In this paper, we examine the implementation of a Japanese teacher educators' lesson, where he applies and, at the same time, inform the students about "structured problem solving". We describe a specific lesson titled "Quantity and Measurement" for elementary school teacher students and we show how the educator make the students aware of the didactic transposition of the material and how he makes the students experience and learn about applying "structured problem solving" in practice. We also show how the Japanese curriculum influences the scale of the mathematical praxeology to be learned and how the students are given opportunities to develop their insight into the PCK during their education in mathematics.

The aim of this paper is to investigate which kind of conditions and constraints affect Japanese and Swedish teacher educators’ pedagogical content knowledge (PCK). We analyse the praxeologies of the lessons in which the educators teach area determination. Our study shows that the Japanese teacher educators’ PCK are more explicitly shared by the community of the teacher educators compared to the Swedish counterpart. Also, the detailed Japanese curriculum and the structured problem solving approach promote to illustrate how to construct rich mathematical and didactical organisations for prospective teachers.

In this study, we have observed three different teacher educators’ lessons, concerning area determination of polygons in primary school teacher training courses in Japan, Finland and Sweden. The aim of this paper is to investigate the main elements of the lessons and to compare the differences between the countries. We focus on how the teacher educators relate the didactic construction of the lessons for prospective teachers to the school mathematical and didactical organisations by applying Chevallard’s anthropological theory of the didactic (ATD). The analysis shows how the curricula and the different traditions of teaching practice in each country influence the mathematical and didactical construction of the lessons.

The aim of this paper is to investigate and compare lessons given in primary school teacher education in Japan, Finland and Sweden. We analyse one lesson from each country and compare them using a common framework. Chevallard’s anthropological theory of the didactic (ATD) is used to frame this analysis and in particular to model teacher educators' didactic organization of the lessons. The focus is on how the didactic organizations of the teacher educators relate to the mathematical and didactic organizations of primary school. Based on official documents and viewpoints of the teacher educators, we also discuss how the contents and descriptions of the national curricula, and the different traditions of the teaching practices in each country, influence the didactic organizations found in the lessons. 

The purpose of this study is to analyse what kind of conceptions secondary school teachers in mathematics have about equations and how these conceptions are related to the formal definition of the concept of equation. Data was gathered by interviews and questionnaires. Both newly graduated and experienced secondary school teachers were participated in this study. The phenomenographic research approach in order to analyse research outcomes was applied in the investigation. From a phenomenographic analysis of the interview transcripts I found that some patterns could be identified in them and the three qualitatively distinct categories of description about equations could be discerned among the teachers’ conceptions. The research results indicated that equations were apprehended as a procedure, as an answer and as a ‘rewritten’ expression.

In this article student teachers’ conceptions of equations are discussed. Data was gathered from 75 student teachers by interviews and questionnaires. Both the quantitative and qualitative research methods were applied in the study. The research results indicate that some specific examples of concepts, which at first appear in teaching, and learning of mathematics are more central to the student teachers than other examples. The former examples are often called prototypes. The investigation shows that students tend to identify the concept equation with one of the prototypical examples and that the concept image develops from one unique prototypical example to include more examples of increasing distance from the prototypical example. The results also indicate that the student teachers’ concept images include erroneous conceptions of the concept equation.

In this article I discuss the mathematics teachers’ conceptions about concept learning in algebra from school and university time. The study has a focus on the particular topic of equations. The data was gathered by interviews and questionnaires. Both newly graduated and experienced secondary school teachers were participated in this study. The phenomenographic research method in order to analyse research outcomes was applied in the investigation. The research results indicated that the teachers experience the learning of equations from school and university time on four qualitatively different ways. Learning was apprehended as doing routine problems, as memorizing and reproduction of rules and models, as doing applications and as interaction with other students.

