Algebra I Teachers’ Beliefs and Knowledge of Algebra For Teaching

Research indicates that teachers’ mathematical beliefs and mathematical knowledge for teaching impacts practices in the classroom. Research also suggests that success in Algebra I is the gatekeeper to higher-level mathematics. With the increased number of certification pathways in some states, it is important to identify those Algebra I teachers’ beliefs and knowledge of algebra for teaching. A study of current Algebra I teachers revealed that regardless of certification pathway, mathematical beliefs are not significantly different. Additionally, significant differences did exist in regards to the certification pathway and Knowledge of Algebra for Teaching (KAT) levels.

Saberes matemáticos na orientação de professores paranaenses: expertises e contribuições (1970-1980)

A pesquisa aqui apresentada, de caráter investigativo, aponta evidências presentes em documentos produzidos por professores paranaenses para responder a uma demanda de pesquisa manifesta das décadas de 1970 e 1980: como se apresentaram os saberes para ensinar matemática nas orientações para os professores do ensino de 1º Grau no estado do Paraná? Procurou-se responder à essa questão tomando por base a teoria e metodologia da história cultural, a leitura e análise de documentos como a Lei de Diretrizes e Bases da Educação (1971), o Projeto HAPRONT (modelo curricular de formação de professores), “Revista Currículo” (PARANÁ, 1977; 1978; 1979) da SEC do PR, e a “Revista A voz da Escola” (PARANÁ, 1981) do IEP. A organização e abordagem das propostas educativas que fizeram parte do contexto de produção de manuais didáticos para a orientação de formação, tanto inicial quanto continuada de professores, tiveram a participação de educadores cuja expertise associou os saberes para ensinar às teorias educacionais e métodos de ensino em voga na educação brasileira, traduzindo as expectativas para o período em questão. A fundamentação teórico-metodológica da história cultural nas teorias de Julia, Chervel e Chartier deram sustento às argumentações aqui expostas.Mathematical knowledge in the teacher’s orientation of Paraná: expertises and contributions (1970-1980)The research presented here, of an investigative nature, points to evidence present in documents produced by teachers from Paraná to respond to a clear demand for research from the 1970s and 1980s: how was presented the knowledge for teaching mathematics in the guidelines for teachers of elementary school in the state of Paraná? We sought to answer this question based on the theory and methodology of cultural history, the reading and analysis of documents such as Lei de Diretrizes e Bases da Educação (1971), the HAPRONT Project (curricular model for teacher education), “Curriculum magazine” (1977, 1978, 1979), from Education Secretary of Paraná, and “The voice of the school magazine” (1981) from Educational Institute of Paraná. The organization and approach of educational proposals that were part of the context of the production of textbooks to guide both initial and continuing teacher education had the participation of educators whose expertise associated the knowledge to teach with the educational theories end teaching methods in vogue in Brazilian education translating expectations for the period in question. The theoretical methodological foundation of cultural history in the theories of Julia, Chervel and Chartier supported the arguments presented here.Keywords: Education; Teachers’ training; Mathematics knowledge.

TEACHERS’ CONCEPTIONS AND PROFESSIONAL KNOWLEDGE OF VARIABILITY FROM THEIR INTERPRETATION OF HISTOGRAMS: THE CASE OF VENEZUELAN IN-SERVICE SECONDARY MATHEMATICS TEACHERS

Many studies have reported on the influence of teachers’ conceptions of variability on different aspects of their professional knowledge for teaching statistics and their classroom practices. However, research on these kind of conceptions is still scarce, particularly in Latin American countries like Venezuela. In an effort to help fill this gap, a qualitative study was conducted that aimed to characterize the different ways in which Venezuelan in-service secondary school mathematics teachers conceptualize variability. For that purpose, a survey instrument was developed and administered to 27 teachers working at the metropolitan area of Caracas. This paper focuses on the participants’ answers to two items in which interpretation of histograms was necessary. It was found that about a third of the participants exhibited a sophisticated recognition of variability (e.g., gave answers connecting both middles and extremes), whereas about half of them exhibited misconceptions of variability, such as acknowledging variability from the viewpoint of idiosyncratic ideas, or the degree of symmetry (or lack thereof) of a histogram. Moreover, it was also found that about two-thirds of the participants were unable to correctly match real-life contexts to their corresponding histograms, while about two-fifths were unable to correctly determine the accuracy or inaccuracy of descriptions of the variability in a histogram. The author discusses possible reasons for the obtained results, in order to identify relevant implictions for teacher education in the area of statistics. Abstract: Spanish Diversos estudios han reportado que las concepciones de los docentes sobre variabilidad influencian tanto su conocimiento profesional para la enseñanza de la estadística, como sus prácticas en el aula. Sin embargo, investigaciones sobre este tipo de concepciones son aún escasas, particularmente en países latinoamericanos como Venezuela. Intentando satisfacer esta necesidad, se condujo un estudio cualitativo para identificar y caracterizar las diferentes maneras en que maestros venezolanos de matemáticas a nivel de secundaria conceptualizan la variabilidad. Con tal propósito, un cuestionario fue desarrollado y administrado a 27 docentes en el área metropolitana de Caracas. Este artículo se centra en las respuestas dadas por los participantes a dos ítems del cuestionario, en los que era necesaria la interpretación de histogramas. Se descubrió que aproximadamente un quinto de los participantes demostró un reconocimiento sofisticado de la variabilidad (e.g., considerar simultáneamente valores centrales y extremos de un histograma), mientras que alreadedor de la mitad exhibió concepciones erróneas, tales como el reconocimiento de la variabilidad a partir de ideas idiosincrásicas, o del grado de simetría de un histograma. Además, unos cuatro quintos de los participantes fueron incapaces de establecer una correspondencia entre contextos de la vida real y sus respectivos histogramas, mientras que unos dos quintos fueron incapaces de determinar si descripciones de la variabilidad en un histograma eran o no correctas. El autor discute las posibles razones de los resultados obtenidos, con el fin de identificar implicaciones relevantes para la formación docente en el área de la estadística.

