Hur matematikundervisning som har vardagliga samband påverkar elevernas lärande

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Natur, matematik och samhälle

Examensarbete i fördjupningsämnet matematik

15 högskolepoäng, grundnivå

Hur matematikundervisning som har vardagliga samband

påverkar elevernas lärande.

How mathematics teaching that has everyday connections

affects pupils' learning.

Kondwelan James Tembo

Kompletterande pedagogisk utbildning 120hp Examinator: Jöran Petersson Datum för slutseminarium (2020-06-07) Handledare: Nils Ekelund

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Preface

In my home country, I worked as a mathematics teacher for 12 years. While studying at Malmö University, I have had the opportunity to teach mathematics at a school where I was assigned for my practice. It is during these years (while working as a mathematics teacher in my home country as well as while teaching mathematics during my studies in Sweden) that I have heard some pupils complain that mathematics is a difficult subject. I have also seen some pupils struggle to solve certain mathematics problems. Could it be because they think mathematics is a difficult subject? Is it because of the way mathematics is taught? Or is it due to various reasons? Such questions are exactly what encouraged me to write this thesis because I am interested in finding solutions in mathematics teaching in order to increase pupils’ understanding,

performance as well as interest in the subject.

I have chosen this area of study because of its relevance to my profession and that it fits well with the curriculum which emphasizes, among other things that;

“The teaching of the subject of mathematics should aim to develop pupils' knowledge of mathematics and the use of mathematics in everyday life and in various subject areas. The teaching of the subject of mathematics should help pupils develop interest in mathematics and confidence in their ability to use mathematics in different contexts. Through the teaching of the

subject of mathematics, pupils should also be given the opportunity to experience aesthetic values in encounters with mathematical patterns, forms and relationships." (Skolverket, 2019).

In conclusion, I would like to thank my supervisor Nils Ekelund for the useful remarks and insightful comments that enabled me to complete this study. Furthermore, I would like to express my sincere thanks to the school where this study was conducted from, the pupils and teachers who participated in the study including those who willingly shared with me their personal views and experiences. Finally, I would also like to thank my family and friends for their patience and for supporting me throughout the entire process.

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Abstrakt – Svenska

Denna studie fokuserar på problemet där flesta elever tappar intresse, motivation och lust att lära sig matematik eftersom de upplever att det är ett svårt ämne. Syftet med detta arbete å andra sidan är att undersöka om hur matematikundervisning som har vardagliga samband påverkar elevernas lärande. Studien vill också belysa hur lärare arbetar så att undervisningen i matematik kan bidra till att elever utvecklar intresse för matematik och förtroende för deras förmåga att använda matematik i olika sammanhang. Resultaten är baserade på en analys av elevers och lärares enkäter. Det finns tjugotvå (22) elever och fyra (4) legitimerade matematiklärare som deltog i denna studie. Eleverna går i årskurs sex (6) och lärarna undervisar olika klasser från årskurs ett (1) till sex (6). Enkäten behandlade olika frågor bland annat; elevernas intresse, motivation och lärande när lektioner har vardagliga samband.

När det gäller motivation, 19 elever svarade att de är mer motiverade att lösa matematiska uppgifter när de använder eller arbetar med material eller något som hjälper dem att lösa uppgifterna. Med samma siffror svarade å andra sidan att de tappar lusten eller motivation när lektionerna inte är kopplade till saker som de vet eller gillar. När det gäller inlärning/

undervisning gick både elever och lärare i samma håll. Alla 22 eleverna svarade att de lär sig bättre när läraren använder vardagliga samband för att lösa eller förklara matematiska problem eller begrepp medan 3 lärare svarade att dem anser att deras elever lär sig matematik bäst när lektionerna är kopplade till saker de vet eller gillar. Resultaten av denna studie visar emellertid att elevernas intresse, motivation och prestation i matematik ökar när lektioner är kopplade till vardagliga samband.

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Abstract – English

This study focuses on the problem where most pupils lose interest, motivation and the desire to learn mathematics because they perceive it to be a difficult subject. The purpose of this thesis on the other hand is to investigate how mathematics teaching that has everyday life connections affects pupils' learning. The study also wants to shed light on how teachers work so that the teaching of mathematics will contribute to pupils developing interest in mathematics and confidence in their ability to use mathematics in different contexts. The results are based on an analysis of pupils’ and teachers’ questionnaires. There are twenty (22) pupils and four (4) licensed mathematics teachers that took part in this study. The pupils are in grade six (6) and the teachers teach different grades from grade one (1) to six (6). The questionnaires addressed different issues among them; pupils' interest, motivation and learning using everyday connections.

When it comes to motivation, 19 pupils answered that they are more motivated to solve math tasks when they use or work with materials or something that helps them solve the tasks. The same number on the other hand answered that they lose the desire or motivation when the lessons are not connected to things they know or like. On learning/teaching, both pupils and teachers went into the same direction. All the 22 pupils answered that they learn better when the teacher uses everyday connections to solve or explain mathematical problems or concepts while 3 out of 4 teachers answered that their pupils learn mathematics best when the lessons are connected to things they know or like. The results of this study however show that pupils' interest, motivation and performance in mathematics increases when lessons are linked to everyday connections.

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1. Introduction

This study shows that pupils' understanding and interest in the subject can be increased if

mathematics lessons are linked to everyday connections. My supplementary education at Malmö University has had three practical periods (verksamhetsförlagd utbildning- VFU in Swedish). It was during these three practical periods that I met different teaching methods, of which two of them are the most prominent ones. The first one is called traditional teaching and the other one is called varied teaching (Malmer, 1999).

According to Malmer (1999), traditional teaching is the king of teaching where teachers present lessons and later on pupils work individually. Berggren and Lindroth (2011) define varied teaching as a method of teaching where teachers use different teaching methods, such as group work, digital tools, open dialogue, assignments etc. It was during the same practical periods that I also saw and heard how some pupils lost interest and motivation during mathematics lessons as well as when they wrote exercises in their mathematics book. According to Malmer (1999), pupils begin to show difficulties in subjects when there is an erosion of self-confidence. This gradually leads to pupils losing motivation for the subject as it is the case with many pupils who perceive mathematics as a difficult subject.

