Secondary students, especially those with learning disabilities, often lack an understanding of computations involving fractions. Much of the secondary math core, especially algebra, requires an understanding of fractions to be able to successfully complete core classes. Instruction on fraction concepts in not part of the secondary core standards. These students are expected to already have this knowledge. There is a need for students with learning disabilities who struggle with fraction computations to receive instruction on fraction concepts in addition to their core instruction. This study used direct instruction and fluency practice as an intervention to teach basic fraction skills to two secondary students with learning disabilities. A multiple probe multiple baseline design was used. Results suggest that fluency training has a positive impact on secondary students' acquisition and retention of basic fraction skills. The implications of this study suggest that this intervention is a viable option to help students acquire fraction skills in a minimal amount of time.

A local urban school district recently reported that 86% of third graders did not demonstrate proficiency on the Math Standardized Test, which challenges students to solve problems and justify solutions. It is beneficial if these skills are developed prior to third grade. Students may be more academically successful if kindergarten teachers have moderate to high self-efficacy when teaching lessons that focus on justifying solutions. Bandura's self-efficacy theory was incorporated into this study as the conceptual framework lens. Research questions were designed to investigate kindergarten teachers' instruction in mathematics that focused on justifying solutions, their self-efficacy in challenging students to justify solutions, and the identification of professional development. Voluntary participants for this study were selected from the 11 elementary schools in the district. Within the 11 elementary schools, there were 33 lead teachers who were invited to participate in the study and 7 agreed to participate in interviews and observations. The data were analyzed using both situation and strategy coding. The analysis of the data revealed a connection between professional development, self-efficacy, and instructional strategies. A relationship was identified between professional development and the teachers' ability to challenge students to problem solve and justify solutions. These findings may be valuable for early childhood stakeholders within the education field. Professional development tends to improve the self-efficacy of teachers and the instructional strategies they incorporate.

I det svenska undervisningslandskapet har extraundervisning, från både privata och offentliga aktörer, under senaste decenniet fått en mer och mer betydelsefull och naturlig plats. Där både ideella och vinstdrivande organisationer under de senaste årtionden erhållit statliga subventioner eller bidrag. Det blir därför viktigt att skapa en förståelse för vilken typ av matematikundervisning som ges av dessa organisationer. Av särskilt intresse för denna studie är de ideella organisationerna, och dess volontärer, som erbjuder kostnadsfri undervisning i form av räknestugor i matematik för elever från gymnasieskolan. Tidigare forskning om de svenska ideella räknestugornas undervisning och dess volontärer lyser med sin frånvaro. Syftet med detta arbete är att skapa förståelse för verksamheten inom de ideella räknestugorna, dess volontärer och vilken matematikundervisning som där ges. För att undersöka volontärernas undervisning så genomfördes dels intervjuer med volontärerna, om deras föreställning om matematik som ämne och om matematikundervisning, dels observationer av volontärernas undervisning inom räknestugans verksamhet. Analys och tolkning av observationer och intervjuer tar utgångspunkt i tidigare forskning om lärares föreställning om matematik, lärande och dess undervisning. De intervjuade och observerade volontärerna gav en samstämmig bild, där föreställningen av matematik som kom till uttryck i intervjuer stämde väl överens med vad som observerades i undervisning. Av den tidigare forskningens tre föreställningar av matematiken gav de intervjuade och observerade volontärerna uttryck för två av de tre föreställningarna. Matematik som verktyg var ett förhållningssätt som uttrycktes av samtliga volontärer i både intervju och undervisning. Perspektivet matematik som problemlösning uttrycktes inte av någon av volontärerna i undervisning, och hade en tämligen tillbakaskjuten roll i den föreställning om matematik som uttrycktes i intervjuer. Slutsatsen är att utifrån vilka föreställningar av matematik som synliggörs inom undervisning kan räknestugor medföra vissa begränsningar sedda som undervisningsform, då den observerade undervisningen primärt gav uttryck för en föreställning om matematik som verktyg. Medan de platonska synsätten förekom till viss del och föreställningen av matematik som problemlösning ej kom till utryck i undervisningen. / In the Swedish educational landscape, supplementary tutoring, from both private and public organizations, has in the last decade gained a more and more significant and natural place. Where both non-profit and for-profit organizations have received government subsidies or grants in recent decades. It is therefore of important to gather an understanding of the type of mathematics education provided by these organizations. Of particular interest for this study are the non-profit organizations, and its volunteers, offering free tutoring in mathematics for student in the upper secondary school.Previous research on the non-profit supplementary tutoring in Sweden, and its volunteers, is lacking. The purpose of this work is to create an understanding of the activities in the non-profit supplementary tutoring, its volunteers and the mathematics teaching that is there provided. In order to investigate the volunteer’s educational practice, both interviews with the volunteers regarding their beliefs about mathematics as a school subject and their beliefs about mathematical education where conducted, and also on observations of the volunteer’s educational practice where done. Analysis and interpretation of observations and interviews are based on previous research about teachers’ beliefs of mathematics, learning and its teaching. The volunteers gave a coherent picture, where the beliefs of mathematics expressed in interviews was in agreement with what was observed in teaching practice. Of the previous research’s three beliefs of mathematics, the interviewed and observed volunteers, expressed two of the three beliefs. The Instrumentalists beliefs, mathematics as a tool, was expressed by all volunteers in both interview and teaching. The perspective of mathematics as problem solving was not expressed by any of the volunteers in their tutoring practice, and was almost invisible in the beliefs of mathematics expressed in interviews. The conclusion is that, from the beliefs of mathematics expressed in tutoring practice, the supplementary tutoring can be seen to have certain limitations as a form of education, since the observed practice primarily expressed the belief of mathematics as a tool. The platonic perspective appeared to some extent and the notion of mathematics as problem solving was not expressed in teaching.

