Thesis (Educat.D.)--Fairleigh Dickinson University, 1981. / Source: Dissertation Abstracts International, Volume: 42-08, Section: A, page: 3484.

Thesis (Ph.D.)--Indiana University, School of Education, 2006. / Source: Dissertation Abstracts International, Volume: 67-06, Section: A, page: 2088. Adviser: Frank K. Lester. "Title from dissertation home page (viewed June 21, 2007)."

Algebra is considered a linchpin for success in secondary mathematics, serving as a gatekeeper to higher-level courses. Access to algebra is also considered an important lever for educational equity. Yet despite its prominence, large-scale examinations of algebra instruction are rare. In my dissertation, I endeavor to better understand what contemporary algebra instruction looks like. I explore instructional practices across a large sample of video recorded algebra lessons from 5 urban districts. To do this, I draw on video and other data from the Measures of Effective Teaching (MET) Project. In the first study, I utilize grounded analysis to describe the format and features of instruction in lessons in the sample. I find that most lessons are teacher-centered with some opportunity for student engagement in mathematical thinking; however, very few lessons provide significant opportunities for student exploration or discovery of mathematical concepts. Looking beneath the surface, I find specific instructional practices teachers employ in algebra lessons and argue that improving these practices may be a promising lever for instructional improvement. Next, I describe the development and validation of an observational instrument oriented toward algebra and designed to measure the nature and quality of these practices. Finally, in the third paper, I use the observational instrument to describe the frequency and quality of these practices in algebra lessons in the sample. I present both descriptive results and qualitative cases of algebra lessons to illustrate these instructional features. / Education Policy, Leadership, and Instructional Practice

In 1998 Paul Black and Dylan Wiliam published the article, Inside the Black Box: Raising Standards Through Classroom Assessment (Black & Wiliam, 1998b). They asserted that formative assessments were the strongest way of raising student achievement. There are a number of empirical studies that document positive impacts of formative assessment on student learning (Brookhart, 2004; Allal & Lopez, 2005; Köller, 2005; Brookhart, 2007; Wiliam, 2007; Hattie & Timperley, 2007). There are also critics of much of the existing research (Shute, 2008; Dunn and Mulvenon, 2009; Bennett, 2011; Coffey Hammer Levin and Grant, 2011). The literature points to the need for more research in this area. Shavelson (2008), who looked at formative assessment in the science classroom stated, “[a]fter five years of work, our euphoria devolved into a reality that formative assessment, like so many other education reforms, has a long way to go before it can be wielded masterfully by a majority of teachers to positive ends. This is not to discourage the formative assessment practice and research agenda.” In this study I examined how teachers understand formative assessment in relation to their instruction, and how they actually implement formative assessment in their math classrooms. I used a thematic analysis research design, analyzing interviews, and observation recordings and field notes. I created a Depth of Implementation Framework, based on both a definition of formative assessment constructed from the review of literature and on the data gathered from teachers, to help make sense of the interplay between teachers’ understanding and use of formative assessment. Based on the data, teachers’ use of formative assessment was characterized as deep, developing or superficial. Teachers’ understanding of formative assessment, especially the definitions that they constructed for themselves, had an impact on both how they used formative assessment and how they saw themselves improving their use of formative assessment.

