Learning Trajectories in Mathematics Education

Last four/five decades have seen revolutionary development in the field of electronics, computer and automation. Naturally avionics and C.N.S facilities also have adopted these technologies to the best of their advantage. The present paper shows how these technologies have modernized the aircraft cockpit and how C.N.S facilities have been modernised to give smooth and safe flying. The description is based on author's observation of the development in civil aviation for the last more than four decades and future trends in this field.

The significant development and use of digital technologies has opened up diverse routes for learners to construct and comprehend mathematical knowledge and to solve problems. This implies a revision of the pedagogical landscape in terms of the ways in which students engage in learning, and how understandings emerge. In this chapter we consider how the availability of digital technologies has allowed intended learning trajectories to be structured in particular forms and how these, coupled with the affordances of engaging mathematical tasks through digital pedagogical media, might shape the actual learning trajectories. The evolution of hypothetical learning trajectories is examined, while the transitions learners make when traversing these pathways are also considered. Particular instances are illustrated with examples in several settings.

Young children in many US classrooms have limited opportunities to learn mathematics because their teachers are not well prepared. We developed a learning trajectories-based professional development (PD) approach that has shown large significant effects on early childhood teachers’ beliefs and practices, as well as these teachers’ positive effects on children from low-resource communities. We describe the PD strategies we used, including an online tool that was central to the PD, “Building Blocks Learning Trajectories” (BBLT). We conclude that this approach provided a coherent program of teaching and learning, which promoted significant levels of high-quality practice. Teachers developed pedagogical competence based on specific competencies from the learning trajectories’ PD. Finally, we describe our efforts to disseminate these effective PD practices and systems to far greater numbers of teachers through the creation of a new generation of the online tool that leads to successful results in our scale-up project. Building on the content of the BBLT and adding many new research-based features, “Learning and Teaching with Learning Trajectories” (LTLT, or [LT]2) at LearningTrajectories.org is a PD tool for teachers and caregivers to support early math instruction from birth to grade 3.

tested the specific contributions of LTs to teaching compared with instruction provided without LTs (Frye et al., 2013). The goal of the present study was to compare the learning of kindergarteners who received instruction on length measurement following an empirically validated LT to those who received an equal amount of time on the same instructional activities that were not sequenced along the LT's developmental progression. Theoretical Framework and Background Learning Trajectories: Definition and Assumptions. Our theoretical framework is hierarchic interactionalism (Sarama & Clements, 2009). This term reflects the influence and interaction of global and local (domain specific) cognitive levels and the interactions of innate competencies, internal resources, and experience (e.g., cultural tools and teaching). LTs synthesizing these interactions stand at the core of this theory. Different fields name similar constructs differently, such as the use of "learning progressions" in science education. The structure of LTs is built on the assumption that to be optimally useful to educators, LTs must include and integrate educational standards, empirical research on how children think and learn, as well as teaching strategies

Uno de los problemas que se presenta en la enseñanza del Cálculo consiste en que prevalece el trabajo mecánico en las aulas, dejando a un lado la construcción de los conceptos, así como la evasión de con-sideraciones visuales al abordarlos. Este trabajo se enfoca en propo-ner elementos que permitan al alumno visualizar situaciones en las que está inmerso el concepto de derivada, como es el caso de la razón de cambio instantánea. Para ello se recurre a la tecnología disponible para los jóvenes, como los dispositivos móviles y aplicaciones de cómputo de uso específico. También se empleó realidad aumenta-da debido a que esta tecnología ha sido reconocida como apoyo en ciertos aspectos educativos tales como: la atención y relación viso-espacial. Para evaluar el sistema se consideró una muestra de 2 grupos de 30 estudiantes cada uno, un grupo fue el de control (GC) y el otro el de estudio (GE), ambos fueron evaluados después de haber trabaja-do una secuencia de enseñanza, en donde se utilizó la misma prueba y diferentes herramientas. Los resultados mostraron que el grupo de estudio identificó el concepto de derivada resolviendo de forma correcta un porcentaje mayor de reactivos en el cuestionario final en relación al grupo de control. El incremento en promedio del GE fue de 2.6 a 7.4 y del GC fue de 2.08 a 4, por lo que se concluye que actividades apoyadas con dispositivos móviles contribuyen a mejorar la visualización y la construcción de los conceptos.

