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.

<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 för detta examensarbete är att beskriva hur matematikkunskap i termer avpraxeologier utvecklas och får liv hos eleverna in en undervisningssituation. Studien ärinspirerad av aktionsforskning och beskriver ett försök att introducera en lösningsteknik,kallad proportionsmetoden, i samband med undervisning inom området procent i kurs 1apå gymnasiet. Proportionsmetoden grundar sig på proportionalitetsbegreppet ochproportionella ekvationer. Huvudfrågan i denna fallstudie är vilka typer avlösningstekniker grundade på proportionalitet bland de som var tillgängliga för elever ärrepresenterade i elevlösningar av uppgifter som handlar om procent. Resultatenanalyseras med utgångspunkt i teorin om den didaktiska transpositionen av kunskap därden kunskap som undervisas sammanställs mot den kunskap som faktisk lärs av eleverna.Under denna studie har det visat sig att proportionsmetod är inte oftast förekommande ielevlösningar och att undervisningen av metoden kräver längre tid för att kunna integrerasi olika områden i matematik.

Proportionalitet är ett centralt begrepp i skolmatematiken. Begreppet introduceras i de lägre stadierna och återkommer i så gott som samtliga kurser från årskurs 9 tillsista kursen på gymnasiet. Det övergripande syftet med denna studie är att undersöka hur det matematiska begreppet proportionalitet hanteras i den svenska gymnasieskolan. En generell problematik kopplad till detta syfte är hur skolans styrdokument realiseras i läromedel och nationella prov. Fokus för denna avhandling har varit hur proportionalitet hanteras i det svenska gymnasiet i kursen Matematik A i några läromedel och nationella prov. För att undersöka detta utvecklades ett analysverktyg utifrån det teoretiska ramverket i ATD (Anthropological Theory of the Didactic). Av intresse är här relationer mellan de olika nivåerna i den didaktiska transpositionen, som berör just hur skolans styrdokument realiseras i läromedel och nationella prov. För det empiriska studiet av materialet användes från ATD begreppet matematisk organisation, genom att använda ett analysverktyg för att granska typer av uppgifter om proportionalitet, lösningstekniker och teoretiska modeller för proportionalitetsbegreppet. De data som presenterats i denna studie ger en ganska ostrukturerad bild av de matematiska organisationer av begreppsområdet proportionalitet som presenteras i läromedel och i nationella prov och de ser även olika ut när det gäller hur proportionalitet hanteras i läromedlen respektive det nationella provet för Matematik A. Resultatet visar att ungefär var fjärde uppgift i de studerade kapitlen och de nationella proven berör proportionalitet men att begreppet hanteras ensidigt vad avser uppgiftstyp. Skillnader observerades mellan läromedel och nationella prov när det gäller hur lösningstekniker rekommenderas för olika typer av proportionalitetsuppgifter. De två teoretiska modeller för proportionalitet som har undersökts, dvs. statisk och dynamisk proportionalitet, finns representerade i ungefär lika omfattning i både läromedel och nationella prov. Vid uppgifter inom geometri handlar det dock ofta om statisk proportionalitet medan det inomområdet funktioner är vanligare att använda dynamisk proportionalitet. Lärare bör få kunskap om skillnader mellan läromedel och läroplaner, och hur dessa tolkas i nationella prov, så att de i sin verksamhet kan välja det undervisningsinnehåll, inklusive övningsuppgifter, som ger en god variation för eleven kopplat till kursplanernas mål / Proportionality is a key concept in school mathematics. It is introduced in the primary grades, and reappears in almost all mathematics courses from Grade 9 to the last course in upper secondary school. The overall aim of this study is to investigate how the mathematical concept of proportionality is handled in the Swedish upper secondary school. A general problem connected to this end is how the national curriculum is realised in textbooks and national examinations. The focus of this thesis is on how proportionality is handled in the first Swedish upper secondary course in mathematics in some textbooks and national examinations. To examine this an analysis tool based on the theoretical framework of the ATD (Anthropological Theory of the Didactic) was developed. Of interest here are relations between the different levels in the didactic transposition, concerned with exactly how the national curriculum is realised in textbooks and national examinations. For the empirical study of the material the ATD concept of mathematical organisation was used, employing an analytical tool to examine the types of tasks dealing with proportionality, techniques for solving these tasks, and theoretical models for the concept of proportionality. The data presented in this study gives a fairly unstructured picture of the mathematical organisations of the conceptual field of proportionality, as presented in textbooks and in the national tests. They also look different when it comes to how proportionality is handled in the textbooks and the national test for "Matematik A". The result shows that about every fourth task of the chapters and the national tests studied involved proportionality, but that there was a low variation in terms of types of tasks. Differences were observed between textbooks and national tests in terms of how solution techniques are recommended for different types of proportionality tasks. The two theoretical models of proportionality that were studied, ie. static and dynamic proportionality, are represented to approximately the same extent in both textbooks and national examinations. In geometry, it is often static proportionality, while in the field of functions it is common to use dynamic proportionality. Teachers should have access to knowledge of the differences between textbooks and curricula, and how they are interpreted in the national tests, so that they can make deliberate choices in their teaching activities, including exercises, to support a good variety for students linked to curriculum objectives.