Designing Professional Learning Tasks for Mathematics Learning Trajectories

In this paper, we present an emerging set of learning conjectures and design principles to be used in the development of professional learning tasks that support elementary teachers’ learning of mathematics learning trajectories. We outline our theoretical perspective on teacher knowledge of learning trajectories, review the literature concerning mathematics professional learning tasks, offer a set of initial conjectures about teacher learning of learning trajectories, and articulate a set of principles to guide the design of tasks. We conclude with an example of one learning trajectory professional learning task taken from our current research project

Alternative learning trajectories

Lesson planning usually involves the generation of a hypothetical learning trajectory. This paper illustrates a teaching strategy that is one focus of a major research project. Alternative learning trajectories with different entry level prompts were used to enable students to access the concepts and procedures necessary for their joining the main learning trajectory. The strategy is being trialled in primary classrooms that have a large proportion of lower SES students, with the aim of maximising success in mathematics for all students.<br /

Characterizing Graduateness in Computing Education

In my research, I employ a highly qualitative, narrative methodology to explore the sense students make of their own educational experiences within their wider learning trajectories. By taking such a holistic perspective on a Computing Education, I hope to be able to identify and distil aspects of successful Computing programs, whose effects may only emerge over time

Learning Trajectories are Generalization Indicators

This paper explores the connection between learning trajectories of Deep Neural Networks (DNNs) and their generalization capabilities when optimized using (stochastic) gradient descent algorithms. Instead of concentrating solely on the generalization error of the DNN post-training, we present a novel perspective for analyzing generalization error by investigating the contribution of each update step to the change in generalization error. This perspective allows for a more direct comprehension of how the learning trajectory influences generalization error. Building upon this analysis, we propose a new generalization bound that incorporates more extensive trajectory information. Our proposed generalization bound depends on the complexity of learning trajectory and the ratio between the bias and diversity of training set. Experimental findings reveal that our method effectively captures the generalization error throughout the training process. Furthermore, our approach can also track changes in generalization error when adjustments are made to learning rates and label noise levels. These results demonstrate that learning trajectory information is a valuable indicator of a model's generalization capabilities

An Examination Of Preservice Teachers\u27 Use Of Learning Trajectories To Guide Instruction

In order for teachers to support students’ mathematical thinking, Battista (2004) believed they must identify core mathematical concepts, recognize conceptual frameworks for understanding children’s thinking, and use appropriate assessment tasks. In his view, learning trajectories provide teachers with information on children’s cognitive abilities as well as a structure for assessment. The present study investigated the ways in which preservice teachers used, and reflected on their use of, learning trajectories to assess, plan, and instruct during a one-on-one tutoring project focused on geometric shapes. In addition, preservice teachers were asked to reflect on the ways in which they might use learning trajectories during small-group and whole-group instruction. This study employed collective case study methodology as a qualitative research design methodology. The goal of the research was to understand how three preservice teachers interpreted their experience. Participants were preservice teachers seeking K-8 teaching certification with a minor in mathematics education. Throughout the twelve-week study, digital recordings of fieldwork and interviews were collected, along with journal entries, lesson plans, and fieldnotes. The data analysis strategy followed Stake’s (2006) methodology for collective case study analysis. Trustworthiness was accomplished through thick description, triangulation of data, and member checks. Supported by constructivist learning theory as the theoretical framework guiding the research, the study found that during the assessment phase, learning trajectories gave preservice teachers flexibility in identifying their students’ level of mathematical thinking. While planning, preservice teachers created lesson plans that encouraged active learning and were within their students’ zone of proximal development. And during instruction, learning trajectories were used as a tool for formative assessment. When asked to reflect on how learning trajectories might be used during small-group and whole-group instruction, preservice teachers surmised that learning trajectories could be used to create hetero-homogeneous groupings and used to ask questions that increase in sophistication, respectively

Visual perturbances in digital pedagogical media

Several studies have investigated how the formation of informal conjectures, and the dialogue they evoke, might influence young children’s learning trajectories, and enhance their mathematical thinking. In a digital environment, the visual output and its distinctive qualities can lead to interpretation and response of a particular nature. In this paper the notion of visual perturbance is explored, and situated within the data obtained, when ten-year-old children engaged in number investigations in a spreadsheet environment

Learning trajectories and fractions: primary teachers’ meaning attributions

International audienceA learning trajectory constitutes a hypothesis and a description of students’ thinking related to learning a mathematical notion. The study reported here, employing a multiple case study approach, investigates the use of learning trajectories in the teaching of fractions by three 5th grade Greek primary school teachers. Particularly, the research problem pursued concerns the teachers’ meaning making and use of the concept of learning trajectories introduced by a recent Mathematics Curriculum reform. The results of analyzing the teachers’ answers to two semi-structured interviews and the transcripts of a non-participant observation of their lessons show that learning trajectories were understood as means of planning teaching and used as ‘maps’ of navigating classroom instruction, predominately successful when students’ previous knowledge and thinking, and not just their difficulties, are taken into consideration

Building learning opportunities in classrooms of disadvantage: rethinking the learning trajectories

Researching learning trajectories in classrooms of disadvantage introduces specific challenges as well as opportunities. Situating our work within design research, we illustrate the power of theoretical approaches in which close attention is paid to both learners' conceptual developments and the means capable of supporting such developments for all learners. We illustrate how considerations of teachers' learning can and should inform the formulation of (students') learning trajectories, if these are to become viable outside of research studies