Social Justice Mathematics, or SJM, is a mathematics-specific form of Social Justice Pedagogy (Frankenstein, 1983; Gutiérrez, 2002), that aims to teach mathematics content while developing conscientização (Freire, 1970), or sociopolitical consciousness (Gutstein, 2006). Research on SJM has generally focused on teachers’ implementation of SJM, finding that teachers struggle to meet the dual goals of teaching mathematics content while developing students’ sociopolitical consciousness (e.g. Bartell, 2013; Gregson, 2013). The literature that explores students’ experiences with SJM yields conflicting findings, where some studies indicate student resistance (Brantlinger, 2007, 2014; Frankenstein, 1990) and other studies indicate students feeling empowered by SJM (Gutstein, 2006; Yang, 2009). In addition, students’ reactions to Social Justice Pedagogy (of any subject area) appear to differ substantially depending on students’ level of privilege and/or marginalization (e.g. Camagnian, 2009; Seider, 2008; Swalwell, 2013). This comparative case study focuses on two sixth grade mathematics classrooms, one in an elite private school and the other in a Title I public school. The present study investigates how teachers’ and students’ backgrounds and their experiences with privilege and/or marginalization influence how they make sense of SJM, with consideration of the fluid and context-dependent nature of privilege and marginalization (Hulko, 2009). Findings indicate the two case study teachers’ SJM goals were influenced by their own lived experiences and by the populations they teach. Similarly, students’ takeaways of SJM differed by background, where students of privilege learned to empathize with others, gaining a more theoretical understanding of social justice as relevant to the lives of others. On the other hand, students of historically marginalized backgrounds responded to SJM activities with strong emotional reactions (e.g. anger, sadness) because the social issues explored in the activities were intimately related to their own lives. These results suggest different supports are appropriate for different students for SJM to be successful. For students of historically marginalized backgrounds, the teacher’s sociopolitical consciousness is fundamental to his or her ability to develop meaningful SJM activities relevant and sensitive to students’ backgrounds. For students of privileged backgrounds, SJM work is supported with a school-wide social justice focus.

Se situant d'une part, à un niveau d'âge qui a déjà fait l'objet de recherches en apprentissage des mathématiques et d'autre part, dans un contexte social connu; la présente recherche vise à cerner le processus suivi par l'apprenant haïtien de la troisième année du primaire d'âge normal pour résoudre un problème de type additif. Une revue exhaustive de la littérature appropriée a permis de déduire que l'apprenant du primaire, en situation de résolution de problèmes, se fait une représentation de l'énoncé, laquelle est influencée par la taille des nombres enjeu, le vocabulaire utilisé, la place de la question posée. Cela a permis de dresser une typologie différente de problèmes de type additif selon que l'on se situe au pôle sémantique ou de calcul relationnel. Malheureusement l'état actuel des connaissances n'a pas permis de cerner le processus suivi par l'apprenant pour aboutir à cette représentation. C'est ce que la présente recherche vise à vérifier. Pour y parvenir, cette recherche a emprunté le modèle de Bergeron et Herscovics (1988), lequel définit de manière satisfaisante le processus mental de l'apprenant en pareille situation. (Abstract shortened by UMI.)

Deux facteurs principaux ont ete retenus afin d'expliquer le rendement en mathematiques des eleves d'origine immigrante. Le premier est celui des attentes des enseignants. Le deuxieme facteur retenu est celui de concept de soi scolaire de l'eleve. Ce projet de recherche vise a clarifier la relation entre les attentes des enseignants, le concept de soi scolaire et le rendement en mathematiques des eleves d'origine immigrante qui frequentent les ecoles canadiennes. Trois questions de recherche ont ete verifiees aupres d'eleves de 6, 7 et 8$\sp\circ$ annees inscrits aux ecoles de langue francaise de la region d'Ottawa-Carleton. Aut total, 75 eleves, dont 40 d'origine immigrante, ont participe a l'etude. Les attentes cognitives et normatives, le concept de soi scolaire ainsi que le rendement en mathematiques ont ete mesures a l'aide d'instruments possedant une cetaine forme de validite. L'analyse des donnees demontre que la relation entre les deux types d'attentes et le rendement en mathematiques des eleves d'origine immigrante peut etre etablie. (Abstract shortened by UMI.)

Abstract Not Available.

