In 1911, mathematician Max Dehn posed three decision problems for finitely presented groups that have remained central to the study of combinatorial group theory. His work provided the foundation for geometric group theory, which aims to analyze groups using the topological and geometric properties of the spaces they act on. In this thesis, we study group actions on Cayley graphs and the Farey tree. We prove that a group has a solvable word problem if and only if its associated Cayley graph is constructible. Moreover, we prove that a group is finitely generated if and only if it acts geometrically on a proper path-connected metric space. As an example, we show that SL(2, Z) is finitely generated by proving that it acts geometrically on the Farey tree.

I takt med att den svenska skolan blir allt mer mångkulturell ökar behovet av inkluderande undervisning. Flerspråkiga elever möter både text och tal i matematikundervisningen, vilket ställer krav på läs- och matematikförmåga. Denna studie undersöker hur flerspråkiga elever upplever kontexten i textbaserade problemlösningsuppgifter, med fokus på hinder i processen och hur stöd kan hittas i uppgiftskontexten. Pólyas problemlösningsmodell användes för att identifiera var problem kan uppstå, och medieringstriangeln användes för att identifiera mönster i hur elever finner stöd. Studien använder en kvalitativ metod med semistrukturerade intervjuer och problemlösningsuppgifter som komplement. Resultatet visade att eleverna hade svårigheter vid övergången mellan tolkningsfasen och planeringsfasen. Under granskningsfasen hade eleverna svårt att motivera sina val. Eleverna fann stöd i uppgifter med mindre komplex kontext, där de lättare kunde urskilja matematiska relationsbegrepp och symboler. Intresset för kontexten spelade också en viktig roll för deras problemlösningsförmåga. / As the Swedish school system becomes increasingly multicultural, the need for inclusive education is growing. Multilingual students encounter both text and numbers in mathematics education, which demands both reading and mathematical ability. This study examined how multilingual students experience the context of text-based problem-solving tasks, focusing on obstacles in the process and how support can be found in the task context. Pólya's problem-solving model was used to identify where problems may arise, and the mediation triangle was employed to identify patterns in how students find support. The study employed a qualitative method with semi-structured interviews and supplementary problem-solving tasks. The results showed that students struggled with the transition between the interpretation phase and the planning phase. During the review phase, students had difficulty justifying their choices. Students found support in tasks with less complex contexts, where they could more easily discern mathematical relational concepts and symbols. Interest in the context also played a significant role in their problem-solving ability.

In this thesis we prove the following: The semigroup which is a disjoint union of two or three copies of a group is a Clifford semigroup, Rees matrix semigroup or a combination between a Rees matrix semigroup and a group. Furthermore, the semigroup which is a disjoint union of finitely many copies of a finitely presented (residually finite) group is finitely presented (residually finite) semigroup. The constructions of the semigroup which is a disjoint union of two copies of the free monogenic semigroup are parallel to the constructions of the semigroup which is a disjoint union of two copies of a group, i.e. such a semigroup is Clifford (strong semilattice of groups) or Rees matrix semigroup. However, the semigroup which is a disjoint union of three copies of the free monogenic semigroup is not just a strong semillatice of semigroups, Rees matrix semigroup or combination between a Rees matrix semigroup and a semigroup, but there are two more semigroups which do not arise from the constructions of the semigroup which is a disjoint union of three copies of a group. We also classify semigroups which are disjoint unions of two or three copies of the free monogenic semigroup. There are three types of semigroups which are unions of two copies of the free monogenic semigroup and nine types of semigroups which are unions of three copies of the free monogenic semigroup. For each type of such semigroups we exhibit a presentation defining semigroups of this type. The semigroup which is a disjoint union of finitely many copies of the free monogenic semigroup is finitely presented, residually finite, hopfian, has soluble word problem and has soluble subsemigroup membership problem.

Pupils' difficulties in solving selected word problems from TIMSS research. (Diploma Thesis.) Abstract The theoretical part of the diploma thesis describes international comparative surveys, namely PISA and TIMSS, and analyses results of Czech pupils. Some areas are distinguished in which our pupils were unsuccessful and from them, the area of word problems and their mathematisation was selected for further work. Next, a solving strategy is characterised and some relevant research from this area is given. The core of the work lies in the experimental part whose goal was to find out what strategies pupils use when solving selected problems from TIMSS research and why they fail in them, via the analysis of pupils' written solutions complemented by interviews with them. Causes of failure of our pupils in these problems in TIMSS 2007 are looked for in mistakes pupils make, while it is also followed in what phase of the solving process they appear. The participants of research were pupils of Grade 9 of a primary school who solved three selected word problems from TIMSS research. Their written solutions were complemented by interviews with the experimenter focused on their mistakes and lack of clarity of the solutions. Four pupils participated in a pilot study. The atomic analysis of their solutions confirmed...