Learning materials that I focused on during this study are those that have a connection of mathematics to life outside school as such provide pupils with tools that enable them to analyze problems in different ways. I will describe different contexts such as concretized teaching, pupils' interest/motivation, learning environments, interactive teaching/ permissive learning climate and mathematics in different contexts. These different contexts will be the starting point for my study, by highlighting the influence of these situations on pupils' understanding, learning as well as performance. Results from the pupils’ questionnaire (question number 4) shows that all pupils learn better when teachers use everyday life connection to solve/explain mathematical problems/concepts (figure 4). The teachers are also in agreement with the pupils. The results from the teachers’ questionnaire (question number 1) show that most teachers consider that their pupils learn mathematics best when lessons are connected to things pupils already know or like (figure 7).

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2. Previous research

In this section, the concepts of everyday play activities, Mathematics education in its cultural context, variation in teaching and reality-related experience will be explained.

2.1 Everyday play activities

From life in memorial, games and other everyday activities have been a great way of developing mathematical knowledge and skills. It does not matter whether these everyday activities are conducted digitally or physically. One thing in common is that many of these activities use and build on mathematical abilities. According to Good and Ottley (2019), play and other everyday activities provides children with the opportunity to explore mathematical concepts, express mathematical knowledge, and see mathematical relationships.

In Swedish educational system, play is considered to be a very important element for children’s development and learning. According to Skolverket (2011), conscious use of play to promote the development and learning of each individual child is advised to be present in all school systems starting from preschool activities. This is so because play and enjoyment in learning, in all its various forms stimulate the imagination, insight, communication and the ability to think symbolically, as well as the ability to co-operate and solve problems.

2.2 Mathematics education in its cultural context

According to Bishop (1988), Mathematics in this context is therefore conceived of as a cultural product, which has developed as a result of various activities such as counting, locating, measuring, designing, playing and explaining. Below is the explanation of what each fundamental activity entails according to Bishop (1988).

Counting; The use of a systematic way to compare and order discrete phenomena. It may involve tallying, or using objects or string to record, or special number words or names. Locating; Exploring one’s spatial environment and conceptualizing and symbolizing that environment, with models, diagrams, drawings, words or other means.

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8 Measuring; Quantifying qualities for the purposes of comparison and ordering, using objects or tokens as measuring devices with associated units or ‘measure-words’.

Designing; Creating a shape or design for an object or for any part of one’s spatial environment. It may involve making the object, as a ‘mental template’, or symbolizing it in some

conventionalized way.

Playing; Devising and engaging in, games and pastimes, with more or less formalized rules that all players must abide by.

Explaining; Finding ways to account for the existence of phenomena, be they religious, animistic or scientific.

According to Bishop (1988), Mathematics as cultural knowledge derives from humans engaging in these six universal activities in a sustained and conscious manner. The activities can either be performed in a mutually exclusive way or, perhaps more significantly, by interacting together, as in ‘playing with numbers’ which is likely to have developed number patterns and magic squares, and which arguably contributed to the development of algebra. Bishop (1988) argues further that the six fundamental activities appear to be carried out by every cultural group ever studied, and are also necessary and sufficient for the development of mathematical knowledge.

2.3 Variation in teaching

According to Berggren and Lindroth (2011), variation in teaching is the common factor among schools whose pupils do not lose interest in mathematics. This is so because pupils receive varied teaching which is based on investigative and laboratory activities, where the teachers focus on problem solving and communication as this provides a good environment for pupils to

communicate and exchange experiences with each other. Berggren and Lindroth (2011) conclude that a varied teaching increases pupils' desire to learn.

2.4 Reality-related experience

Wistedt, Brattström and Jacobsson (1992) appear to be in the same line with Berggren and Lindroth (2011) as they raise the concept of experimental activities. This is the kind of teaching mathematics that focuses on laboratory material and the interaction between mathematics and practical school subjects such as art, home and consumer studies, physical education and health.

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9 The experimental activities yielded positive results and confirmed that pupils think that

mathematics is fun when there is collaboration not only between school subjects but also to everyday activities because it helps to increase their imagination and also it stimulates new ways for them to present different things.

According to Skolverket (2019), the purpose of teaching the subject of mathematics is that pupils should develop the knowledge about mathematics and the use of mathematics in everyday life and in various subject areas. The curriculum emphasizes further that through the teaching of mathematics, pupils should be given the opportunity to develop awareness that there are several different ways to solve problems (Skolverket 2019). When pupils feel confident in their ability to use mathematics in different contexts, they dare to test new methods that enable them to reflect on what they do and what the result will be. There is no doubt that when pupils are given the opportunity to formulate their own problems and solve them while working with richer

problems, their creativity develops to greater heights. This is indeed achievable if the teaching of mathematics is designed in a way where there is an interaction between theory perspective and sensory experience, which in turn contributes to harmonious development. Harmonic

development means that pupils are given so many opportunities as possible in order to try, explore and acquire knowledge in different ways.

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3. Purpose

The purpose of this thesis is to investigate how mathematics teaching that has everyday connections affects pupils' learning.

3.1 Research questions

På vilket sätt påverkas elevernas intresse av att matematikundervisningen knyts till vardagliga situationer?

Inom vilka områden anser eleverna att de använder matematik i vardagliga sammanhang?

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4. Theoretical perspective

This study is based on learning by doing theory of John Dewey (1859 – 1952) and Realistic Mathematics Education (RME) of Hans Freudenthal.

4. 1 Learning by doing - John Dewey (1859–1952)

According to Kroksmark, (2003), Dewey is considered to be a true representative of

philosophical pedagogy whose focus was on learning theories. Dewey believed that education was ‘a crucial ingredient in social and moral development.’ (Kroksmark, 2003). Dewey's main idea is that pupils can learn by doing. Furthermore, Dewey describes that teaching can be

enriching if teachers succeed in creating an enriching environment where pupils handle different materials (e.g. during concrete teaching) which favors their interaction with each other.

According to Dewey, teachers have the task of encouraging pupils with various activities that reinforce their knowledge. Dewey believed that laboratory materials lead to enriching

development.