What Mathematical Knowledge Do Prospective Teachers Reveal When Creating and Solving a Probability Problem?

The teaching of probability is conditioned by teachers’ mathematical knowledge. In this paper, an exploratory study is carried out with prospective teachers. A training task was designed requiring them to create and solve a probability problem using the values of euro coins, which was adapted to students aged 11 to 12. The study aimed at determining what mathematical knowledge prospective teachers show when dealing with the task. The data were collected through the Moodle online Campus. We framed the data analysis in the Mathematical Knowledge for Teaching model and we used content analysis as the methodological approach. The results indicate that, despite finding evidence of adequate common and specialised mathematical knowledge, in approximately half of the prospective teachers participating in the study, too many of them still show a lack of knowledge in both subdomains. There was also little evidence of knowledge of the curriculum. The main finding of the research is that, when prospective teachers get involved in complex creative tasks, they mobilised together specialised and common mathematical knowledge, working into different mathematical processes such as problem posing and solving, communication, and argumentation, which reinforces the need to continue working on these types of complex tasks.

Pre-Service Primary School Teachers’ Knowledge and Their Interpretation of Students’ Answers to a Measurement Division Problem with Fractions

During the last decades, research in teacher noticing has increased since its development is considered important in teacher training programs. An issue that needs more research is the relationship between teachers’ mathematical knowledge for teaching in a specific mathematical domain and their ability to notice. This study focuses on how pre-service primary school teachers (PPTs) solve a measurement division problem with fractions and interpret (score and justify) students’ answers to this problem. The participants were 84 PPTs who answered two tasks. Task 1 consisted of solving a measurement division problem with fractions. Task 2 involved interpreting (scoring and justifying) the answers of four primary school students to the problem. Responses to Task 1 were classified based on their accuracy and the procedure used. For Task 2, the scores given along with their justifications were analyzed. The results show that PPTs’ knowledge of division with fractions is limited and that they had difficulties in identifying conceptual errors in students’ answers. This study provides information on the relationships between PPTs’ knowledge of these types of problems and how PPTs interpret students’ answers. This information could aid in adjusting mathematical teaching knowledge in training programs.

Exploring Latent Topics and Research Trends in Mathematics Teachers’ Knowledge Using Topic Modeling: A Systematic Review

Mathematics teachers’ knowledge is considered one of the most critical factors in instruction and student achievement. As such, various studies have focused on mathematics teachers’ knowledge. Despite the expansion of the field, however, a systematic review was rarely implemented. Therefore, this study aimed to identify major research topics and trends on mathematics teachers’ knowledge by analyzing abstracts of 3485 scholarly articles published from 1987 to 2021. Using a text-mining technique, 11 underlying topics were found in the articles. The topics were classified based on their relationships and the following four groups were identified: “assessment”, “teachers’ knowledge for teaching”, “students’ knowledge and understanding”, and “teachers’ professional learning”. Over time, the analysis of research trends showed that professional development is the most popular topic, followed by pedagogical content knowledge and students’ mathematical understanding. Moreover, the popularity of these topics has not changed considerably over time. This study provides implications based on these results.

Specialised Knowledge for Teaching Geometry in a Primary Education Class: Analysis from the Knowledge Mobilized by a Teacher and the Knowledge Evoked in the Researcher

Deepening the specialised knowledge of mathematics teachers is necessary for teaching, and it is one of our concerns as researchers; accordingly, in this study we analyze the knowledge that a teacher mobilizes when giving a first lesson on geometry to a class of fifth year of primary education at a Spanish school. We provide this analysis with a complementary perspective that leads us to broaden our focus of attention to other aspects of classroom management, thus identifying the specialised knowledge evoked in the researcher by the opportunities observed in the classroom. We take opportunities to mean classroom situations that occur as a result of the teacher’s management for which the researcher considers that an alternative management of such situations would be supported by relevant specialised knowledge for the teaching of the given content. Considering the possible alternative management of specific classroom situations makes this specialised knowledge emerge, which, together with that mobilized by the teacher, generates specialised knowledge useful for guiding the content of the training of primary education teachers.

Two Replication Studies of the Relationships between Mathematical Knowledge for Teaching, Mathematical Quality of Instruction, and Student Achievement

In this paper, we share two conceptual replications of Hill et al.’s (2012c) study linking Mathematical Knowledge for Teaching (MKT), Mathematical Quality of Instruction (MQI), and student assessment scores. In study 1, we share data from 4th and 5th grade teachers in an urban school district. In study 2, we share data from middle school teachers in a school district with a relatively high proportion of emergent bilingual students. By varying contexts, we found that Hill et al.’s (2012c) suggested use of the MKT cutoff points was not warranted in our differing settings. Further, we found some significant relationships between MKT, MQI, and student assessments; however, we were not able to reproduce these consistently with our data. We suggest that the relationship between teaching practice and MKT may be quite sensitive to contextual factors including grade level, demographics, school effects, and assessments. We recommend that policymakers and researchers take caution when using such instruments to evaluate program initiatives and identify teachers for remediation or leadership positions. The impact sheet to this article can be accessed at 10.6084/m9.figshare.16610080.