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This study was based on our common interest in teachers’ use of language in algebra teaching. The purpose of the study is to investigate the teachers’ use of concepts in algebra teaching and to gain in-depth knowledge of how the teachers use formal mathematical language and informal everyday language. To produce results for the study, five teachers were studied in two separate math lessons, which was followed up with an interview to answer the following research questions: (1) How do the teachers use formal and informal concepts in the teaching of algebra? (2) What impact do the teachers think their language use has on the student's learning and own language use? (3) What conscious language choices do teachers make to support students' conceptual learning and to connect informal and formal language use?  The first research question, which was answered with the help of the study's observations and interviews, showed that the participating teachers use both informal and formal concepts in algebra teaching, in a varied and coexisting way. Both the language of mathematics and everyday language were considered necessary to combine in order to make mathematics teaching understandable for the students.  The second and third research questions, which were answered using the study's interviews, showed that most of the participating teachers believed that their language use has a high impact on students. One participating teacher mentioned that many students speak as the teacher does and therefore, the teacher's language becomes crucial for the student's language development. To support the students, most of the participating teachers used to confirm and repeat the informal concept that emerged and linked to the formal concept. / Denna studie grundades i vårt gemensamma intresse för lärares språkbruk i algebraundervisningen. Studiens syfte är att undersöka lärares användning av begrepp i algebraundervisningen för att få fördjupad kunskap om hur lärare använder det formella matematikspråket respektive det informella vardagsspråket. För att framställa resultat till studiens syfte studerades fem lärare vilka observerades vid två separata lektionstillfällen, som följdes upp med en intervju för att besvara följande forskningsfrågor: (1) Hur använder lärare formella respektive informella begrepp i undervisningen om algebra? (2) Vilken påverkan anser lärare att deras språkbruk har på elevers inlärning och eget språkanvändande? (3) Vilka medvetna språkval gör lärare för att stötta elevers begreppsinlärning och för att koppla samman informellt och formellt språkbruk?  Första forskningsfrågan, som besvarades med hjälp av studiens observationer samt intervjuer, visade att de deltagande lärarna använder både informella och formella begrepp i algebraundervisningen, på ett varierande och samexisterande sätt. Både matematikspråket och vardagsspråket ansågs nödvändiga att kombinera för att göra matematikundervisningen begriplig för eleverna.  Den andra och tredje forskningsfrågan, som besvarades med hjälp av studiens intervjuer, visade att majoriteten av de deltagande lärarna ansåg att deras språkbruk har stor påverkan på eleverna. En av de deltagande lärarna nämnde att många elever pratar som lärare gör och därför blir lärares språk avgörande för elevers språkutveckling. För att stötta elever bekräftade och upprepade majoriteten av de deltagande lärarna de informella begrepp som uppkom och kopplade till de formella begreppen.

This thesis investigates how the sub-structure of words can be accounted for in probabilistic models of language. Such models play an important role in natural language processing tasks such as translation or speech recognition, but often rely on the simplistic assumption that words are opaque symbols. This assumption does not fit morphologically complex language well, where words can have rich internal structure and sub-word elements are shared across distinct word forms. Our approach is to encode basic notions of morphology into the assumptions of three different types of language models, with the intention that leveraging shared sub-word structure can improve model performance and help overcome data sparsity that arises from morphological processes. In the context of n-gram language modelling, we formulate a new Bayesian model that relies on the decomposition of compound words to attain better smoothing, and we develop a new distributed language model that learns vector representations of morphemes and leverages them to link together morphologically related words. In both cases, we show that accounting for word sub-structure improves the models' intrinsic performance and provides benefits when applied to other tasks, including machine translation. We then shift the focus beyond the modelling of word sequences and consider models that automatically learn what the sub-word elements of a given language are, given an unannotated list of words. We formulate a novel model that can learn discontiguous morphemes in addition to the more conventional contiguous morphemes that most previous models are limited to. This approach is demonstrated on Semitic languages, and we find that modelling discontiguous sub-word structures leads to improvements in the task of segmenting words into their contiguous morphemes.

O presente trabalho tem por objetivo divulgar e explorar as Provas sem Palavras no âmbito do Ensino Fundamental e Médio para, posteriormente, servir de possível material de consulta por parte de professores, na busca de melhores métodos de explicação sobre importantes assuntos desenvolvidos em sala. Dessa forma, imagina-se que a compreensão dos alunos através da visualização fortalecerá de maneira significativa a aquisição do conhecimento. Para tanto, são sugeridos diversos temas onde a visualização é desenvolvida e explicada, seja através de relato, seja através de demonstrações matemáticas. / The aim of the present work is to disseminate and explore the Proof Without Words in the scope of elementary and high school to be used later as a possible reference material for teachers in the search for better methods of explanation on important subjects developed in the classroom. In this way, it is imagined that students’ comprehension through visualization will significantly strengthen the acquisition of knowledge. In order to do so, several themes are suggested where the visualization is developed and explained, either through storytelling or through mathematical demonstrations.