4.2 Realistic Mathematics Education (RME) - Freudenthal.

According to Treffers and Beishuizen (1999), Realistic Mathematics Education (RME) involves a complete reversal of the teaching/learning processes than the common one which starts with the demonstration of a topic, followed by the consolidation through exercises and problem application towards the end. With RME, context problems are used as both a starting point (a route 'into' the mathematics) and the medium through which pupils develop understanding (a route 'through' the mathematics) where rich, “realistic” situations are given a prominent position in the teaching/learning process. These situations serve as a source for initiating the development of mathematical concepts, tools, and procedures and as a context in which pupils can in a later stage apply their mathematical knowledge, which then gradually has become more formal and general and less context specific. This relates strongly to Freudenthal's view that 'mathematics must be connected to reality; stay close to children, and be relevant to society in order to be of human value’ (Treffers & Beishuizen, 1999).

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5. Method

The curriculum (Skolverket, 2019), says among other things that;

Teaching in mathematics should aim at helping the pupils to develop knowledge of mathematics and its use in everyday life and in different subject areas. Teaching should help pupils to develop

their interest in mathematics and confidence in their own ability to use it in different contexts.

Based on the purpose of my research, I wanted to know how teachers teach according to the above goals and how such kind of teaching affects pupils' learning.

5.1 Selection and implementation

During the survey, data was collected from a grade 6 class of 22 pupils and 4 licensed

mathematics teachers who teach grade 1 to 6 (see appendices 1 and 2). The pupils were informed by their class teacher about how the survey would go, and that all the pupils and teachers

involved would be anonymous as outlined by The Swedish Research Council (2017). I also introduced myself and read aloud through all the survey questions (see appendix 1). The pupils were awarded an opportunity to ask questions or seek clarification where needed before they answered the questionnaire individually. I also encouraged them to answer the questions as honestly as possible so as to avoid drawing false conclusions from the study. The survey

questions were formulated in various ways to gather information that was sufficient to build this study. According to Dysthe (1996), using different questioning techniques is one of the most effective ways of gathering the intended information from the pupils. The first five questions in the pupils’ questionnaire are aimed at examining how mathematics teaching that has everyday connections affects their learning. Through the closing question, pupils are given the opportunity to state when they use mathematics in their everyday life (appendices 1). Teachers’ questionnaire has only five questions focusing among other things on how pupils' interest, understanding and learning is affected when mathematics lessons have everyday connections. All the questions are open questions except for question number one (1) which has multiple choice answers

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13 I decided to give a questionnaire to pupils in grade six because according to Skolverket (2019), pupils at this age (at the end of the sixth school year) have acquired mathematical knowledge that can enable them to describe, handle situations and solve concrete problems in their everyday life. They should thus also be able to use more of their mathematical knowledge both in school and outside school. I also believe that these pupils have reached a maturity stage in their personal development that they can take into account their mathematical learning in the present and future compared to grade one pupils. The grade six pupils can in some ways put more school work in relation to life outside the school.

Using questionnaires as the method for my survey provided all the participants with the same conditions rather than being influenced or led by the interviewer, which is mostly the case during interviews. During interviews, participants may be unknowingly influenced by the interviewer to provide the answers they believe are expected from them (Patel & Davidson, 2019). The other disadvantage of an interview is that it is difficult to avoid asking questions that are linked to an interview's outcome. Furthermore, the disadvantage of interviews is that interviewers have their own values and expectations from an interview hence making it difficult to avoid asking

questions that have yes or no answers. According to Johansson and Svedner (2004),

unknowingly expressing the interviewer’s own values and expectations when asking questions affects the interview’s outcome. These are some of the risks that I managed to eliminate because of using questionnaires only.

A further reason why I felt that questionnaires benefited me more than conducting interviews was that I wanted as much material as possible to study. By using questionnaires, my survey achieved a high degree of standardization as everyone answered the same questions under similar conditions (Trost, 2001). This helped me to reduce the risk of misinterpretation or mispronunciation. I consider the degree of structuring of the questionnaires to be at an average level as the response alternatives are both closed and open in nature. The term structuring can be used to describe details of the questions or answer options in a survey (Trost, 2001). The term structured is used when the question in a questionnaire has fixed answer alternatives. If the answer possibilities are open, then the question is unstructured. However, I realized that my method does not allow me to ask in-depth follow-up questions.

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5.2 Data Collection methods

The database used for collection of literature review of the theory and previous research in this study is the library’s database called Libsearch and Malmö University Electronic Publishing (MUEP). Libsearch and MUEP being the university's database ensures that the literature and articles obtained from there are reliable and of high quality of the material. The words used to find the literature and articles that fit in this study are Learning Mathematics through Everyday Play Activities, vardagsmatematik, laborativ matematik, undersökande arbetssätt and laborativt arbetssätt. The main tool used was the questionnaire for pupils and teachers. My research questions were aimed at examining teachers' and pupils' perceptions of the importance of everyday life connections in mathematics teaching. To investigate this, I needed to have contact with both teachers and pupils in order to gather broader information addressing my research questions. Through the pupils' answers I learnt about their experience of using mathematics in everyday life and how lessons connected to everyday life affect their interest, motivation, understanding and performance. Teachers showed that incorporating everyday life connections in mathematics lessons is a vital teaching methodology as it highlights different areas of learning which in turn arouses pupils' interest and also nuances their development.

5.3 Ethical considerations

The basis for research ethics is to strike a balance between the research requirement and the individual protection requirement. The research requirement means that researchers strive to build studies that have a high quality. It should be that research includes a clear purpose and information that helps to improve and develop society. It must be implemented in a way that must not offend people (The Swedish Research Council, 2017). In this work, both surveys were conducted according to the Swedish Research Council's research ethical principles (2017). The guardians of the pupils involved were first informed about the intended questionnaire. Both the pupils and teachers were also informed about the purpose of the questionnaire and that

participation was both voluntary and anonymous. It was also made clear to guardians, pupils and teachers that the information collected will be used only for the purpose of the study and that the forms will be destroyed according to the Malmö University's guidelines upon the completion of the study.

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6. Results

In this chapter, I will present the results of my survey. There are twenty (22) pupils and four (4) licensed mathematics teachers that took part in this study. The pupils are in grade six (6) and the teachers teach different grades from grade one (1) to six (6). The results are divided into two parts, i.e. pupils' results and teachers' results.