abstract
One of the problems presented in the teaching of calculus is that mechanical work prevails in the classrooms, reducing the time for the construction of concepts, as well as preventing visual consider-ations when they are approached. This paper focuses on proposing elements that allow students to visualize situations that incorporate the concept of the derivative, such as the case of the reason for instantaneous change. To this end, we used the technology available to young people, such as mobile devices and specific computer  applications. We also used Augmented Reality since this technology has been considered a successful support in certain educational aspects, such as attention and visual-spatial relationships. To evaluate the system a sample of two groups of 30 students each was used; one was a control group (CG) and the other a research group (RG). Both were evaluated after having worked through a sequence of teaching in which the same test was used with different tools. The results showed that the research group identified the concept of the derivative and correctly solved a higher percentage of problems in the final questionnaire compared with the control group. The aver-age increase of the RG was from 2.6 to 7.4, and the CG was from 2.08 to 4, which led to the conclusion that activities supported by mobile devices contribute to the improvement of visualization and the con-struction of concepts.

Measurement is a critical component of mathematics education, but research on the learning and teaching of measurement is limited. We previously introduced, refined, and validated a developmental progression-the cognitive core of a learning trajectory-for length measurement in the early years. A complete learning trajectory includes instructional activities and pedagogical strategies, correlated with each level of the developmental progression. This study evaluated a portion of our learning trajectory, focusing on the instructional component. We found that the instruction was successful in promoting a progression from one level to the next for 40% of the children, with others developing positive new behaviors (but not sufficient to progress to a new level). We identified specific characteristics of the interventions' tasks and teacher-child interactions that engendered learning. The use of standard or multiple nonstandard units may play different roles at different points in the learning trajectory.

The growing national attention to students’ learning trajectories (LTs) renews the opportunity to explore the ways that teachers may use students’ thinking in their instruction. In this article, we examine teachers’ learning of two frameworks, one for students’ thinking in a particular domain and one for broad student-centered instructional practices, in the context of an elementary grades mathematics professional development setting. As a part of a retrospective analysis of a design experiment, we analyzed 19 lessons of teachers who participated in 60 hr of professional development designed to support their learning of one LT and one framework about student-centered instructional practices. Our findings describe the ways in which teachers brought together these frameworks to enact instructional practices that elicit and use students’ mathematical
thinking in classroom instruction. We conclude by arguing that LTs can serve as a referent for student-centered instructional practices, bridging guidelines for student-centered instruction with domain-specific understandings of students’ thinking for
teachers.

This case study, which is a part of a larger design study, examines teachers' uses of learning trajectories (LTs) in diverse classrooms. Specifically data were collected and analyzed for evidence of equitable use of LTs and learning trajectory-based instruction (LTBI). Qualitative analysis revealed that the use of LTBI did not guarantee equitable instructional practices. Moreover, a number of deficit-oriented themes emerged in one case.

We tested a specific theoretical assumption of a learning trajectories (LTs) approach to curriculum and teaching in the domain of early length measurement. Participating kindergartners (n = 189) were assigned to one of three conditions: LT, reverse-order (REV), or business-as-usual (BAU). LT and REV students received one-on-one instruction using the same activities from a length LT, while the REV condition reversed the LT order. At posttest, LT and REV children exhibited significantly greater learning relative to BAU peers. But importantly, LT children outperformed their REV peers. We conclude that instruction following LTs (i.e., providing instruction just beyond a child’s present level of thinking, progressing through the levels in order as the child advances) may promote more learning than an equivalent amount of instruction using the same activities but that are not theoretically sequenced.

The development of two middle school students' reasoning about quotients in arithmetic and algebraic contexts was studied through a series of individual teaching experiments. Initially, when children were confronted with algebraic contexts, they viewed algebraic fractions as a new case. As they progressed, it appeared that understanding of quotients of algebraic expressions depended heavily on the students' understanding of arithmetical quotients. They used analogy to arithmetic fractional reasoning as a tool for initial understanding, followed similar developmental routes when constructing more advanced knowledge, and folded back to arithmetic analogies when new information was presented. In particular, similar to students' development of understanding in numeric fractions, their thinking began with an understanding of Algebraic Fractions as Part Whole, moved to a notion of Algebraic Fractions as Division, then to seeing Division of Polynomials as Algebraic Fractions. At the end of the teaching experiments, they approached situations involving algebraic fractions or division of polynomials flexibly and used multiple-constructed methods.

This case study, which is a part of a larger design study, examines teachers’ uses of learning trajectories (LTs) in diverse classrooms. Specifically data were collected and analyzed for evidence of equitable use of LTs and learning trajectory-based instruction (LTBI). Qualitative analysis revealed that the use of LTBI did not guarantee equitable instructional practices. Moreover, a number of deficit-oriented themes emerged in one case.