During the last several decades, mathematics reform has emphasized the goal of ensuring that students develop both conceptual and procedural understanding in mathematics. The current mathematics reform, Common Core State Standards for Mathematics (National Governors Association and the Council of Chief State School Officers [NGA Center & CCSSO], 2010), promotes this goal, with procedural knowledge building upon a strong conceptual base. This study uses content analysis to investigate the extent and ways in which Houghton Mifflin Harcourts Go Math! K-8 (HMH, 2016) supports teachers in building procedural fluency from conceptual understanding when teaching equivalence of fractions.Krippendorfs (1980) framework for content analysis guided this study. I identified a priori codes, and allowed for emergent codes, that characterize quality mathematical instruction. Careful analysis of the teacher editions of the textbook series revealed that, if the teacher instructions are to be followed with fidelity, students are not given opportunities to create and share their own strategies for solving tasks designed to help them learn equivalence of fractions. Neither are they given opportunities to make connections among strategies. All connections are introduced by the teacher. Although the teacher editions promote transitions from visual models to algorithms, they provide inconsistent use of problem-solving practice tasks and equal-sharing problems, two methods that are strongly supported by the research literature for developing procedural fluency from conceptual understanding in equivalence of fractions. Finally, the teacher materials include multiple instances in which the same or similar language and terms are used for mathematical and pedagogical practices found in mainstream research and professional literature, yet the practices were to be implemented in ways contrary to mainstream interpretations.Overall, Go Math! K-8 (HMH, 2016) provided little support to teachers in helping students build procedural fluency from conceptual understanding. A teacher-driven, rather than student-driven, approach to instruction was emphasized, thus minimizing opportunities for students to engage in the kinds of tasks and discourse recommended in the literature. The ways in which mathematical language and terms were implemented contrary to mainstream research interpretations can easily cause confusion among educators. The dearth of authentic problem-solving practice was inconsistent with quality mathematics instruction that supports students conceptual and procedural understanding.

Digital technology has made great progress in a relatively short time, which has led to new needs emerging to take advantage of the opportunities provided thanks to the digitization of society. The digital progress has resulted in teaching conditions being changed. The curriculum standards that all pupils should have equal opportunities to be good democratic citizens, demand today that digital competence needs to be provided to pupils. In the revised curriculum for the subject of mathematics it is written that mathematical activities must be linked to digital progress. It also says in the aim of the subject that teaching should give pupils the opportunities to develope knowledge of using digital tools. The result of the study states that the pupils´digital competence is of great significance for their development in mathematics when using digital tools and digital learning recources. The result also shows that one needs to compensate for pupils´previous knowledge regarding digital competence. / Den digitala tekniken har på relativt kort tid gjort stora framsteg, det har lett till att det uppstått nya behov för att ta tillvara på de möjligheter som ges tack vare samhällets digitalisering. Den digitala utvecklingen har gjort att förutsättningarna för undervisning har förändrats. Läroplanskravet att alla elever ska få likvärdiga förutsättningar att vara fullvärdiga demokratiska medborgare gör att lärare idag behöver ge eleverna en digital kompetens. I den reviderade läroplanen för ämnet matematik står det att de matematiska aktiviteterna ska vara kopplade till den digitala utvecklingen. Det står också i ämnets syfte att undervisningen ska utveckla kunskaper i användandet av digitala verktyg. Resultatet av studien visar att elevernas digitala kompetens har stor betydelse för deras utveckling i matematiken vid användandet av digitala verktyg och digitala lärresurser. Resultatet visar även att lärare behöver kompensera för elevers olika förkunskaper gällande digital kompetens.