The aim of this study is to describe and to clarify the mathematics teachers’ subject matter and pedagogical content conceptions about equations. As the basis of these conceptions, the teachers’ experiences of the concept learning of equations from their own school time are described. The early research of conceptions has been concentrated on pupils’ conceptions of the topic as a contrast to scientific conceptions since the middle of the 1970s. Research of teachers’ conceptions of mathematics and mathematics teaching and learning has grown during the last decade. However, in these studies teachers’ conceptions of a specific content area in mathematics have not been investigated. In the theoretical background of the research, different traditions of school mathematics learning and teaching are treated. By using theories of experiential learning, it has been possible to study such learning situations and experiences, which may lead to the development of subjective conceptions of mathematical concepts. In order to understand difficulties concerning the concept formation in mathematics the theory of the concept image and the concept definition as well as the theory based on the duality of mathematical concepts have been studied. The acquired experiences from school time seem to lay the basis of both the teachers’ subject matter and pedagogical content-specific conceptions and decisions. Different components in teacher knowledge base together with current research both in teachers’ subject matter and pedagogical content knowledge are therefore presented at the end of the theoretical framework. By combining different kinds of methods like questionnaires, recorded interviews, videotape recording of six lessons in mathematics and observations the research empirical material was collected. In this investigation, five novice, five expert and 75 student teachers in mathematics participated. The preliminary investigation included 30 student teachers. In the study the phenomenographic approach is used in order to reveal differences between the teachers’ conceptions and experiences about equations. The research results indicate that equations are not apprehended as complete, static objects. Conceptions about equations reveal that equations are closely related to the symbols x and y and solving procedures. The teachers’ experiences of learning and teaching of mathematics may have formed their conceptions. The conceptions about equations seem to be based on the teachers’ experiences in arithmetic and their first impressions of learning the process of solving equations. The teachers apprehend equation teaching as a study of procedures rather than as a study of central ideas and concepts of algebra. Both aspects are however equally important at compulsory school, since the teaching of algebra should develop pupils’ ability both to use and to understand the basic algebraic concepts. Some of the teachers do not have a clear conception what the pupils should attain in algebra at compulsory school according to the specific goals in Swedish mathematics curriculum. The research results further show that both the expert and the novice teachers have various apprehensions of the pupils’ difficulties concerning equations.

The purpose of this study is to analyse what kind of conceptions secondaryteachers in mathematics have of the equation concept and what kind of experiencesgave rise to their own concept learning. Ten secondary school teachers ofmathematics participated in the study. Data was gathered by interviews andquestionnaires. The ‘phenomenographic’ research method was applied ininterpreting the results of the investigation. The results indicate that the teachers’conceptions of the equation concept differ from the formal definition of the equationconcept. At school they spent most of their time developing procedural skills insteadof mathematical understanding.

Syftet med min studie är att beskriva hur matematiklärare uppfattar begreppet ekvation. Min målsättning är också att undersöka lärarnas erfarenheter av ekvationsinlärning från grundskolan till universitetsnivån. Tio lärare från grundskolans högstadium har deltagit i undersökningen. Fem är nyutexaminerade lärare med mindre än ett års yrkeserfarenhet. Data i undersökningen samlades in genom enkäter och intervjuer. Undersökningsresultat analyserades genom fenomenografisk metod. Resultatet pekar på att lärarnas uppfattningar av ekvationsbegreppet avviker från den formella begreppsdefinitionen. De känner osäkerhet inför matematiska symboler, bokstavsuttryck och lösningsprocedurer. Deras skolerfarenheter visar att de har använt största delen av tiden till att utveckla algoritmiska färdigheter istället för matematisk förståelse.

We report on a teaching experiment regarding Taylor polynomial approximations at the level of university mathematics teaching. The experiment was carried by using the free dynamic mathematics software GeoGebra. A student group (n = 19) was taught Taylor polynomials with the assistance of GeoGebra while a control group (n = 18) was taught in a traditional way. The theoretical assumptions of this study rest on Variation theory. The data were gathered by doing a post test concerning Taylor polynomials. Our experiment revealed that the answers from the GeoGebra group in the post test were more varied compared with the results in the control group.

In this paper, we report a teaching experiment regarding the theory of polynomial approximations at the university mathematics teaching in Sweden. The experiment was designed by applying Variation theory and by using the free dynamic mathematics software GeoGebra. The aim of this study was to investigate if the technology-assisted teaching of Taylor polynomials compared with traditional way of work at the university level can support the teaching and learning of mathematical concepts and ideas. An engineering student group (n = 19) was taught Taylor polynomials with the assistance of GeoGebra while a control group (n = 18) was taught in a traditional way. The data were gathered by video recording of the lectures, by doing a post-test concerning Taylor polynomials in both groups and by giving one question regarding Taylor polynomials at the final exam for the course in Real Analysis in one variable. In the analysis of the lectures, we found Variation theory combined with GeoGebra to be a potentially powerful tool for revealing some critical aspects of Taylor Polynomials. Furthermore, the research results indicated that applying Variation theory, when planning the technology-assisted teaching, supported and enriched students’ learning opportunities in the study group compared with the control group. 

