan ilmapiiri ja kulttuuri vaikuttavat siihen ratkaisevasti. Vaikka suomalaisoppilailla on vähemmän matematiikka-ahdistusta kuin useimpien muiden maiden oppilailla, ei matematiikka myöskään herätä iloa tai kiinnostusta. Vaikka sukupuolierot matematiikan kokonaisosaamisessa ovat kadonneet, on pojilla edelleen tyttöjä kansainvälisesti vertaillen paljon korkeampi luottamus omiin matemaattisiin kykyihinsä. Tämä heijastuu selvästi lukiomatematiikan kurssivalinnoissa sekä erityisesti teknisille aloille hakeutumisessa. Artikkelissa esitellään myös aihepiirin tutkimusmenetelmiä keskittyen kyselytutkimuksissa käytettäviin mittareihin.

Research in mathematics related affect uses a variety of theoretical frameworks. Three different dimensions have been suggested as significant to characterize concepts in this area: (1) emotional, motivational, and cognitive aspects of affect, (2) state and trait aspects of affect, and (3) physiological, psychological, and sociological level of theorizing affect. In this study, we used the information in reference lists and graph theory to identify Graph Communities (coherent clusters) of research papers published in the affect groups of CERME conferences. The four main Graph Communities identified in the analysis were Foundation (beliefs, attitudes, emotions), Self-Efficacy, Motivation, and Teacher Development. There were six smaller Graph Communities that may suggest emerging new frameworks: Academic Emotions, Metacognition, Teacher Beliefs, Resilience, Meaning, and Identity. These results suggest that of the three possible dimensions to structure the area, the distinction between cognition (beliefs), motivation, and emotions is the most important one.

This study investigates behavioral executive functions (EFs) in the mathematics classroom context among adolescents with different mathematics performance levels. The EF problems were assessed by teachers using a behavioral rating inventory. Using cutoff scores on a standardized mathematics assessment, groups with mathematics difficulties (MD; n = 124), low mathematics performance (LA; n = 140), and average or higher scores (AC; n = 355) were identified. Results showed that the MD group had more problems with distractibility, directing attention, shifting attention, initiative, execution of action, planning, and evaluation than the LA group, whereas the differences in hyperactivity, impulsivity, and sustaining attention were not significant. Compared to the AC group, the MD group showed more problems with all behavioral EFs except hyperactivity and impulsivity, while the LA group showed more problems only with shifting attention. Male adolescents showed more behavioral EF problems than female adolescents, but this gender difference was negligible within the MD group. The practical implications of the results are discussed.

ISSN 0771-100 Proceeding volume: Volume 1 In this short note we consider three major challenges to the idea that it is possible to teach problem solving: do the students learn to be better solvers of novel problems, or do they just learn standard solutions for new sets of routine tasks; developing problem solver's affective disposition is not teaching of "problem solving"; and simply solving tasks is not "teaching". Moreover for each challenge we outlined several particular issues that need to be clarified. In this short note we consider three major challenges to the idea that it is possible to teach problem solving: do the students learn to be better solvers of novel problems, or do they just learn standard solutions for new sets of routine tasks; developing problem solver's affective disposition is not teaching of "problem solving"; and simply solving tasks is not "teaching". Moreover for each challenge we outlined several particular issues that need to be clarified.

The purpose of this study was to investigate Finnish mathematics teachers’ beliefs about teaching and teachers as expressed through metaphors. Because teachers’ beliefs play a significant role in their teaching, it is important to recognize those beliefs. Metaphors provide insights into beliefs that are not explicit or consciously held. In this study we investigated what kind of metaphors Finnish mathematics teachers in different schools and in different stages of their careers use. This study focused on Finnish 7-9 grade mathematics teachers’ (n=70) metaphors about teacher. The metaphors were classified into five categories: teacher as a subject specialist, teacher as a pedagogue, teacher as a didactics expert, self-referential and contextual metaphors. Teacher as a didactic expert was the most frequently used metaphor (49%). The information gained from this metaphor analysis show teachers’ beliefs about themselves. Changing teachers’ beliefs can help to change teachers’ behaviours and in such way improve teaching and learning process.

The aim of this study was based on pupils’ drawings to find out what kind of emotional atmosphere dominates in third graders’ mathematics lessons. Pupils’ (N = 133) drawings were analyzed by looking for content categories related to a holistic evaluation of mathematics lesson. As a summary we can conclude that the emotional atmosphere in the mathematics lessons is positive as a whole even though the differ-rences between the classes are great. Furthermore, it can be said that drawings is a good and many-sided way to collect data about the emotional atmosphere of a class.