Diante da afirmação recorrente entre alunos e pesquisadores de que a aprendizagem escolar dos conteúdos matemáticos são carentes de sentido e significado, temos elaborado propostas contextualizadoras para o ensino dessa disciplina. Na primeira parte deste trabalho, procuramos esclarecer a significação das palavras sentido, significado e contextualização, do modo como têm sido por nós interpretadas e a relação que essas palavras têm entre si. Para isso, nos embasamos em teorias da aprendizagem, as quais nos apontam caminhos para acreditar que a aprendizagem escolar pode ser relevante e significativa para os estudantes. A partir dessas teorias, argumentamos que a aprendizagem de um conteúdo escolar deve estar conectada às necessidades do indivíduo e a outros conteúdos, os quais compõem os contextos do conteúdo principal. Na segunda parte, através de uma pesquisa-ação, buscamos atingir o nosso objetivo de investigar se determinada sequência didática, elaborada a partir de uma concepção específica de contextualização, contribui para conferir sentido e significado para a aprendizagem de um conteúdo matemático específico. O conteúdo matemático escolhido foi Progressões Geométricas e a sequência didática contextualizadora foi a oficina Matemática e Música, aplicada a alunos do Ensino Médio numa escola estadual de São Paulo. Utilizamos, como ferramentas de investigação, entrevistas com os alunos e avaliações diagnósticas. As análises dos resultados contribuem para exemplificarmos a nossa concepção de práticas contextualizadoras, bem como atingirmos o objetivo da pesquisa-ação, acima enunciado. / Faced with the recurring affirmation between students and researchers that school learning of mathematical contents is short of sense and meaning, it was elaborated contextualized teaching purposes for that subject. In the first part of this work, we sought to clear the signification of the words sense, meaning and contextualization, the way that we have been making interpretations about them and the connection among those words. Thus, we based on learning theories that point us out some ways to believe that school learning can be relevant and meaningful for the students. Originating from those theories, we discuss that learning a school content must be connected to the needs of an individual and other contents that compose the contexts of the main content. In the second part, through an action research, we tracked down our objective of investigating if specific didactical sequence, elaborated from a particular conception of contextualization, contributes for giving sense and meaning for learning an explicit mathematical content. The chosen mathematical content was Geometrical Progression, and the didactical sequence was a workshop about Mathematics and Music, applied for high school students of a state school of Sao Paulo. We used, as investigations tools, interviews with students and diagnostic evaluations. The analysis of the results contributes to exemplify our conception of contextualized practices, as well as to reach the aim of action research, as proposed above.

Computer Algebra Systems (CAS) have been changing the mathematics instruction requirements for many years. Since the tendency of using CAS in mathematics instruction has been rising for decades and reports have often been positive, the implementation of notebook classes seems to be the consequent next step of mathematics instruction supported by computers. Experiences that have been made with the use of CAS in PC-rooms can be transformed directly into the classroom. Hence the use of CAS is no longer limited to certain rooms. The permanent availability of the notebook with installed CAS offers the chance to realize these concepts that have already been approved with the use of CAS so far. The following speech shall show what these concepts could look like and that the use of notebooks is not only the further development of teaching in PC-classes. Examples from personal experience in teaching will especially show meanders and thought-provoking impulses in order to support teachers finding their way into teaching mathematics instruction in notebook classes successfully. Please allow me to point out two things in the beginning: (1) Yes, I am a vehement supporter of the use of notebooks (and the use of CAS in particular) in mathematics instruction. (2) No, I do not believe that teachers who have chosen another path (or at least partly) are teaching badly.

Philosophiae Doctor - PhD / In this study the construct of personal theories is used to represent the teacher's conceptions, which are interpreted as the consciously held beliefs. The teacher's personal theories encompass beliefs, images, values and attitudes as well as understanding about teaching and learning. This study investigates the influence of the teacher's conceptions of mathematics, of the teaching and learning of mathematics and of the context before and after a structured learning experience. The interest in the teacher's conceptions is derived from the assumption that these serve as a primary component that influence how teachers think about their professional responsibilities and how they act in their classrooms. Furthermore, the extent of implementation of a new curriculum has been linked to the scope of congruence between the teachers' conceptions and the underpinning philosophy of the intended curriculum. The study of the teacher's conceptions is especially relevant during a time of educational reform, such as the current transition to an Outcomes Based Education curriculum in South Africa. The participants in this study consist of four primary school mathematics teachers with various educational backgrounds, who teach at schools situated in different physical environments. The conceptions that these teachers have of mathematics, of the teaching and learning of mathematics and the influence of the context are investigated using a variety of instruments. Data collection was done with a questionnaire, a repertory grid, a semi-structured interview and lesson observations. The teachers participated in the Teaching Intervention and Support Programme (TISP), as a structured teacher learning experience. The programme is centred on the integration of the developmental and socio-cultural perspectives on teacher learning. With the developmental perspective the focus is on the acquisition of intellectual skills, while the socio-cultural perspective emphasizes participation in social practice. Both are directed at effecting conceptual change. With the developmental approach the process of conceptual change involves the development of new conceptions from existing conceptions. From the socio-cultural perspective the context is paramount and conceptual change is seen as new ways of being and acting within a particular context. The teachers were invited to attend a two-week intervention session, followed by a six months support programme that was aimed at establishing a teacher learning community. The learning experiences provided during the intervention session were drawn mainly from Realistic Mathematics Education. On completion of the programme, the teachers' conceptions of mathematics, of the teaching and learning of mathematics and the influence of the context were again investigated. The results of this study show that two of the participants had highly mechanistic conceptions of mathematics, and the teaching and learning of mathematics. The remaining two had a more empiristic approach with its high focus on environmental activities. After the programme, the teachers with the mechanistic views adopted a mixed. conception with some of the mechanistic conceptions retained, but now interspersed with some empiristic and realistic conceptions. The participants with the empiristic conceptions adopted a more realistic conception, but again to varying degrees. Thompson's (1991) hierarchical structure for the development of conceptions was also used to describe the extent of conceptual change. However, it was found that a concentric, rather than a hierarchical representation is a more appropriate to describe these changes. With regards to the socio-cultural view of conceptual change, all the participants perceived the context differently. The teachers' actions were also more commensurate with the practices associated with teachers that encourage learner autonomy, mathematical investigations and a facilitative role for the teacher.

