I argue that common beliefs regarding mathematics originate in the practices of elementary mathematics instruction rather than in science. While learning how to solve mathematical problems in school, we come to believe that mathematics has a set of properties in itself, for instance that it is useful in everyday life, even though this is not necessarily so. I call this object of belief the mathematics of schooling (skolans matematik), while the system of practices by which the belief is produced is called mathematics education (skolmatematik). I introduce a terminology inspired by psychoanalytic theory to describe the peculiar properties of the mathematics of schooling and suggest that it can be understood as a sublime object of an ideology propagated by the system of education. To substantiate this claim I give an overview of the history of Swedish mathematics education, based on curricula, textbooks, discussions in teacher magazines, and other published material – covering in general terms the eighteenth to the twenty-first century. The historical narrative moves between social factors determining the practices of mathematics education, the changing ideas about mathematics expressed in these contexts, and the interplay between external social factors and internal “ideological” meaning. My conclusion is that while elementary arithmetics is, and should be, a part of common knowledge, the mathematics of schooling is something quite different. This object is thoroughly ideological and plays a central part in society mainly by making the social effects of mathematics education – keeping children away from production while sorting them – to appear as something else, namely as most often failed attempts to give children a necessary knowledge of mathematics.

<p>Abstract</p><p>The purpose of this essay is to investigate if elementary school pupils’ attitudes towards mathematics, from a teacher’s perspective, have changed since 2001-2002. I also intend to investigate if the whys and wherefores among the pupils attitude are the same and what influences pupils’ attitude to mathematics. The reason why I chose to deepen myself into elementary school pupils’ attitude to mathematics, and things that relate, is that I find it very interesting. As I shall soon be an upper school mathematics teacher I hope to discover some crucial factors for making mathematics learning an inspiring adventure.</p><p>To collect materials for this study I interviewed six teachers living in the midst of Sweden. To create a picture of elementary school pupils’ attitude to mathematics I interviewed two teachers from each lower, intermediary and upper school.</p><p>The study shows that the teachers I interviewed have observed an attitude towards mathematics similar to the trend for the whole country. A higher level of abstraction and increasing difficulties to diversify teaching is said to be the reason. The study also shows that variation and the need to succeed are two important factors that affect pupils’ relation to mathematics. It is however, hard to make any generalization from my study, as it is too small.</p><p>Key words: attitudes, mathematics, pupils, crucial factors</p>

<p>ABSTRACT</p><p>The purpose of this essay is first to get a general idea about a group of working teachers thoughts of mathematics and the teaching in mathematics. I also would like to find out what they think affect pupils relationship to the subject. The surveys main purpose is to compare teachers in grade 4-6 and teachers in grade 7-9 view of what you as a teacher can do to further the learning in mathematics. I have done eight interviews. Four of them are with teachers in grade 4-6 and the other four interviews are with teachers in grade 7-9.</p><p>Concerning the result of my examination it is difficult to come to a general conclusion. It is also difficult to generalise the answers I got but I think that many teachers can know in oneself minds and valuations about mathematics and how the subject could be teach. I would also like to declare that the conclusion I have made only concerns my respondents, which means that they not can be compared with another or bigger selection group.</p><p>The result of my examination have shown that the views of what you can do as a teacher to further pupils learning in mathematics are separated in some ways. Both groups think that the variation in teaching is central for the motivation of learning. The respondents that teach in grade 4-6 say that it is important for the pupils to see the meaning of mathematics in ordinary situations. The teachers in grade 7-9 mean that it is central to build up the pupils</p><p>self-confidence and their trust in themselves in the subject of mathematics. Nowadays it is the textbooks that roles the teaching in all grades. All respondents say that it is also important to work without the textbook. The turn-over in the practice are different. According to the teachers the pupils relationship to mathematics are both positive and negative. The respondents that teach in grade 4-6 say that their pupils motivation lack when the formal and abstract thinking is introduced. The respondents that teach in grade 7-9 think that the pupils meanings differs but the lack of motivation shows when their pupils begin in grade 7.</p><p>My personal opinion about the subject is that to many pupils create a negative attitude to mathematics in early stages of life. As a future teacher my question is how you should form your teaching to further the pupils learning and interests in mathematics.</p>