The purpose of this study was to investigate sixth grade students&rsquo / conceptual and procedural knowledge and word problem solving skills in the domain of length, area, and volume measurement with respect to gender, previous mathematics achievement, and use of materials. Through the Conceptual Knowledge test (CKT), the Procedural Knowledge Test (PKT), and the Word Problems test (WPT) and the Student Questionnaire, the data were collected from 445 sixth grade students attending public schools located in four different main districts of Ankara. Both descriptive and inferential statistics techniques (MANOVA) were used for the data analysis. The results indicated that the students performed relatively poor in each test. The lowest mean scores were observed in the WPT, then CKT, and PKT respectively. The questions involving length measurement had higher mean scores than area and volume measurement questions in all tests. Additionally, the results highlighted a significant relationship not only between the tests but also between the domains of measurement with a strong and positive correlation. According to the findings, whereas the overall performances of students on the tests significantly differed according to previous mathematics achievement level, gender did not affect the students&rsquo / performance on the tests. Moreover, a wide range of mistakes were found from students&rsquo / written responses to the length, area, and volume questions in the tests. Besides, the results indicated that use of materials in teaching and learning measurement was quite seldom and either low or non-significant relationship between the use of materials and the students&rsquo / performance was observed.

The aim of this document is to familiarize first grade elementary school readers with the critical points in solving word problems in mathematics and to analyze various problem- solving strategies in concrete tasks. Primarily, the document is focused on determining pupils' difficulties and acquiring information about the problem-solving process through a written record of the solution and subsequent in-depth interviews. The theoretical part is divided into six sections dealing with terminology (e.g. word problem) and the areas influencing success in the solution of word problems and partial processes. The practical part contains an analysis of each pupil's solutions and a description of phenomena found in them, together with the grading teacher's help. KEYWORDS Word problem, pupil's solutions, analysis of interviews, questions, solution process

The aim of this work is to determine the extent to which the cultural context of a word problem influences its solution by pupils of the 9th grade of secondary schools. Two research questions were set. The first: Will the different cultural context affect the solution of the word problem? How? The second: What do pupils think about word problems with different cultural contexts? The thesis consists of two parts: the theoretical and the practical. In the theoretical part, the general concepts such as culture and the word problem are addressed, followed by a discussion of the cultural context of word problems and their analysis in chosen textbooks. Furthermore, a significant part is devoted to ethnomathematics and multiculturalism in Czech schools. Finally, the difference between the word problem with a typical Czech context and the word problem with a different cultural context is established based on the results of textbooks analysis and the pre-research. The research, which is described in the practical part was conducted qualitatively, and the data were obtained from written tests and semi-structured interviews. The results have shown that the different cultural context of a word problem affects the time needed to solve the problem. In addition, the choice of the type of solution and the type of...

京都大学 / 新制・課程博士 / 博士(教育学) / 甲第24353号 / 教博第283号 / 新制||教||214(附属図書館) / 京都大学大学院教育学研究科教育科学専攻 / (主査)教授 MANALO Emmanuel, 教授 楠見 孝, 准教授 野村 理朗 / 学位規則第4条第1項該当 / Doctor of Philosophy (Education) / Kyoto University / DGAM

Given a presentation of G, the word problem asks whether there exists an algorithm to determine which words in the free group, F(A), represent the identity in G. In this thesis, we study small cancellation theory, developed by Lyndon, Schupp, and Greendlinger in the mid-1960s, which contributed to the resurgence of geometric group theory. We investigate the connection between Van Kampen diagrams and the small cancellation hypotheses. Groups that have a presentation satisfying the small cancellation hypotheses C'(1/6), or C'(1/4) and T(4) have a nice solution to the word problem known as Dehn’s Algorithm.

<p>This thesis concerns itself with drawing out some interesting connections between the fields of group theory and formal language theory. Given a group with a finite set of generators, it is natural to consider the set of generators and their inverses as an alphabet. We can then consider formal languages such that every group element has at least one representative in the language. We examine what the structure of the language can tell us about group theoretic properties, focusing on the word problem, automatic structures on groups, and generalizations of automatic structures. Finally we prove new results concerning applications of languages of nested words for studying the word problem.</p> / Master of Science (MSc)