In this paper we report on a teaching experiment regarding the definite integral concept in university mathematics teaching. The experiment was carried out at a Swedish university by using the free dynamic mathematics software GeoGebra. In our theoretical framework we apply Variation Theory, originating in the phenomenographic research tradition. The data of this study consist of the lecture plan and the engineering students’ answers to pre and post tests. In the analysis of the data we applied statistical methods. The experiment revealed that by using GeoGebra it is possible to create learning opportunities of the definite integral concept that support the students’ learning.

In recent years, there have been several studies in mathematics education basing on the variation theory and the model of Learning Study that involves co-operation between teachers and researchers in an iterative process. Most of these studies have focused on the teaching and learning of elementary school mathematics rather than topics in advanced mathematics. In this paper, we discuss some challenges and possibilities of the Learning Study model and the variation theory when developing the teaching of mathematics at advanced levels. More precisely, we report on a series of teaching experiments on the definite integral concept. The experiments were carried out at a Swedish university. The data of this study consists of the documents on the observation of three lectures and the students’ answers to pre and post tests. Both engineering and teacher students participated. In the analysis of the data, we applied statistical methods. Although the series consisted only of three lectures, it revealed that the students’ understanding about certain – but not necessarily all – aspects of the definite integral concept can be enhanced by using the Learning Study model.

In this paper, we report on a collaborative teaching experiment based on the Learning Study model (LS model) which grounds on the Variation Theory. Until today, most of such studies have focused on the teaching and learning of elementary school mathematics; ours was carried out in undergraduate mathematics education. In the following, we discuss how we managed to promote students’ conceptual learning by varying the treatment of the object of learning (the concept of definite integral and the Fundamental Theorem of Calculus) during three lectures on an introductory course in calculus. We also discuss the challenges and possibilities of the LS model and the Variation Theory in the development of the teaching of tertiary mathematics in general. The experiment was carried out at aSwedish university. The data of the study consists of the documents of the observation of three lectures and the students’ answers to the pre- and post-tests of each lesson. The analysis of learning results revealed some critical aspects of the definite integral concept and patterns of variations that seem to be effective to a significant degree. For example, we found several possibilities to use GeoGebra to enrich students’ learning opportunities.

This study is a part of an ongoing research that attempts to explain the relationship between the teachers’ instructional practise and students’ learning in the context of functions and function inverses. The question in this paper is how the use of technology as a pedagogical tool may contribute to the understanding of the inverse function concept. An engineering student group (n =17) was taught functions and inverse functions with the assistance of GeoGebra. In our theoretical framework we apply Variation theory together with the theory of Concept image and Concept definition. The data were gathered by doing a pre and post test concerning inverse functions. Our experiment revealed that students’ concept images in the post test were more developed compared with the results in the pre test.

The present study investigates how technology assisted and designed teaching influences engineering students’ understanding of the connection between the graph of a function and its derivatives. An engineering student group (n = 27) was taught with the assistance of GeoGebra while a control group (n =20) was taught in a traditional way. The data of the study consist of the documents and photos of the observation of two lectures and the students’ answers to the pre and post tests. In our theoretical framework we discuss the distinction between conceptual and procedural knowledge. When creating the teaching sequences we applied variation theory. In the analysis of the students’ pre and post tests results we applied statistical methods. Our experiment revealed that the GeoGebra-assisted teaching design created more opportunities for students to grasp the connection between a function and its derivatives.

Här beskrivs ett antal undervisningstillfällen kring begreppen area och omkrets i en förskoleklass. I planeringen har lärarna utgått från att barnen ska samarbeta, kommunicera med varandra, använda flera sinnen, reflektera över sitt och andras tänkande samt dokumentera och sätta egna ord på det de varit med om.

The aim of this article is to describe a design and implementation of a school-university action research project about teaching and learning biology and mathematics in primary school. Nine teachers in grades 1 to 6, in collaboration with two researchers, were using content representation (CoRe) in learning study (LS)-inspired cycle as pedagogical tools when planning, implementing, and reflecting on lessons and pupil learning. By using pre- and post-tests, the teachers acquired knowledge about pupil subject-specific knowledge and learning. Some examples are given on how the tools in the project influenced the teaching practices and pupil learning. This research design brought together university and school practitioners to work collaboratively in a professional learning community, which developed teaching and learning in biology and mathematics.

The present study is part of the EU-Fp7 project SECURE. The research question in the present study is: How is the current Swedish MST curriculum for preschool implemented by the teachers, and perceived by the 5-year old learners?