Non

Tutkimuksen tavoitteena on selvittää, minkälainen yhteys on opettajan toiminnan ja kolmasluokkalaisille uuden ja epätavanomaisen ongelman ratkaisujen välillä. Ongelmana on aritmagon-tehtävän ratkaiseminen, jolloin pitää löytää kolmion kärkiin sopivat luvut, kun niiden summat on annettu sivujen keskellä. Oppilaiden tehtävänä oli lisäksi keksiä mene-telmä, millä aritmagon-tehtävän tietynlaisen erikoistapauksen voi aina ratkaista. Oppitunnit (8 luokkaa) videoitiin ja litteroitiin, oppilaiden (N=102) ratkaisut kerättiin ja tarkastettiin. Opettajan tehtävänannon luokittelussa käytetään lähtökohtana Polyan (1945) mallista kehitettyä opetusmallia.

Infinity is an important concept in mathematics, which students find difficult to learn. This paper will report Finnish elementary teacher students’ understanding of density at the beginning of their studies and the development of that understanding during a mathematics methods course. The results show that even quite limited teaching can initiate significant improvement. Moreover, students can make progress even if their initial level of performance is low. Yet, only 60 % of elementary education students reach satisfactory content knowledge and pedagogical content knowledge by the end of the course.

Volume: 2011 According to the Finnish curriculum (NBE 2004), improving pupils’ problem solving skills is an important objective in teaching mathematics. Teachers’ role is crucial in carrying out the objectives of the curriculum. Especially their conceptions influence decisions they make when teaching mathematics. The purpose of this paper is to find out what kind of conceptions elementary teachers (teachers for grade 3) have concerning problem solving and its teaching in mathematics, as well as what is their understanding about problem solving and how they implement it. The data was gathered with a questionnaire consisting of open questions during November 2010. The questionnaire was sent to a random sample (N = 100) of all Finnish grade 3 teachers. Two rounds of posting produced 63 answers. According to teachers´ responses, problem solving in mathematics means in the first place word problems, and problem solving is mainly done collaboratively.

Non

Proceeding volume: Volume 1 This study reports results from a Finnish survey for grade 4 and grade 8 students (N = 927). The survey measured students’ mathematics related achievement goal orientation, attitude and beliefs. The results confirm the reliability of used instruments and they indicate that the structure of grade 8 student’s view of mathematics is more coherent than on grade 4. Students’ mastery goal orientation is primarily correlated to students’ positive attitude and self-efficacy. Moreover, students’ performance orientation is positively correlated with self-efficacy and this relationship is stronger among older students. On grade 8, performance orientation has a positive correlation also with attitude.

The article presents an introduction to the concept of affect in mathematics education research.

One of the goals of the working group is to enhance discussion in the CERME - conferences and research between the conferences. Working Group 2 “Affect and mathematical thinking - This includes the role of beliefs, emotions, and other affective factors“ at the CERME 4 -conference was succesful in creating an atmosphere of collaboration among its16 participants. In preparation to the conference took place a call for paper and as a consequence of a reviewing process, 11 papers were accepted for presentation at the conference. The conference program scheduled 7 sessions, each 105 minutes, for work in the group. The chair of the organizing team worked out a concept for this 7 sessions. In 6 of these 7 sessions should take place a presentation of the key ideas and results of the accepted papers followed by a general discussion to the papers. Each session was extended by further activities (small group discussions to various themes, role play, analysis of data, problem solving, etc.). The last session was used for a summary of activities during the conference and highlighting important research questions for the following years.

The 13th International Congress on Mathematical Education [ICME-13] was organized in Hamburg, Germany in 2016. This chapter is the conclusion to the book composed of the selected papers presented in the Topic Study Group 28 on “Affect, beliefs and identity in mathematics education”. The book reflects the wide theoretical and conceptual variety in the field of affect, from embodied perspectives to social theories. This richness is reflected in the varied terminology and methodology apparent in this book. To evaluate the terminology, we analyzed the 4,739 affect term occurrences found in the chapters. We found belief to be the most frequently appearing affect concept, and that many motivational concepts (e.g. motivation, interest, value, goal, and meaning) have high prominence in the current volume. With respect to methodology, we found great variation from in-depth qualitative case studies to very large longitudinal and comparative studies using quantitative methods. Based on the contributions to this volume questionnaires and interviews are the dominating research methods for mathematics-related affect. In this chapter, we also reflect how the various chapters address the different roles of affect as a relatively stable trait and as the dynamically evolving state. Moreover, we reflect how the articles address the special nature of mathematics with its rigorous logic, cumulative structure, highly formalized special language, abstractness, and its role as a gatekeeper for many careers.