Abstract In this dissertation the most common methods of solution, in which The Rule of Three is included, are presented. The Rule of Three tasks have been considered in various manners during the centuries. This dissertation examines how The Rule of Three, as a rhetorical method of calculus, is presented in Swedish textbooks from the 17th century till the beginning of the 20th century. Moreover, a comparative analysis, of The Rule of Three tasks and their solutions, is presented. In addition, a variety of other rules are presented and treated as examples of rhetorical mathematics. Similar to The Rule of Three, these rules are built on proportionality. The tradition that Björk started in using proportionality was continued by Celsius, Beckmarck and Forsell and later by Björling and Zweigbergk. Zweigbergk’s textbook is excellent from a mathematical as well as from a pedagogical point-of-view. In Zweigbergks’s account of Regula de Tri the emphasis is on comprehension and mechanical skills. The results of this thesis show that the textbooks can be divided into three groups, based on their description of The Rule of Three. Group: 1. Include the (textbook) authors who strongly emphasize algorithms and mechanical arithmetics. 2. Include the (textbook) authors who build their descriptions of The Rule of Three on the theory of proportion. 3. Include the (textbook) authors who solve The Rule of Three problems through going back to the unit.

Abstract The purpose of this essay is to investigate if elementary school pupils’ attitudes towards mathematics, from a teacher’s perspective, have changed since 2001-2002. I also intend to investigate if the whys and wherefores among the pupils attitude are the same and what influences pupils’ attitude to mathematics. The reason why I chose to deepen myself into elementary school pupils’ attitude to mathematics, and things that relate, is that I find it very interesting. As I shall soon be an upper school mathematics teacher I hope to discover some crucial factors for making mathematics learning an inspiring adventure. To collect materials for this study I interviewed six teachers living in the midst of Sweden. To create a picture of elementary school pupils’ attitude to mathematics I interviewed two teachers from each lower, intermediary and upper school. The study shows that the teachers I interviewed have observed an attitude towards mathematics similar to the trend for the whole country. A higher level of abstraction and increasing difficulties to diversify teaching is said to be the reason. The study also shows that variation and the need to succeed are two important factors that affect pupils’ relation to mathematics. It is however, hard to make any generalization from my study, as it is too small. Key words: attitudes, mathematics, pupils, crucial factors

ABSTRACT The purpose of this essay is first to get a general idea about a group of working teachers thoughts of mathematics and the teaching in mathematics. I also would like to find out what they think affect pupils relationship to the subject. The surveys main purpose is to compare teachers in grade 4-6 and teachers in grade 7-9 view of what you as a teacher can do to further the learning in mathematics. I have done eight interviews. Four of them are with teachers in grade 4-6 and the other four interviews are with teachers in grade 7-9. Concerning the result of my examination it is difficult to come to a general conclusion. It is also difficult to generalise the answers I got but I think that many teachers can know in oneself minds and valuations about mathematics and how the subject could be teach. I would also like to declare that the conclusion I have made only concerns my respondents, which means that they not can be compared with another or bigger selection group. The result of my examination have shown that the views of what you can do as a teacher to further pupils learning in mathematics are separated in some ways. Both groups think that the variation in teaching is central for the motivation of learning. The respondents that teach in grade 4-6 say that it is important for the pupils to see the meaning of mathematics in ordinary situations. The teachers in grade 7-9 mean that it is central to build up the pupils self-confidence and their trust in themselves in the subject of mathematics. Nowadays it is the textbooks that roles the teaching in all grades. All respondents say that it is also important to work without the textbook. The turn-over in the practice are different. According to the teachers the pupils relationship to mathematics are both positive and negative. The respondents that teach in grade 4-6 say that their pupils motivation lack when the formal and abstract thinking is introduced. The respondents that teach in grade 7-9 think that the pupils meanings differs but the lack of motivation shows when their pupils begin in grade 7. My personal opinion about the subject is that to many pupils create a negative attitude to mathematics in early stages of life. As a future teacher my question is how you should form your teaching to further the pupils learning and interests in mathematics.