6.1 Pupils’ results

The pupils' questionnaire has six (6) questions. See appendix 1. The first question examines when best do pupils like mathematics lessons in relation to when lessons are connected to things they know or like. Out of 22 pupils, 17 (77%) answered that they like mathematics best when the lessons are connected to things they know or like, 2 (9%) answered that they like mathematics best when the lessons are not connected to things they know or like, 3 (14%) answered that it doesn't matter if the lessons are connected to things they know or like and 0 (0%) answered that he/she doesn’t know.

Figure 1. När gillar du matematiklektioner bäst?

The second question addresses motivation with regards to using laboratory materials to solve mathematics problems. Out of 22 pupils, 19 (86%) answered that they are more motivated to solve mathematics tasks when they use or work with materials or something that helps them solve the tasks, 1 (5%) answered that he/she is more motivated to solve math tasks when he/she is not using or working with materials or something that helps him/her solve the tasks, 2 (9%) answered that it doesn’t matter whether they are using or working with materials or something that helps them solve the tasks and 0 (0%) answered that he/she doesn’t know.

0 2 4 6 8 10 12 14 16 18 A nt al e le ve r Svarsalternativ

När lektioner är kopplade till saker som jag vet/gillar.

När lektioner inte är kopplade till saker som jag vet/gillar.

Det spelar ingen roll om lektionerna är kopplade till saker.

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Figure 2. När är du mer motiverad av att lösa matematikuppgifter?

The third questions addresses the moment when pupils lose their desire or motivation during mathematics lessons. Out of 22 pupils, 2 (9%) answered that they lose their desire or motivation when the lessons are connected to things they know or like, 19 (86%) answered that they lose the desire or motivation when the lessons are not connected to things they know or like, 0 (0%) answered that it doesn't matter if the lessons are connected to things they know or like and 1 (5%) answered that he/she doesn’t know.

Figure 3. När tappar du lusten eller motivation under matematiklektioner?

The purpose for the fourth question was to find out the moment when pupils feel that they learn better during mathematics lesson. On this question, all the pupils (22 - 100%) answered that they learn better when the teacher uses everyday connections to solve or explain mathematical

problems or concepts.

Figure 4. Jag lär mig bättre när läraren använder vardagliga samband för att lösa/förklara matematiska problem/begrepp.

0 2 4 6 8 10 12 14 16 18 20 An ta l e le ve r Svarsalternativ

När jag arbetar med material eller något som hjälper mig att lösa uppgifter.

När jag inte arbetar med material eller något som hjälper mig att lösa uppgifter.

Det spelar ingen roll om jag arbetar med material eller något som hjälper mig att lösa uppgifter. Vet ej. 0 2 4 6 8 10 12 14 16 18 20 An ta l e le ve r Svarsalternativ

När lektioner är kopplad till saker som jag vet/gillar.

När lektioner inte är kopplade till saker som jag vet / gillar.

Det spelar ingen roll om lektioner är kopplade till saker som jag vet/gillar. Vet ej. 0 5 10 15 20 25 An ta l e le ve r Svarsalternativ Det stämmer inte. vet ej. Det stämmer mycket bra.

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17 The fifth question was aimed at finding out if pupils use mathematics in their everyday life. On this question, all the pupils (22 - 100%) answered that they use mathematics in their everyday life.

Figure 5. Använder du matematik i vardagen?

The last question (sixth) was the only open question (without multiple choices) in the

questionnaire. With this question, I wanted to know exactly where the pupils use mathematics in their everyday life. Pupils had to write their own answers of which I have put them into groups i.e. shopping, leisure activities, household chores and general counting. I have also written down some of the pupils' answers, exactly the way they wrote them and in the original language - Swedish. (Some pupils wrote more than one activity otherwise there were only 22 pupils in class). From this question, out of 22 pupils, 13 (59%) answered that they use mathematics when doing some shopping, 10 (45%) answered that they use mathematics during leisure activities e.g. when playing football, when playing handball, when playing during breaks at school, when swimming, when taking a bus and when cycling to and from school, 8 (36%) answered that they use mathematics during house activities e.g. when baking, when cooking, when fixing coffee and 9 (41%) answered that they use mathematics during general counting e.g. when counting things like shoes, when counting time and random numbers as well as when helping siblings to do their homework.

Figure 6. När använder du matematik i vardagen? 0 5 10 15 20 25 An ta l e le ve r Svarsalternativ Nej. Ja. Vet ej. 0 2 4 6 8 10 12 14 A nt al e le ve r Svarsalternativ När jag handlar. Under fritidsaktiviteter. Under hushållssysslor. Under allmän räkning

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18 Below are some of the sentences which pupils themselves wrote in Swedish;

När använder du matematik i vardagen?

 På rasten när vi ska spela, då räknar jag hur många ska vara i varje grupp. Så ja!  När jag handlar, lagar mat och spelar.

 Jag brukar räkna ihop slumpmässiga tal ibland jag vet inte varför.  När jag ska handla och när min syrra behöver hjälp.

 När min simlärare säger att jag ska simma 25×4 då vet jag att det är 100 meter.  Till exempel, om jag ska handla något och det är rea så räknar jag ut hur mycket det

kostar.

 När jag spelar handboll.

 När jag handlar då kollar jag på priset. Om det står 9,90 kr, avrundar jag och då vet jag att priset är nästan 10 kr.

 När jag ska fixa kaffe, räknar jag hur många kaffeskedar jag ska lägga.  När jag handlar med min mamma.

 När jag räknar saker. Till exempel, skor.  När jag går till affären.

 Jag brukar räkna hur länge bullar ska vara i ugnen när jag bakar och hur många jag har bakat.

 När jag kanske räknar priset i affären.

 På fotboll. Till exempel,, spelar vi elva mot elva.  Om jag ska köpa något och kollar priser.

 När jag kollar tiden. Om nån säger att jag ska hem så kollar tiden så jag vet hur länge jag kommer ta.

 När jag ska åka buss, kollar jag tid när bussen kommer.

 Jag använder matte i butiker, hemma och utomhus när jag ska räkna hur många timmar tills jag ska hemma.

6.2 Teachers’ results

The teachers' questionnaire has five (5) questions. See appendix 2. All the questions are open questions except for question number one (1) which has multiple choice answers. The answers

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19 will be written in order of the grades teachers teach and in the original language (Swedish) i.e. exactly how the teachers themselves wrote. Grade 6 has two teachers who teach mathematics in different classes. I will name one as teacher A and the other as teacher B.

