An Investigation of Student Learning in Beginning Algebra Using Classroom Teaching Experiment Methodology and Design Research

Students in grades 6-8 often struggle with learning expressions, equations, and functions (NCTM, 2011). This study investigated what sense sixth-grade students make when solving algebra tasks presented in a whole class teaching experiment (Lamberg & Middleton, 2009; Middleton, Gorard, Taylor, & Bannan-Ritland, 2008; Steffe & Thompson, 2000) using design research (Cobb, 2000; Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003; Cobb & Yackel, 1996; Gravemeijer, 1994). The teaching episodes were video recorded and all student work was documented and analyzed. The teaching experiment was an iterative process conducted in three phases. Data collection and analysis was a parallel process with prospective analysis occurring throughout the teaching experiment and retrospective analysis occurring after the teaching experiment. This research developed theories about the students' learning process in algebra, as well as techniques designed to support their learning. The instructional unit that was used in this study was developed with the goal of promoting student learning of the algebra tasks and was modified to further student understanding of the tasks. The realized learning trajectory for extending arithmetic to algebraic expressions, solving one-variable equations, and representing functions was also documented

Towards democratic culture and political practice in early childhood education: The case for transformative change at a time of converging crisis // Rumo a uma cultura democrática e prática política na Educação de Infância: o caso para uma mudança transformadora num momento de crise convergente

O artigo argumenta a importância da escolha de uma cultura democrática para a educação em geral e para a educação de infância em particular. Situa a sua discussão numa visão de cultura democrática no contexto actual de crises existenciais convergentes e de fim da hegemonia neoliberal, tornando a mudança transformadora, embora difícil, ao mesmo tempo urgente e possível. O conceito de “cultura democrática” é elaborado e é enfatizada a sua importância para a produção de significado para permitir uma mudança de práticas e pensamentos técnicos para práticas e pensamentos políticos. São consideradas algumas possíveis consequências da escolha de uma cultura democrática para a educação de infância, em termos das escolhas políticas que podem ser feitas no que diz respeito aos propósitos da educação, da pedagogia e da imagem das escolas, dos professores e das crianças. O artigo termina considerando possíveis processos de mudança transformadora para a educação de infância, fornecendo exemplos de projetos locais que evoluíram em direção a uma cultura democrática. PALAVRAS-CHAVE educação de infância, cultura democrática, prática política, crise convergente.// The paper argues the case for choosing a democratic culture for education in general and early childhood education in particular. It locates its discussion, and the case made for democratic culture, in a current context of converging existential crises and the ending of the neoliberal hegemony, making transformative change, though difficult, both urgent and possible. The concept of ‘democratic culture’ is elaborated and its importance for the production of meaning and for enabling a turn from technical to political thinking and practice emphasised. Some possible consequences of choosing a democratic culture for early childhood education are considered, in terms of the political choices that may be made with respect to the purposes of education, pedagogy and the image of schools, teachers and children. The article ends by considering possible processes of transformative change for early childhood education, providing examples of local projects that have evolved towards a democratic culture

A Re-emergent Analysis of Early Algebraic Learning

In this paper, we discuss a novel approach for collaborative retrospective analysis. One researcher was directly involved in a classroom teaching experiment, adopting an emergent perspective as an interpreter-witness of classroom interactions during a four-week algebra instructional unit with sixth-grade students. The other researcher experienced and analyzed the data in reverse chronological order. We describe how this re-emergent perspective revealed aspects of students’ early algebraic reasoning

Using a Noticing Framework in a Mathematics Methods Course

A noticing framework was introduced to prospective teachers (PTs) as a tool to use for analyzing student work. The purpose of this study was to determine the impact of PTs’ use of a noticing framework for: 1) interpreting students’ mathematical thinking; and 2) reflecting on and discussing future implications for teaching. The study also sought to determine where PTs needed, if any, further support in engaging in the process of noticing. Using a coding schema that reflected three levels of understanding (periphery, transitional, and accomplished), a frequency table was constructed that allowed PTs’ use and understanding of a noticing framework to be analyzed

Pre-Service Elementary School Teachers’ Perceptions Of Themselves As Learners Of Mathematics And Science

Access the online Pressbooks version of this article here. This study investigated how prospective elementary teachers view themselves as learners of mathematics and science during their last year in a teacher preparation program at an American university. Using drawing and reflections as the method for collecting data, prospective teachers were prompted to draw themselves and reflect on learning mathematics and draw themselves and reflect on learning science prior to and after their mathematics and science methods courses. Drawings (n = 147) were coded according to the presence or absence of several themes including physical objects, teachers, students, and environment. The drawings and reflections indicated that the experience of participating in mathematics and science methods courses taught from a social constructivist perspective positively impacted prospective teachers’ conceptions of themselves as learners and in ways consistent with current research-based pedagogies. The research study described here proposes that prospective teachers’ learning experiences in mathematics and science methods classes might impact how they will teach mathematics and science in their future elementary classrooms

The earliest stages of second language learning:a behavioural investigation of long-term memory and age

A study with 40 L1 Italian 8-9 year old children and its replication with 36 L1 Italian adults investigated the role of declarative and procedural learning ability in the early stages of language learning. The studies investigated: (1) the extent to which memory-related abilities predicted L2 learning of form-meaning mapping between syntax and thematic interpretation, word order and case marking; and (2) the nature of the acquired L2 knowledge in terms of the implicit/explicit distinction. Deploying a computer game in incidental instruction conditions, the participants were aurally trained in the artificial language BrocantoJ over three sessions. Standardized memory tasks, vocabulary learning ability, and an alternating serial reaction time task provided measures of visual/verbal declarative and procedural learning ability. Language learning was assessed via a measure of comprehension during practice and a grammaticality judgment test. Generalized mixed-effects models fitted to both experimental datasets revealed that, although adults attained higher accuracy levels and were faster learners compared to children, the two groups did not differ qualitatively in what they learned. However, by the end of the experiment, adults displayed higher explicit knowledge of syntactic and semantic regularities. During practice, declarative learning ability predicted accuracy in both groups, but procedural learning ability significantly increased only in children. The procedural learning ability effect emerged again significantly only in the child grammaticality judgment test dataset. In the practice data declarative learning ability and vocabulary learning ability interacted negatively with procedural learning ability in children, whereas declarative learning ability interacted positively with procedural learning ability in adults. Moreover, the positive interaction in adults only obtained for a subset of practice stimuli, i.e. sentences where the processing of linking between morphosyntax and thematic interpretation was required. Overall, the findings support age-related differences and linguistic target differences in the way abilities related to long-term memory predict language learning