Syftet med denna uppsats är att undersöka hur lärare ser på det konkreta arbetets betydelse för elevers taluppfattning i årskurs 1, samt att undersöka om det finns någon skillnad i elevernas taluppfattning i två klasser som arbetat olika mycket konkret inom matematiken den första tiden i grundskolan. Den metod som använts är kvalitativa intervjuer, samt en undersökning av elevers taluppfattning i två olika klasser. Intervjuerna visade att samtliga intervjuade lärare använde sig av konkret arbete för att eleverna ska få en god taluppfattning. Undersökningen i klasserna visade att den klass som arbetat mindre konkret inom matematiken presterade bättre på den skriftliga diagnosen, medan elever från båda klasserna förklarade på ett bra sätt med hjälp av ett material hur de löser en uppgift. En slutsats som dragits av denna undersökning är att elever kan lära på olika sätt, och att lärare tycker att det konkreta arbetet i matematik är viktigt för att ge eleverna en god taluppfattning.

<p>Abstract</p><p>In this dissertation the most common methods of solution, in which</p><p>The Rule of Three is included, are presented. The Rule of Three tasks</p><p>have been considered in various manners during the centuries. This</p><p>dissertation examines how The Rule of Three, as a rhetorical method</p><p>of calculus, is presented in Swedish textbooks from the 17th century</p><p>till the beginning of the 20th century. Moreover, a comparative</p><p>analysis, of The Rule of Three tasks and their solutions, is presented.</p><p>In addition, a variety of other rules are presented and treated as</p><p>examples of rhetorical mathematics. Similar to The Rule of Three,</p><p>these rules are built on proportionality.</p><p>The tradition that Björk started in using proportionality was continued</p><p>by Celsius, Beckmarck and Forsell and later by Björling and</p><p>Zweigbergk. Zweigbergk’s textbook is excellent from a mathematical</p><p>as well as from a pedagogical point-of-view. In Zweigbergks’s</p><p>account of Regula de Tri the emphasis is on comprehension and</p><p>mechanical skills.</p><p>The results of this thesis show that the textbooks can be divided into</p><p>three groups, based on their description of The Rule of Three. Group:</p><p>1. Include the (textbook) authors who strongly emphasize</p><p>algorithms and mechanical arithmetics.</p><p>2. Include the (textbook) authors who build their descriptions of</p><p>The Rule of Three on the theory of proportion.</p><p>3. Include the (textbook) authors who solve The Rule of Three</p><p>problems through going back to the unit.</p>

Syftet med arbetet var att främja barns lärande i matematik genom att skapa lustfyllda aktiviteter som hade sin grund i lek och estetik. Vi använde oss av olika forskningsmetoder såsom observationer, loggböcker och intervjuer för att beskriva resultatet. Arbetet hade en teoretisk bakgrund där representerades läroplaneras kunskapsmål i matematik vi beskrev även forskningar kring matematikdidaktik och lekteorier med koppling till estetik. Resultat delen omfattade tre cyklar där vi hade observerat, intervjuat, planerat aktiviteter, genomfört aktiviteterna, reflekterat över och dokumenterat barns lärande process och utveckling. I skolan hade vi arbetat med räknesätten och multiplikation medan i förkolan handlade arbetet om taluppfattning och geometrisk former. Resultat redovisningen belyste hur matematik undervisning kräver en lustfylld atmosfär för utveckling och lärande. Barnen/eleverna under arbetets gång utvecklade sitt matematiska tänkande med lust, intresse och engagemang. Att variera och åstadkomma lustfyllda arbetsätten med hänsyn till inlärarans intresse krävs i arbetet med matematik.

Problemlösning kan beskrivas på olika sätt och ges olika betydelser. Problemlösning är en viktig del av matematiken och matematikämnet i skolan. Ordet har positiva konnotationer. Det ses som viktigt, eftersträvansvärt och önskvärt att kunna lösa problem. Varför är det så och vilken substans har ordet problemlösning? Genom att tolka vad som sägs och hur det diskuteras kring och skrivs om problemlösning kan ordet ges andra innebörder än bara det sagda eller skrivna. I denna uppsats synliggör jag och ger exempel på olika sätt att se på och beskriva området problemlösning i matematikämnet i grundskolan. Jag synliggör också vilka spår av lärandeteorier jag kan se i beskrivningarna av problemlösning. Studien bygger på litteraturstudier. Undersökningsmaterialet är litteratur inom matematikämnets didaktik och läromedel i matematik i grundskolan.