The introduction of a revised preschool curriculum in 2011 has given MST a prominent place in Swedish preschool, and may be intended by the authorities to change the weak positions and declining trends in the PISA and TIMSS surveys for Swedish school children. The new teaching task seems to be highly appreciated by the preschool teachers. However, the curriculum is quite ambitious, and a major concern is how the implementation will be done. The concept of ”emergent science” is discussed in order to induce positive attitudes towards MST.

In this study, semi-structured interviews with teachers in 15 Swedish preschools are the main source of information combined with some interviews with learners and photographic materials. The method for the analysis uses a category system of which results from some of the categories are reported.

The preschool teachers are aware of the new role as teachers rather than just child minders. The revised curriculum seems to increase the self-confidence for MST of the learners as well as the teachers. The teachers say that the learners are interested in MST and that it is a case of catching opportunities, which may lead to emergent science. The 5-year old learners confirm that they like the activities related to MST.

Working with hypotheses and storyline activities is shown as an example how emergent science can be promoted in 5-year old children. Activities connected to ”emerging science” have been common in Swedish preschools for some time, but after the implementation of the revised curriculum teachers also verbalize this for themselves and the learners.

Svenska elever hamnar efter i PISA- och TIMSS-undersökningarna. I matematik har matningarna visat på sjunkande kunskaper sedan början av 2000-talet och i naturvetenskap ligger de svenska eleverna ständigt under det internationella genomsnittet. Detta är alarmerande för landet och skälen bör utredas.Som en del i EU-projektet SECURE inom ramprogrammet Fp7 presenteras resultat från lokal forskning om genomförandet av MST-kursplanerna och attityder till ämnena bland lärare och elever. Med MST avser vi matematik, naturvetenskap och teknik. Intervjuer och enkäter har utförts bland lärare och elever i 47 klasser i 15 skolenheter i några mellansvenska kommuner. De åldersgrupper som studerats är elever från åldersgrupperna 5, 8, 11 resp. 13 år. Vi fokuserar här på förskolans verksamhet, d.v.s. arbetet med 5-åringar.Som en följd av införandet 2011 av en reviderad läroplan för förskolan, kan vi se att MST har fått en framträdande plats i den svenska förskolan som verkar vara mycket uppskattad av förskollärarna. I den reviderade läroplanen framgår att förskollärare ska ansvara för att arbetet i barngruppen genomförs så att barnen stimuleras och utmanas i sin matematiska utveckling, samt stimuleras och utmanas i sitt intresse för naturvetenskap och teknik. Exempel på hur de skall gå tillväga finns också i läroplanen. Detta kan i praktiken uppfattas som en informell tidigareläggning av den obligatoriska skolan i Sverige, eftersom förskolan inte är obligatorisk men utnyttjas av de allra flesta. Det skulle därmed kunna innebära en vändpunkt för resultaten i grundskolan och förhoppningsvis ge avtryck i kommande PISA- och TIMSS-undersökningar.De förskollärare som intervjuats är medvetna om vikten av att undervisa MST i förskolan. Detta tydliggörs av några uttalanden från intervjuade förskollärare. 'Det som är nytt nu är att man pratar om lärare och att vi utbildar barnen.' 'Det här lärandet är för livet. Det är inte bara för individen. Verksamheten på förskolan bidrar till hela samhällsutvecklingen.' Även föräldrarnas förväntningar på förskolan har förändrats. 'Föräldrarna uppskattar resultaten i matematik och ser förskolan som en skolform snarare än förvaring av barn.'I denna presentation ges exempel på hur MST-ämnena undervisas på ett integrerat sätt i förhållande till den praxis som råder i svenska förskolan, samt i förhållande till ålder och mognad hos barnen.

I detta dokument diskuterar vi pågående projekt i matematikdidaktik vid Högskolan i Gävle. Syftet med projektet är att utveckla och testa metoder för att förbättra lärande i matematik från förskolan till universitetsnivån. Dessutom är syftet att utveckla en forskningsmiljö för lärande i matematik som kan fungera som en plattform för att utveckla lärarnas kompetens i matematik. Den övergripande forskningsfrågan är: Hur påverkar lärarens sätt att hantera lärandeobjekt (t.ex. funktionsbegrepp, integralbegrepp etc.) i klassrummet elevens/studentens lärande? Vilka är de kritiska aspekterna för studenternas lärande? Som ett teoretiskt ramverk använder vi Learning Study modellen designad av Marton et al. (2004). Modellen bygger på variationsteorin (Marton et al. 2004), som härrör från den fenomenografiska forskningstraditionen och som introducerades i Sverige under senare delen av 1960-talet och i början av 1970-talet. Resultaten av studierna antyder bl.a. att det är betydligt enklare att hitta olika typer av konkreta men användbara representationer av lärandeobjektet i undervisningen av den elementära matematiken jämfört med undervisningen av den högre matematiken. Troligen beror det på att universitetsmatematik är mer abstrakt och bygger på den tidigare erhållna kunskapen.

