Distinguishing schemes and tasks in children's development of multiplicative reasoning

We present a synthesis of findings from constructivist teaching experiments regarding six schemes children construct for reasoning multiplicatively and tasks to promote them. We provide a task-generating platform game, depictions of each scheme, and supporting tasks. Tasks must be distinguished from children’s thinking, and learning situations must be organized to (a) build on children’s available schemes, (b) promote the next scheme in the sequence, and (c) link to intended mathematical concepts

Distinción de esquemas y tareas en el desarrollo del razonamiento multiplicativo de los niños

We present a synthesis of findings from constructivist teaching experiments regarding six schemes children construct for reasoning multiplicatively and tasks to promote them. We provide a task-generating platform game, depictions of each scheme, and supporting tasks. Tasks must be distinguished from children’s thinking, and learning situations must be organized to (a) build on children’s available schemes, (b) promote the next scheme in the sequence, and (c) link to intended mathematical concepts.Presentamos una síntesis de hallazgos de experimentos de enseñanza constructivistas en relación con seis esquemas que los niños construyen para razonar multiplicativamente y tareas para promoverlos. Proveemos una plataforma de juego generadora de tareas, descripciones de cada esquema y tareas para apoyarlos. Las tareas deben distinguirse del pensamiento de los niños, y las situaciones de aprendizaje deben organizarse para que (a) se basen en los esquemas que los niños tienen disponibles, (b) promuevan el siguiente esquema en la secuencia y (c) se relacionen con los conceptos matemáticos pretendidos.This research was supported by the US National Science Foundation under grant DRL 0822296, which funded the activities of the Nurturing Multiplicative Reasoning in Students with Learning Disabilities in a Computerized Conceptual-Modeling Environment (NMRSD) project. The opinions expressed in this article do not necessarily reflect the views of the Foundation

Locally integrating theories to investigate students’ transfer of mathematical reasoning

International audienceTo theorize the study of students’ transfer of mathematical reasoning, we integrate Lobato’s theory of actor-oriented transfer, Thompson’s theory of quantitative reasoning, and Marton’s variation theory. Linking theory and method, we argue for an expansion of design possibilities for researchers’ investigations of students’ transfer of reasoning. To bolster our argument, we provide empirical data of a secondary student’s transfer of a particular form of reasoning—covariational reasoning. Our research has implications for researchers’ development of new theoretical and methodological approaches to investigate complex phenomena, such as students’ mathematical reasoning

Leveraging difference to promote students’ conceptions of graphs as representing relationships between quantities

International audienceBy creating different Cartesian graphs to represent the same relationship between quantities, students can expand their conceptions of what graphs represent. Interweaving Marton’s variation theory and Thompson’s theory of quantitative reasoning, we designed digital task sequences topromote students’ conceptions of graphs as representing relationships between quantities. In the tasks, which link animations and dynamic graphs, students created different graphs to represent the same relationship between quantities. We report results of a qualitative study (n=13) investigatingsecondary students’ interactions with the digital tasks in an individual interview setting. We found that the digital tasks were viable for promoting students’ creation of graphs to represent relationships between quantities. Our findings have implications for task design. Namely, students should have opportunities to create different graphs to represent the same relationship between quantities

Understanding Students' Learning through their Activity: Toward a Basis for a Scientific Approach to Task Design and Sequencing

No abstract available

Understanding Students' Learning through their Activity: Toward a Basis for a Scientific Approach to Task Design and Sequencing

No abstract available

PNA

Resumen basado en el de la publicaciónSe presenta una síntesis de hallazgos de experimentos de enseñanza constructivistas en relación con seis esquemas que los niños construyen para razonar multiplicativamente y tareas para promoverlos. Se provee una plataforma de juego generadora de tareas, descripciones de cada esquema y tareas para apoyarlos. Las tareas deben distinguirse del pensamiento de los niños, y las situaciones de aprendizaje deben organizarse para que (a) se basen en los esquemas que los niños tienen disponibles, (b) promuevan el siguiente esquema en la secuencia y (c) se relacionen con los conceptos matemáticos pretendidos.Ministerio de Educación. Subdirección General de Documentación y Publicaciones; Calle San Agustín, 5; 28014 Madrid; Tel. +34917748000; Fax +34917748014; [email protected]

Supplement 1. Examples of R code to fit models.

<h2>File List</h2><blockquote> <a href="FalsePositiveLikelihoods.R">FalsePositiveLikelihoods.R</a><br> <a href="FPdatasets.Rdata">FPdatasets.Rdata</a> </blockquote><h2>Description</h2><blockquote> <p>The FalsePositiveLikelihoods.R file contains scripts for R (ver 2.12.1) to calculate each of the basic likelihoods presented in the paper. The script calls the example datasets FPdatasets.Rdata which give examples fitting the same data set using each of the models. The script also gives an example for estimating occupancy when detection is a function of a covariate (dist).</p> </blockquote

Appendix E. Additional example application for experimental data with known occupancy probability.

Additional example application for experimental data with known occupancy probability