The first question is the only multiple choice question. Its purpose was to get the direct view of the teachers on whether or not they think that their pupils learn mathematics best when the lessons are connected to the things they already know or like. Out of 4 licensed mathematics teachers, 3 (75%) answered that their pupils learn mathematics best when the lessons are connected to things they know or like, 1 (25%) answered that it doesn't matter if the lessons are connected to the things pupils know or like, 0 (0%) answered that it doesn't matter if the lessons are connected to things they know / like and 0 (0%) answered that he/she doesn’t know.

Figure 7. På vilket sätt anser du att dina elever bäst lär sig matematik

The purpose for the second question was to find out the importance of linking mathematics teaching to everyday connections.

Grade 1 - 3

Det är viktigt att undervisningen har ett tydligt syfte, att eleven har förståelse/får förståelse för vikten av vad hen ska lära sig.

Grade 4 & 5

Det är viktigt att det handlar om vardagliga situationer, i alla fall ibland. Det blir lättare för eleverna att förstå och se samband.

Grade 6 Teacher A

När man kan koppla matematiken till vardagliga saker är det lättare för eleverna att förstå

meningen med att lära sig matematik. Det är viktigt för att de blir mer intresserade och har lättare att förstå. 0 1 2 3 4 A nt al e le ve r Svarsalternativ

När lektioner är kopplade till saker som de vet/gillar.

När lektioner inte är kopplad till saker som de vet/gillar.

Det spelar ingen roll om lektionerna är kopplade till saker som dem vet/gillar. Vet ej

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20 Teacher B

Det är viktigt att anknyta till vardagliga samband så de förstår hur de kan använda och ha nytta av matematiken.

The aim for the third question was to find out how pupils' interest, understanding and learning is affected when mathematics lessons have everyday connections.

Grade 1 – 3

Det kan bli tydligare för eleven om det kan knytas till verkligheten. Grade 4 & 5

Håller intresset uppe och gör att de lättare ser samband. Grade 6

Teacher A

Vardagliga samband gör matematiken mer konkret och eleverna känner igen situationer där de kan behöva den.

Teacher B

När man anknyter till saker/situationer du känner till förstår de bättre meningen med att lära sig matematiken.

The purpose for the fourth question was to find out ways in which teachers link mathematics to everyday connections in their teaching.

Grade 1 – 3 Buss tidtabeller Area

Avstånd Grade 4 & 5

Matematikproblem som hänger ihop med olika teman vi arbetar med eller aktuella händelser knyter dem till. t.ex studiebesök.

Ber exempel från deras vardag. Grade 6

Teacher A

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21 Startar ofta upp ett nytt område i det vardagliga.

Utgår från vardagliga samband under genomgångar. Teacher B

Vi har nyligen pratat om hastighet t.ex. om vi åker till Lund i 90 km/h. Hur lång tid tar det? (ca 20 km)

Enheter t.ex. volym enheter, anknyta till recept.

Längd enheter: steg är 10 m - hur många steg är det för mig? Stega sedan t.ex. skolgården, diagonalt till matsalen. Hur många meter är sträckan? Mät hur nära kom du?

With the last (fifth) question, I wanted to find out how teachers link mathematics teaching to everyday situations.

Grade 1 – 3

Gör det verklighetsanknutet och förklara syftet samt praktisk/konkret undervisning. Grade 4 & 5

Försöker ha intressanta, roliga och omväxlande lektioner. Försöker peppa och berömma och se så mycket individuellt stöd som möjligt.

Grade 6 Teacher A

För att alla ska lita på sin förmåga är det viktigt att individualisera. Jag försöker ha olika nivåer på uppgifter så att alla känner att de klarar av det de håller på med men också att det finns utmaningar till de som behöver det. För att skapa intresse försöker jag att variera undervisningen och prata mycket matematik t.ex. uppgifter som diskuteras i par. Vi arbetar med många olika moment så att man inte bara fastnar i att följa ett läromedel, matteboken uppfattas lätt som tråkig så det är viktigt att göra uppgifter utanför den.

Teacher B

Jag är själv intresserad av matematik och vill försöka förmedla skönheten och hur spännande det kan vara! Att försöka upptäcka mönster. Jag vill också försöka stärka elevernas självförtroende. Att försöka förmedla att det är möjligt för alla att komma in i matematikens värld.

Vi pratar en hel del om olika matematiska ’’kluringar’’ och diskutera hur man på olika sätt kan komma fram till lösningar. Att försöka förklara samband och olika tankesätt på ett så enkelt sätt som möjligt är en stor utmaning!

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7. Discussion and analysis

In this chapter, the above issues will be addressed through discussion and analysis of the results perspective. Each heading represents one of the problems addressed by the research questions.

7.1 Concretized teaching

Talking about concrete teaching, the results from the pupils’ and teachers’ questionnaire shows that pupils learn mathematics best when lessons are connected to things they already know or like (figure 4 & 7). Concrete teaching in mathematics can have different meanings. It can mean working with real things (laboratory materials) in investigative situations, explaining

mathematics concepts using real situations as well as using children’s leisure activities. Three things which are vivid according to the teachers who took part in this study are that;

1. Concrete teaching makes it easier for pupils to understand the meaning of learning mathematics.

2. Concrete teaching awakens pupils’ interest, desire and motivation.

3. Concrete teaching helps pupils to understand how they can use and benefit from mathematics.

Besides the above views, it is without doubt that concrete teaching helps pupils to understand mathematics more easily because they are able to visualize and practice the concepts. This can be aligned to Dewey’s learning by doing theory where pupils are able to practice the visualized concepts with the help of laboratory materials (Kroksmark, 2003). With the same view,

Freudenthal talks about the use of rich, “realistic” situations that can initiate the development of mathematical concepts where pupils can apply their mathematical knowledge (Treffers & Beishuizen, 1999). This makes it more effective as pupils develop a language for their own observations and experience. It is evident that through Dewey’s learning by doing theory and Freudenthal's RME, the child is perceived to be an active learner solving problems that are presented by the environment (both at school and outside school). It is without doubt that concrete objects influence pupils’ mental development because pupils are actively and

continuously thinking about different ways of solving problems presented to them (Kroksmark, 2003).