[pt] Este trabalho visa a estudar a influência da Reforma Benjamin Constant no currículo de matemática do Colégio Pedro II. Benjamin Constant foi professor e Ministro da Instrução, dos Correios e Telégrafos, em 1890-1. Ele quis reformar o ensino brasileiro, desde a escola primária até os cursos superiores. Essa Reforma teve caráter enciclopédico, inchando de conteúdos os programas das disciplinas, sobretudo os de matemática. O Colégio Pedro II foi criado em 1837-8 para servir de modelo ao ensino secundário brasileiro, que se caracterizou por dirigir-se à elite socioeconômica; na prática, o Colégio não correspondeu bem à expectativa inicial. A Reforma alterou os programas de matemática do Colégio; dentre outras modificações, introduziu neles, a partir de 1895, noções de cálculo diferencial e integral. Para analisar a influência da Reforma nos mencionados programas, o trabalho compara os currículos vigentes de 1880 a 1890 aos posteriores (1890- 1900). As bases teórico-metodológicas da pesquisa são Ivor Goodson e Jean- François Sirinelli. O trabalho visa, ainda, a caracterizar Benjamin Constant como intelectual e a examinar se e como esse seu status teria influído na Reforma. A pesquisa usa numerosos documentos: atos normativos (como os decretos que instituíram reformas educacionais no Império e no início da República), documentos curriculares (programas de matemática do Colégio Pedro II) e documentos produzidos pelo próprio B. Constant. / [en] This paper intends to study the influence of the Benjamin Constant Reform in the Mathematics curriculum of the Pedro II School. Benjamin Constant was a Mathematics teacher and the Head of the Instruction, Mail and Telegraph Ministry (1890-1). He intended to reform all levels of Brazilian education and decreed the Reform named after him. The Reform had an encyclopedic nature, inflating the content of the programs for each subject, especially the one for Mathematics. The Pedro II School was founded in 1837-8 to serve as a model for Brazilian secondary education, which was characterized as targeted to the social/economical elite; however, the School does not seem to have fulfilled the initial expectations. The Reform altered the Mathematics syllabus of the Pedro II School; among other changes, it added, from 1895 on, notions of differential and integral calculus. To analyze the influence of the Reform in the aforementioned syllabi, this paper compares the Mathematics curriculum in use in 1880-1890 to the ones used later on (1890-1900). The theoretical and methodological bases for this research were Ivor Goodson and Jean-François Sirinelli. The paper also aims to establish Benjamin Constant as an intellectual, and to examine if and how this status would have affected the Reform. The research uses several documents: normative acts (such as the decrees that established educational reforms in Brazil, during the Empire and in the early Republic), curriculum documents (Mathematics syllabi of the Pedro II School in the 1880-1900 period) and documents produced by B. Constant himself.