In this study we analyse conceptions that prospective teachers in mathematics have about equations and how these conceptions are related to the truth value (of mathematical statements). Our phenomenographic research reveals that there exists a large variety of fundamentally different conceptions about equations and that in students’ actual concept image of equation it is commonly assumed that a mathematical expression must possess the truth value ‘true’ in order to be an equation in spite of the fact that any consideration related to the truth value only rarely appears in concept definitions given by students.

The aim of this study is to describe what kind of knowledge base is needed when pre-school teachers work goal-oriented with children’s mathematical learning. The question of this study aims to answer is: What kind of knowledge base do pre-school teachers need when they work with the object of learning in the pre-school context? Both as a theoretical and an analytical framework, we use variation theory. In all, four pre-school teachers are involved in the study. The research results indicate that it is important for pre-school teachers to have subject and pedagogical content knowledge in order to recognize children’s experiences and be aware of their expressions.

Earlier reports have shown that prospective teachers' conceptions about teaching science to a high degree are resistant and do not change substantially during the teacher-training programme. In our investigation we elucidate the prospective teachers' initial conceptions about pupils' understanding of science and mathematics. We applied 'The Lesson Preparation Method' and used a phenomenographic approach in order to reveal the range of conceptions that the prospective teachers hold. A third of the prospective teachers did not consider pupils' conceptions when planning lessons. The rest of the 32 participants expressed awareness; some of the prospective teachers even referred to subject-specific teaching experience. Also regarding the prospective teachers' conceptions about pupils' knowledge and beliefs, as well as about pupils' difficulties, there was a significant diversity. By raising these issues about pedagogical content knowledge the prospective teachers' conceptions can be extended and developed during the education.

This article describes how some Swedish compulsory schools work to achieve collective development of practices in mathematics and science. The overall aim is to increase knowledge about factors influencing progression to professional learning in different school contexts. Data were collected through four case studies by interviews with teacher teams and principals and were analysed in a meta-perspective by using the components of the Collaborative Action Research model. The findings showed that the schools had reached different phases concerning progression to professional learning. Changes aiming to improve teaching and learning are context-bound. Therefore the authors suggest some crucial questions to support professional learning in the prevailing school culture.

Kan man utveckla undervisningen i biologi och matematik samtidigt? Ett lärarlag samarbetade med två forskare för att undersöka och svara på frågan. De undervisningsområden som valdes ut handlar om ekosystem och övergången mellan aritmetik och algebra.

The aim of this paper is to provide insights into nine primary school teachers’ concerns and instructional needs in biology and mathematics, grades 1 to 6. By using Content Representation, combined with Learning Study in an action research project, teachers were encouraged to reflect on their conceptions, processes of instructing and pupil learning. From concerns articulated by teachers three instructional needs emerged: (i) to make subject progression, especially in biology, and pupil learning more visible, (ii) to develop mathematics teaching in order to change pupils’ views of the subject, and (iii) to develop teachers’ subject matter knowledge and teaching in an ongoing collaborative process.  The paper argues that in order to stimulate teacher professional development it is important to make teacher concerns and thereby needs explicit. They help teachers to identify and reflect on relations between teacher subject matter knowledge, pedagogical content knowledge and pupil learning.

This article presents the evaluation of a two-year action research project in biology and mathematics teaching involving a primary school and a university in Sweden. The aim of the study was to contribute knowledge about a school–university intersection as a professional learning arena. The teachers’ conceptions about the project implementation, the impact on their learning, teaching practices and pupil learning were made explicit by focus group interview. The evaluation revealed that several motivating factors in this specific learning community – the relevance of the project and connection to the continuing education course, mentors from university, planning tools and time for collaboration – were critical for project implementation and for professional learning to occur. Furthermore, it indicated how teacher learning and teaching practices were related to pupil learning in the professional learning community. The results are also discussed in the light of new research on teachers’ work identity and self-reported health.