Face a des problemes a resoudre varies, l'eleve fait appel a des graphismes (dessin, tableau, proposition, formule, etc). Ces graphismes seraient une representation spontanee du texte probleme, manifestee dans des productions ecrites. Ils refleteraient des elements du probleme ou des relations entre ces elements. Il semble que ces graphismes peuvent servir d'aide mnemonique et qu'ils facilitent la comprehension, ce qui permet la planification, si critique pour visualiser les problemes en mathematique. Meme s'ils semblent primitifs et imparfaits, ces graphismes sont importants mais a ce jour, ils ont ete peu examines. L'objectif de cette etude est de definir la nature et la fonction des graphismes produits par l'apprenant alors qu'il solutionne un probleme en mathematique. Plus particulierement, cette etude s'attachera aux relations entre les graphismes et le processus de resolution de problemes, a savoir si ces graphismes inhibent ou facilitent la tache a accomplir. Le cadre conceptuel propose par Holland, Holyoak, Nisbett et Thagard (1986) permet d'identifier les differences individuelles et le dynamisme dans les habiletes cognitives, dans un contexte de resolution de problemes. Ces auteurs ont utilises les modeles mentaux pour expliquer les processus cognitifs qui sous-tendent le rendement dans une tache. L'approche des modeles mentaux accentue la coherence dans les differentes recherches en resolution de problemes en permettant d'interpreter un phenomene inductif de raisonnement, telle la representation spontanee d'un texte probleme. Une tache de resolution de problemes est presentee a 36 eleves de 8e et de 10e annee. Les problemes se rattachent aux trois domaines de la geometrie, de la distance et de l'inclusion. Afin de capter les types d'information cherches (production de graphismes, processus cognitifs de resolution de probleme, fonction des graphismes), la recherche fait appel a une methodologie qui permet d'etudier les processus cognitifs qui sous-tendent le rendement. L'analyse de rapports verbaux emis lors de la resolution d'un probleme permet de degager les structures et les processus cognitifs utilises lors de l'accomplissement de la tache. L'observation directe des eleves en situation de resolution de problemes est enregistree sur video. Une tache de retrospection assistee du retrovisionnement du video donne acces a l'intentionalite de l'eleve en action. La retroaction visuelle utilisee dans la presente recherche est un outil didactique puissant parce qu'elle aide a conscientiser des actions qui restent souvent implicites et qu'elle permet de revenir et de reflechir sur les procedes utilises. (Abstract shortened by UMI.)

Mathematics is a broad field, mastered at a functional level by many preschoolers, and taught in its different forms to students from four years of age to adults in school. A review of the literature revealed a large number of instructional interventions in mathematics. Bruner (1960) has long advocated discovery learning. Collaborative learning, working with peers, was effective in developing strategies (Collins, 1992). Worked examples have been used from kindergarten (Villasenor and Kepner, 1993), to university (Schoenfeld, 1985). Allowing a student to watch a peer solving a problem on videotape combines all three types of intervention, and offers many advantages to the learner. A review of the literature did not find evidence that a videotaped intervention had previously been used in mathematics problem solving at the grade seven level. The current study was exploratory in nature, with both a quantitative and qualitative component. Three isomorphic problems were developed which were in areas of interest suggested by students at the same grade level. These problems were considered to be "real world" problems because they involved construction projects similar to ones which the students might have encountered in school or at home. The videotape of a grade seven student solving one of these problems was obtained and used as an instructional intervention. Thirty-four grade seven students were divided into six groups, three experimental and three control groups. Each of the three experimental groups received a different sequence of problems to solve, but each group saw the videotaped problem solution before solving a common final problem. One control group received the same sequence of problems as each of the experimental groups, but they did not see the video before solving the final problem. The students who viewed the video reported that watching the student on the video solve the problem did have an impact on their work. After watching the videotaped problem solution these students spent less time analyzing the next problem, suggesting that the video had resulted in the development of a mental model for this type of problem. Support for this conclusion was also found in the increased amount of time spent on Global Monitoring. These students appeared to be more aware of their global plan, and in many cases to have a better global plan. The improved global plan was demonstrated in their selection of a higher level strategy to solve the second problem. The performance of both the experimental and control groups might have been influenced by the effects of having practised on the first problem. This study found that having one problem to solve did not have a positive effect on the second problem solution for either group. Practise, combined with seeing a videotaped problem solution which was different from the one just attempted, was also not beneficial. The intervention which was most effective was the one in which the students saw the solution of the problem which they had just attempted. All of the students receiving this treatment developed a more complete global plan, and used a more advanced strategy on their second problem. Two of the six subjects solved the problem correctly when they had not been able to solve their first problem.