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7.2 Pupils' interest/motivation

According to the pupils results (question number 3), most of the pupils answered that they lose the desire or motivation during mathematics lessons if the lessons are not connected to the things they know or like (figure 3). The results of this study show also that most pupils like

mathematics best when lessons are connected to the things they know, have an idea of or like (figure 1). In my preamble (preface) I talked about how I have heard some pupils complain that mathematics is a difficult subject as well as how I have seen some pupils struggle to solve certain mathematics problems. It is through such kind of grumbling that affects pupils’ interest and motivation in mathematics. The oxford learners' dictionaries define interest as (1) the feeling that you have when you want to know or learn more about somebody/something and motivation as (2) the feeling of wanting to do something, especially something that involves hard work and effort. If pupils can see a connection between school assignments and the world outside the school, their inner motivation will be positively affected. It is without doubt that pupils’ interest and

motivation in mathematics can be affected either positively or negatively depending on the way the subject is taught. Back to Dewey (Kroksmark, 2003), it is indeed true that pupils’ interest and motivation rises when teachers encourage pupils with various activities that reinforce their knowledge. This is evident because the pupils who are motivated in mathematics learn mathematics best when lessons have a variety of everyday life connections (figure 2). This enables them to create different opportunities that increase their understanding. With such kind of teaching methodologies, the result is always that such pupils perform better and integrate into the community well by using their mathematical knowledge in different areas outside the school.

7.3 Learning environment

This study shows that pupils learn mathematics in different ways. Answering to the research question, Hur gör du för att undervisningen i matematik ska bidra till att eleverna utvecklar intresse för matematik och tilltro till sin förmåga att använda matematik i olika sammanhang? It is clear from the teachers who took part in this study that the learning environment plays a big role if the teaching of mathematics is to help pupils develop an interest in the subject and confidence in their ability to use it in different contexts. The teachers pointed out among other things that the use of laboratory materials, everyday life connections/activities as well as varied

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24 teaching makes the lessons to be reality-related, interesting and fun. This is one of the reasons why most teachers assume that their pupils learn mathematics best when lessons are connected to the things they know or like (figure 7). The teachers also pointed to the importance of

individualism, educational tours as well as using a variety of teaching materials -rather than following just a mathematics text book. By individualism, the teachers mean giving individual support to the pupils in need of it as well as to have/prepare different levels of tasks so that everyone feels they can handle what they are doing but also that there are also

challenging/difficult tasks for those who need them. I sum up all what the teachers have talked about into one teaching method called varied teaching as explained by Berggren and Lindroth (2011) in my introduction.

According to Boaler (2017), every pupil can acquire a dynamic mindset in mathematics with the right teaching. This is very true because during varied teaching (for example), pupils may be given the opportunity to argue and discuss mathematics. Pupils need something beyond theory to be able to connect reality with mathematics in different ways. Boaler (2011) argues that

mathematics teaching can be more effective when it manages to build relationships between pupils and with different forms like in nature. This is true because nature and other everyday life connections are needed so that pupils can analyze and make connections with reality using the tools they receive from the teaching. This can also lead to pupils getting a better understanding as well as acquire the mathematical knowledge which is beyond numbers and calculations. Think about educational tours which one of the teachers in this study mentioned (See what the grade 4 & 5 said to the question number 6.2.4).This can be one of the ways of linking mathematics to reality, by using nature and phenomena as complementary material to increase the understanding of mathematical concepts. By this it means that the connection to reality creates a feeling among pupils that mathematics is not only about numbers but an important subject that life cannot do without it. With this view, Dewey felt that the society and school is one entity and that it is impossible to separate the learning process from the society (Kroksmark, 2003). Not only that, but as pointed out by the teachers, creating a discursive atmosphere (talk a lot about mathematics e.g. tasks discussed in pairs/groups) in the classroom creates a greater understanding of

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7.4 Interactive teaching/ permissive learning climate

The results of this study show that connecting lessons to everyday life creates a permissive learning climate that makes pupils like mathematics (figure 1 and 7). This study shows also that a permissive or interactive learning environment can be created when pupils use or work with materials or something that helps them solve mathematics problems/tasks. This is evident because connecting mathematics to everyday life enables pupils to discuss mathematics openly. This leads to the exchange of experiences among the pupils which further enhances their understanding, interest and motivation (figure 2). According to Dysthe (1996), a good lesson is characterized by high pupil activity and interactive learning. Dysthe (1996) explains further that the best teaching takes place in a climate based on dialogue and social interaction among the pupils, as well as between the teacher and the pupils. If pupils are allowed to build their understanding of a subject (content) by listening to others and by putting words into their own knowledge, then a permissive learning climate can be developed in the class (Good & Ottley 2019). Teachers should involve pupils in the creation of various projects and have them make checklists with intermediate goals. By dividing the goals into smaller and more manageable parts, the pupils gain a sense of participation and control; while at the same time begin to perceive themselves as more competent.

7.5 Mathematics in different context

The results of this study show that pupils use mathematics in their everyday life and in different situational contexts in their everyday life e.g. shopping, leisure activities, household chores and general counting (figure 5 & 6). Wedege (1999) defines situational context as the context in which one uses, learns and discovers mathematics. With Wedge's definition in mind, it is vivid that the pupils involved in this study are aware of mathematics usage in different situational contexts than the usual school context hence it is important that mathematics teaching is connected to everyday life (figure 6). Continuing with the same view of mathematics in a different context, Bishop (1988), talks about mathematics education in its cultural context which is conceived of as a cultural product due to the result of involving various activities such as counting, locating, measuring, designing, playing and explaining.Boaler (2017) explains that, the school's mathematics teaching differs from the teaching in other subjects because of how the

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26 subject is taught which reinforces the view that only those pupils who are faster and can

remember all the rules and formulas are good at mathematics. This has contributed to pupils having a mindset or attitude that mathematics is a difficult subject or that it is only some pupils who are capable of more advanced mathematics. Skemp (1976) uses the concept of relational understanding when he talks about understanding why, how and what to do, while instrumental understanding is just having an understanding of how to do it. According to Skemp (1976), pupils with instrumental understanding learn formulas for different mathematical concepts but not why they should be used. This is a short-term learning process that becomes difficult to manage over time due to lack of everyday life connections. The risk is that these pupils will not like the subject and will lose interest and motivation because they are not able to know why some methods or formulas are used (figure 1 and 7). On the other hand, pupils with a relational understanding see why formulas exist. Such pupils get to try out ideas that make concepts visible in different ways by creating a relational understanding due to the connection of everyday life.