Project-based learning (PBL) is centred on a challenging, yet meaningful, driving question and culminates in a product that students create or do to showcase their learning to a public audience. Other essential elements of a true PBL experience include: sustained inquiry, authentic tasks, opportunities for students to make decisions about their culminating product, reflection, critique, and revision (Hallermann, Larmer, & Mergendoller, 2011). A well-designed PBL combines curriculum and instructional activities to cultivate 21st century skills in students to prepare them for future success in the workforce. Two teams of Year five teachers designed a week-long PBL unit for students organised around the characteristics of the planets, which integrated science, mathematics, and English. The teachers implemented the PBL with six classes of Year five students, documenting their thoughts on planning and implementation to reflect upon the experience.

As the presence of digital game-based learning increases in United States classrooms, understanding their impact on achievement is critical. Digital games for learning offer many potential benefits, including reducing the number of students trapped in a remediation cycle, a contributor to college dropout. Despite the recognized potential of game-based learning, few researchers have explored the relationships between specific patterns of behaviors and types of digital game-based learning environments. The underlying theory for this study was patterns of gamer behaviors may predict in-game behaviors. Archival, third-party data regarding The Lost Function - Episode 1: Sum of the Forgotten Minds by Advanced Training & Learning Technology, LLC was used in this study. Using 4 case groups at the high school and college levels (n=114), self-reported levels of the 3 patterns of gamer behaviors, gender, and age-band were analyzed using multiple regression to determine relationships to time-on-task in a game-based highly interactive virtual environment, designed for mathematics remediation. While the results were inconclusive, this study supported the existing literature regarding gender differences and the lack of mutual exclusivity in behavior typing. Recommendations include additional research in how the statements used in the 3-factor model may be adjusted to allow for a broader population of game players. The social change implication is that further understanding of the relationship between learner traits and digital learning environment may assist educators that employ digital game-based learning a way to better align learners to the most appropriate digital learning environment, thereby increases their chances at success.

This study used mixed methods to examine middle school mathematics teachers’ beliefs about English Learners’ ability to participate in rigorous, grade-level math instruction as well as beliefs about their own capacity for teaching English Learners. Additionally, the study investigated the influence of teachers’ beliefs on their instructional practices and identified the types of support teachers need to develop as culturally and linguistically responsive educators. Findings revealed some dissonance between teachers’ explicit beliefs about teaching English Learners and their implicit beliefs illustrated through instructional decisions made, as well as previously unrecognized gaps in teachers’ ability to identify and differentiate the needs of different types of English learners. Finally, the study identified teachers’ pressing needs for additional support at the school and district levels to continue to develop skills and knowledge to improve their teaching for English learners. The study concluded with an action plan for developing a robust professional learning system to develop teachers’ self-efficacy as culturally and linguistically responsive educators while also addressing implicit bias through reflection.

What do the Honda Civic, the Mazda 626, and the Boeing 747 have in common? What about Weird Al's song Bob, the first name of Miley Cyrus' alter ego, and the 70s sensation Abba? What do all these things have in common? They all contain palindromes. While some people recognise a palindrome when they see one, fewer realise that a palindrome is a special case of a pattern and that these patterns are all around. Palindromes frequently occur in names, both of vehicles and people, and in music.

<p>У дисертацији се разматра утицај примене рачунара на предавање и учење аналитичке&nbsp;геометрије. Израђени су одговарајући динамични радни листови којима је пропраћено градиво&nbsp;изложено у званичном уџбенику. Приказана је примена рачунара и софтвера GeoGebra у&nbsp;изграђивању рачунарског окружења погодног за изучавање аналитичке геометрије у равни.</p> / <p>U disertaciji se razmatra uticaj primene računara na predavanje i učenje analitičke&nbsp;geometrije. Izrađeni su odgovarajući dinamični radni listovi kojima je propraćeno gradivo&nbsp;izloženo u zvaničnom udžbeniku. Prikazana je primena računara i softvera GeoGebra u&nbsp;izgrađivanju računarskog okruženja pogodnog za izučavanje analitičke geometrije u ravni.</p> / <p>The thesis considers the impact that computer usage makes on analytic instruction&nbsp;process. Appropriate dynamic worksheets are developed in order to ease students&rsquo; understanding of the subject material included in the official textbook. &nbsp;Computers and GeoGebra were used to build a computer empowered learning environment suitable for plane analytic geometry instruction.</p>