Research has shown that perceived social support (PSS) (from parents and teachers) influences achievement. However, little is known about how this relationship operates. This study examines the multiple mediational effects of students’ motivational beliefs in relationship to the association between PSS and mathematics achievement. The sample included the African countries that participated in the TIMSS 2011 (Ghana, Botswana, South Africa, Morocco, and Tunisia). A bootstrap analysis indicated a unique pattern of the role of motivational beliefs in mediating the relationships between PSS and achievement. Moreover, gender was found to moderate the indirect effect in some countries. The findings indicate that total mediation effect of students’ motivational belief on the relationship between PSS and achievement is “culture-fair but not culture-free” Research has shown that perceived social support (PSS) (from parents and teachers) influences achievement. However, little is known about how this relationship operates. This study examines the multiple mediational effects of students’ motivational beliefs in relationship to the association between PSS and mathematics achievement. The sample included the African countries that participated in the TIMSS 2011 (Ghana, Botswana, South Africa, Morocco, and Tunisia). A bootstrap analysis indicated a unique pattern of the role of motivational beliefs in mediating the relationships between PSS and achievement. Moreover, gender was found to moderate the indirect effect in some countries. The findings indicate that total mediation effect of students’ motivational belief on the relationship between PSS and achievement is “culture-fair but not culture-free”.

The 13th International Congress on Mathematical Education [ICME-13] was organized in Hamburg, Germany in 2016. This chapter is the introduction to the book composed of the selected papers presented in the Topic Study Group 28 on “Affect, beliefs and identity in mathematics education”. After the conference, 33 reworked papers had been submitted for evaluation and, based on a peer-reviewing process, 15 chapters were judged to be suitable for inclusion in the book. These contributions were clustered as follows: a stand-alone overview chapter, followed by three sections under the headings of interest, motivation, and values; engagement and flow; and identity. To enrich and broaden the contents of the book even further, the book editors invited three established mathematics educators to each write a commentary and critique on one of the three clusters. Mathematics-related affect is a research area with a variety of theoretical approaches. The different sections of the book provide a rich and diverse spectrum of different approaches, exhibiting both chapters that build on extensive research in the field as well as interesting new openings to the directions less explored in mathematics education to this date. The commentary chapters and the final conclusion offer insightful reflections on this field of research.

The paper argues for a three-dimensional course design structure for future mathematics teacher educators. More precisely we describe the design and implementation of a course basing itself on: the two mathematical competencies of modelling and problem tackling, this being the first dimension; the two mathematical topics of differential equations and stochastics, this being the second dimension; and finally a third dimension the purpose of which is to deepen the two others by means of a didactical perspective.

The article presents results from a cross-cultural NorBa-project “Mathematics teachers’ educational beliefs”. We report Estonian, Latvian and Finnish lower secondary mathematics teachers’ beliefs about good teaching. A principal component analyses identified a two-component structure of teachers’ beliefs about good teaching: (1) Reasoning and conceptual understanding and (2) Mastery of skills and facts. Cross-cultural differences were identified in both of these dimensions. Latvian teachers indicated the strongest agreement with reasoning and conceptual understanding, Estonian teachers with mastery of skills and facts, while Finnish teachers scored lowest on both dimensions. Moreover, we analysed the amount of teachers with different profiles with regard to these two dimensions.

This study compares six teachers’ learning goals for two problem solving lessons and their 9 to 11 year-old students’ solutions to two open-ended problems. Some of the learning aims were explicitly given before the lesson and some were revealed during the lessons. The pupils worked on the first open problem in primary three and the second problem in the primary four. The problem solving lessons were video-recorded and transcribed, and pupils´ solutions were collated and examined. An additional data source was the teachers´ lesson plans, and the videos taken in the teachers´ meeting sessions. The classification of teachers´ actions was based on the teaching model developed from Polya’s problem solving model (planning, introduction, guidance, feedback). This paper examines the significance of the teachers´ aims of the lessons and the introduction of the two problems, with the focus on the pupils´ justification of the solution for both tasks