According to Boaler (2011), pupils who have difficulties in mathematics and who cannot use it outside of school also find it difficult to develop in other areas. Thus, pupils who perform poorly in mathematics can be linked to have lessons which lacking everyday life connections. In most cases this can lead to pupils losing interest in the subject or considering the subject to be difficult just because they do not have an idea or knowledge that connects lessons to things they already know or like. This is why Dewey argues that pupils will be motivated to do school tasks when they discover that there is a connecting in school tasks and the society as well as their future careers (Kroksmark, 2003). Dewey’s view on motivation is well aligned to the pupils who took part in this study. When asked on how motivated or unmotivated they are during mathematics lessons, most pupils answered that they lose interest or motivation when lessons are not connected to things they know or like (figure 2).

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8. Conclusion

My conclusion is based partly on the questionnaire that the pupils and teachers answered not forgetting the literature that deals with the subjects that have been raised. This study shows that everyday life connection is an important tool in awakening pupils' interest, understanding and motivation in mathematics as well as in answering the didactic questions that deal with how, where and why. Furthermore, this study indicates that experience is an essential part of the learning process because it enables the pupils to explore mathematical problems in a special way that facilitates their lives. It is vivid through this study that pupils learn mathematics best through experiences which creates contact between them and different environments both at school and outside school. It is through these different environments that pupils are given the opportunity to explore and develop mathematical concepts in different areas. The experiences pupils make of mathematics, at home, in the society at large and especially at school affects their attitude

towards mathematics hence with the right teaching, all pupils have better prerequisites to develop and become successful in mathematics. Connecting mathematics lessons to everyday life is indeed an important part of mathematics teaching and something all teachers need to develop more because of its cultural existence (as explained by Bishop, 1988), which may help to inculcate rich, “realistic” realization into the pupils (Treffers & Beishuizen, 1999). This is so because everyday life connections create different learning environments that lead to pupils' increased understanding which in turn enables them to develop their full potential within the subject.

According to Good and Ottley (2019), teachers who plan their lessons with the incorporation of everyday activities (play-based) can inspire conversations about mathematics while engaging pupils in different activities such as games as well as other everyday life activities that let them manipulate, count, and add tangible objects. Pupils need to experience mathematics in order to develop and achieve a higher creative level. Mathematics is not only about numbers and endless calculations but also about the world we live in as it explains many phenomena around us. I believe that pupils can understand mathematics better by experiencing/doing it. This will enable them to make contact between mathematics and society and the reality. The social environment and meaningful activities are important issues which Dewey argued for the child to develop his

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28 or her individual conditions with freedom under responsibility(Kroksmark, 2003).Through this connection, pupils feel that mathematics is also an important part of facilitating long life careers. The teachers who participated in this study point out to how important everyday life attachment can be used to increase pupils' understanding. If I compare the teachers’ result with the pupils’, I find that an understanding of mathematics also leads to the subject being perceived as fun. Through this, I can conclude that everyday attachment arouses pupils' interest, motivation, participation, understanding as well as performance in mathematics.

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9. Further research

In this study, I explored how mathematics lessons incorporated with everyday life connections increase pupils' interest, motivation, participation, understanding and performance as well as how these connections create different learning environments. All parts of the survey as well as the results and literature used in this study shows different opportunities for how mathematics lessons incorporated with everyday life can increase pupils' understanding and creating their own self-confidence while working with all parts of mathematics.

This study covers a wider range of everyday life connections including those given by the pupils themselves (figure 6). Suggested further research can be to shorten this study and categorically look at how each everyday life connection named by the pupils i.e. shopping, leisure activities, household chores and general counting (from figure 6) can potentially give pupils concrete mathematics education having the same focus of integrating mathematics with reality.

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10. References

Ahlström, R. (1996). Matematik - ett kommunikationsämne. Mölndal: Institutionen för ämnesdidaktik, Göteborgs universitet. ISBN 91-88450-06-6 Upplaga 1:23.

Berggren, P,. & Lindroth, M. (2011). Laborativ matematik: för en varierad undervisning. 1. uppl. Uppsala: JL utbildning. ISBN 9789197777155.

Boaler, J. (2011). Elefanten i klassrummet- att hjälpa elever till ett lustfyllt lärande i matematik. Stockholm: Liber. ISBN: 9789147099719.

Bishop, A. J. (1988). Mathematics education in its cultural context. Educational Studies in Mathematics. 19(2):179-191. https://link.springer.com/article/10.1007/BF00751231

Boaler, J. (2017). Matematik med dynamiskt mindset : hur du frigör dina elevers potential. Natur & kultur. ISBN 9789127817906.

Dysthe, O. (1996). Det flerstämmiga klassrummet. Lund: Studentlitteratur. ISBN: 91-44-61631-7.

Good, S.C., & Ottley J. R. (2019). Learning Mathematics through Everyday Play Activities. YC Young Children; Washington Vol. 74, Iss. 3, (July, 2019): 73-78.

Johansson, B., & Svedner, P-O. (2004). Examensarbetet i lärarutbildningen- Undersökningsmetoder och språklig utformning. Uppsala: Kunskapsföretaget.

Kroksmark, T. (2003). Den den tidlösa pedagogiken (red). Lund: Studentlitteratur. ISBN 914401564X.

Malmer, G. (1999). Bra matematik för alla (1. ed.). Lund: Studenlitteratur AB. ISBN 9789144012872.

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31 Oxford Learner's Dictionaries.

https://www.oxfordlearnersdictionaries.com/definition/english/interest_1?q=interest Oxford Learner's Dictionaries.

https://www.oxfordlearnersdictionaries.com/definition/english/motivation?q=motivation Patel, R,. & Davidson, B. (2019). Forskningsmetodikens grunder: att planera, genomföra och rapportera en undersökning. Lund: Studentlitteratur. ISBN 9789144126050.

Skemp, R. (1976). Relational and Instrumental Understanding Mathematics Teaching, Bulletin of the Association of Teachers of Mathematics, 77, 20-26.

Skolverket (2011). Curriculum for the compulsory school, preschool class and school-age educare. Revised 2018. Stockholm: Skolverket.

Skolverket (2019). Läroplan för grundskolan, förskoleklassen och fritidshemmet: Stockholm: Skolverket.