A crucial issue for prospective teachers (PTs) in their education is to develop pedagogical content knowledge (PCK; i.e. how to make a topic comprehensible to pupils). However, research has shown that PTs may have tacit ideas about teaching that act as filters preventing consideration of unfamiliar and discrepant ideas. These ideas must be elicited and taken into consideration in order to be modified. Therefore, PTs’ explicit conceptions may constitute a valuable resource in teacher education. The aim of this study was to investigate PTs’ ideas about pupils’ difficulties, at a topic-specific level, upon beginning the teacher education programme. The “Lesson Preparation Method” was used in four case studies to elicit the conceptions of 32 PTs regarding pupils’ difficulties in four specific science and mathematics topics: plant growth, gases, equations and heat and temperature. In all four topic groups (n = 5 – 11), there was a variety of initial conceptions about pupils’ difficulties, which were categorised into two to five topic-specific categories. Although, initially, PTs may not have expressed any notions about pupils’ difficulties, conceptions were elicited by using the Lesson Preparation Method. Furthermore, we found that the initial ideas corresponded with earlier research on pupils’ difficulties, which could provide a potential resource when creating a scaffolding context in teacher education programmes where PCK development is stimulated.

Every child in Finland and Sweden has possibility to get pre-school education, in separate pre-school classes, and after that nine years compulsory primary school education. Concerning their math skills development, it is important that children have opportunities to express their own marks and meanings. In this paper we study six to eight year old Finnish (n=100) and Swedish (n=100) children’s spontaneous expressions measured by a pictorial test. In our cross-cultural comparative study we found some similarities and some differences. Both Finnish and Swedish children expressed most numbers and exercises which they use to practice in their early school years. Swedish children expressed significantly more numbers and exercises than the Finnish ones, whereas Finnish children expressed more word problems and mental models. Our cross-cultural interpretation is mainly based on different emphasis in Finnish and Swedish core curriculums. Finnish children follow an early school education that emphasises more learning skills and social development than academic skills learning. Swedish curriculum emphasises more subject-specific knowledge in mathematics.

We analyze data on mathematics students’ understanding of the concept of equation. A majority of the participants (N = 128) studies in teacher education programs in Finland, Sweden and South Africa. The data reveals a variety of fundamentally different concept definitions of equation, of which only a half can be seen to be correct. In the students’ concept image of equation, it is commonly assumed that every equation must possess the truth value ‘true’ in spite of the fact that any considerations related to the truth value only rarely appear in their concept definitions. Also the presence of a variable is regularly assumed in the participants’ concept images. Both a chain of equations and a pair of equivalent equations are surprisingly often seen being a single equation. Finally, the incomplete understanding of the reflexive and symmetric properties of equality often hindersstudents from identifying equations. The study is based on both phenomenographic andquantitative analyses of the students’ answers to a questionnaire.

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In this paper, we examine the concept definitions a group of South African upper secondary school mathematics teachers (N = 47) express and how their understanding of the truth value, the role of variable and the syntax of expression appear in the participants' explanations for their assessment of examples and non-examples of equations. We use content analysis and standard quantitative methods. The data consists of the participants' answers to a questionnaire reflecting both teachers' concept definitions of equation and their skills in classifying examples and non-examples of equation. Altogether 27 participants were able to give a correct definition of the equation concept. Ten participants' definition was slightly ambiguous yet meaningful and ten participants failed in this task. In general, the participants had very high confidence in the sufficiency of their skills in classifying examples and non-examples of equations. Nevertheless, on average, they only succeeded in correctly identifying an equation in 13 of 24 items, with most of the equations being quite simple and none beyond the upper secondary school level. The findings of this study also reveal a common and dominant conception that equations should always possess the truth value 'true' although truth value is discussed only in one participant's concept definition. Secondly, the participants are quite careless about the syntax and the involved binary relation in particular despite the fact that the correct form of equations and the equality relation were regularly mentioned in their concept definitions of equation. Finally, some participants seem to think that there is only one equation related to each algebraic problem.

This study analyses what kind of concept images a group of engineering and teacher students have of the function concept, and how these concept images are related to the historical development of this concept. The study was conducted using questionnaires, and 34 students at a Swedish university participated. It is found that the students primarily rely on operational conceptions of the function concept, with only a minority of students possessing structural conceptions. The definitions given by the students mostly resemble an 18^th or 19^th century view of functions. The study also indicates that the character of the definitions given in the textbooks used by the students affect their concept images.