The Swedish research council (2017). Good research practice. Stockholm: Vetenskapsrådet. https://www.vr.se/download/18.5639980c162791bbfe697882/1555334908942/Good-Research-Practice_VR_2017.pdf

Treffers, A., & Beishuizen, M, (1999). Realistic Mathematics Education in the Netherlands. In Thompson I. (ed.). Issues in teaching numeracy in primary schools. Buckingham: Open University Press.

Trost, J. (2001). Enkätboken. Lund: Studentlitteratur AB. ISBN: 9789144115450.

Wedege. T. (1999). To know or not to know – mathematics, that is a question of context. Educational Studies in Mathematics, 39 (1-3), 205-207.

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32 Wistedt, I. (1992). Att vardagsanknyta matematikundervisningen. Slutrapport från projektet vardagskunskaper och skolmatematik. 144 s. Stockholms universitet: Pedagogisk institutionen.

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11. Appendix

11.1 Appendix 1 – elevers enkät

1. När gillar du matematiklektioner bäst?

När lektioner är kopplade till saker som jag vet/gillar. När lektioner inte är kopplade till saker som jag vet/gillar.

Det spelar ingen roll om lektionerna är kopplade till saker som jag vet/gillar. Vet ej

2. När är du mer motiverad av att lösa matematikuppgifter?

När jag arbetar med material eller något som hjälper mig att lösa uppgifter. När jag inte arbetar med material eller något som hjälper mig att lösa uppgifter. Det spelar ingen roll om jag arbetar med material eller något som hjälper mig att lösa uppgifter.

Vet ej

3. När tappar du lusten eller motivation under matematiklektioner? När lektioner är kopplad till saker som jag vet/gillar.

När lektioner inte är kopplade till saker som jag vet / gillar.

Det spelar ingen roll om lektioner är kopplade till saker som jag vet/gillar. Vet ej

4. Jag lär mig bättre när läraren använder vardagliga samband för att lösa/förklara matematiska problem/begrepp.

Det stämmer inte. Vet ej. Det stämmer mycket bra. 5. Använder du matematik i vardagen?

Nej Ja Vet ej 6. När använder du matematik i vardagen?

________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

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11.2 Appendix 2 – lärares enkät

Årskurs _____

1. På vilket sätt anser du att dina elever bäst lär sig matematik? När lektioner är kopplade till saker som dem vet/gillar. När lektioner inte är kopplad till saker som dem vet/gillar.

Det spelar ingen roll om lektionerna är kopplade till saker som dem vet/gillar. Vet ej

2. Hur viktigt tycker du att det är att matematikundervisningen har vardagliga samband? Kan du även förklara varför du tycker det är viktigt?

________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

3. På vilket sätt påverkas elevernas intresse, förståelse och lärande när matematikundervisning har vardagliga samband?

________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

4. Skriv tre sätt på hur du i din undervisning kopplar matematik till vardagliga samband? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 5. Hur gör du för att undervisningen i matematik ska bidra till att eleverna utvecklar intresse

för matematik och tilltro till sin förmåga att använda matematik i olika sammanhang? ________________________________________________________________________ ________________________________________________________________________

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11.3 Appendix 3 – Samtycke för vårdnadshavare i årskurs

6A

Datum:____________________

Elevens namn:____________________________ Förälders namnteckning:___________________

Natur, matematik och samhälle

Hej,

Jag heter Kondwelan James Tembo och studerar vid Malmö universitet under utländska lärares och akademikers vidareutbildning (ULV). Detta är min sista termin och jag skriver mitt

examensarbete i matematik. På måndag den 30 mars kommer jag i ditt/dina barns klass för att göra en undersökning om hur matematikundervisning som har vardagliga samband påverkar elevernas lärande. Undersökningen genomförs genom enkäter som kommer att besvaras av både elever och lärare. Jag vill betona att skolan, lärare och elever alla är anonyma och att resultaten endast kommer användas för detta ändamål. Svara senast 20.02.27

Om du som elev inte vill delta i undersökning, sätt kryss i rutan Om målsman inte vill att elev deltar i undersökning, sätt kryss i rutan

Studentens underskrift: __________________

Studentens kontaktuppgifter: 0737 252 844, kjtembo@gmail.com

Ansvarig handledare på Malmö universitet: Nils Ekelund, nils.ekelund@mau.se

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11.4 Appendix 4 – Behandling av personuppgifter

Personuppgiftsansvarig Malmö universitet

Dataskyddsombud dataskyddsombud@mau.se

Typ av personuppgifter Namn, anteckning av lärandesituation, bild och/eller filmklipp samt ditt samtycke till att Malmö universitet behandlar dessa personuppgifter.

Ändamål med behandlingen För att möjliggöra undervisnings- och examinations-situationer i skolmiljö för studenter vid Malmö universitets lärarutbildning.

Rättslig grund för behandling Ditt samtycke.

Mottagare Personuppgifterna kommer endast användas i

utbildningssyfte inom ramen för lärarutbildningen vid Malmö universitet och kommer inte att spridas vidare till någon annan mottagare.

Lagringstid Malmö universitet kommer spara dina personuppgifter så länge de behövs för ovan angivet ändamål eller till dess att du återkallar ditt samtycke. Efter genomförd kurs/program kommer personuppgifterna att raderas. Malmö universitet kan dock i vissa fall bli skyldiga att arkivera och spara

personuppgifter enligt Arkivlagen och Riksarkivets föreskrifter.

Dina rättigheter Du har rätt att kontakta Malmö universitet för att 1) få

information om vilka uppgifter Malmö universitet har om dig och 2) begära rättelse av dina uppgifter. Vidare, och under de förutsättningar som närmare anges i

dataskyddslagstiftningen, har du rätt att 3) begära radering av dina uppgifter, 4) begära en överföring av dina uppgifter (dataportabilitet), eller 5) begära att Malmö universitet begränsar behandlingen av dina uppgifter. När Malmö universitet behandlar personuppgifter med stöd av ditt samtycke, har du rätt att när som helst återkalla ditt samtycke genom skriftligt meddelande till Malmö universitet. Du har rätt att inge klagomål om Malmö universitets behandling av dina personuppgifter genom att kontakta Datainspektionen, Box 8114, 104 20 Stockholm.