Bracha Kramarski

Enhancing Mathematical Reasoning in the Classroom: The Effects of Cooperative Learning and Metacognitive Training

The purpose of this study was to investigate the effects of four instructional methods on students’ mathematical reasoning and metacognitive knowledge. The participants were 384 eighth-grade students. The instructional methods were cooperative learning combined with metacognitive training (COOP+META), individualized learning combined with metacognitive training (IND+META), cooperative learning without metacognitive training (COOP), and individualized learning without metacognitive training (IND). Results showed that the COOP+META group significantly outperformed the IND +META group, which in turn significantly outperformed the COOP and IND groups on graph interpretation and various aspects of mathematical explanations. Furthermore, the metacognitive groups (COOP+META and IND +META) outperformed their counterparts (COOP and IND) on graph construction (transfer tasks) and metacognitive knowledge. This article presents theoretical and practical implications of the findings.

Improve: A Multidimensional Method For Teaching Mathematics in Heterogeneous Classrooms

The purpose of the present research was to design an innovative instructional method for teaching mathematics in heterogeneous classrooms (with no tracking) and to investigate its effects on students’ mathematics achievement. The method is based on current theories in social cognition and metacognition. It consists of three interdependent components: metacognitive activities, peer interaction, and systematic provision of feedback-corrective-enrichment. The method is called IMPROVE, the acronym of which represents all the teaching steps that constitute the method: Introducing the new concepts, Metacognitive questioning, Practicing, Reviewing and reducing difficulties, Obtaining mastery, Verification, and Enrichment. The research includes two studies, both implemented in seventh grades: One focused on in-depth analyses of students’ information processing under the different learning conditions (N = 247), and one investigated the development of students’ mathematical reasoning over a full academic year (N = 265). Results of both studies showed that IMPROVE students significantly outperformed the nontreatment control groups on various measures of mathematics achievement. The theoretical and practical implications of the research are discussed.

Online Discussion and Self-Regulated Learning: Effects of Instructional Methods on Mathematical Literacy.

ABSTRACT The authors investigated the effects of online discussion embedded within metacognitive guidance on mathematical literacy and self-regulated learning (SRL). They compared 4 instructional methods: online discussion embedded within metacognitive guidance (Online+Meta), online discussion without metacognitive guidance (Online), face-to-face discussion with metacognitive guidance (Ftf+Meta), and face-to-face discussion without metacognitive guidance (Ftf). Participants were 86 seventh-grade Israeli boys and girls who practiced problem solving in 4 classes. Results showed that the Online+Meta students significantly outperformed the Ftf+Meta students, who, in turn, significantly outperformed the Online and Ftf students on mathematical literacy of standard tasks, real-life tasks, and various aspects of self-regulated learning. Regarding Online students, results were mixed; these students outperformed the Ftf students on part of the criteria for standard problem-solving standard tasks and real-life tasks. The authors found no significant differences between the Online and Ftf students on SRL. Theoretical and practical implications of the findings are discussed.

Effects of Multilevel Versus Unilevel Metacognitive Training on Mathematical Reasoning

Abstract The effects of 3 instructional methods on mathematical reasoning were investigated. The methods are (a) cooperative learning embedded within multilevel metacognitive training (MMT), (b) cooperative learning embedded within unilevel metacognitive training (UMT), and (c) learning in the whole class with no metacognitive training. MMT was implemented in mathematics and English classrooms; UMT was used only in mathematics classrooms; and the whole class with no metacognitive training served as a control group. Results indicated that students who were exposed to MMT significantly outperformed their counterparts who were exposed to UMT who, in turn, significantly outperformed the control group. Effects of MMT were observed on students while they solved mathematical problems. Theoretical and practical implications of the study are discussed.

Using computer algebra systems in mathematical classrooms

This paper describes a research whose main focus is the use of Computer Algebra Systems (CAS) in mathematical classrooms and the didactical possibilities linked with its use. The possibilities of integrating Self-Regulated Learning (SRL) within the CAS environment are brought into focus. Forty-three Israeli students (mean age 13.3) were assigned to two learning algebraic groups. The first group received explicit meta- cognitive SRL with CAS (CAS + SRL); the second group was exposed to CAS without SRL (CAS). Empirical results from the experimental and case study designs revealed that (CAS + SRL) students outperformed (CAS) students on algebraic thinking and that (CAS + SRL) students regulated their learning more effectively.

Using Errors as Springboards for Enhancing Mathematical Reasoning With Three Metacognitive Approaches

The authors examined effects of 3 metacognitive approaches and 1 control group on mathematical reasoning, conceptual errors, and metacognitive knowledge. The metacognitive approaches were (a) diagnosing errors (DIA), (b) improvement via self-questioning (IMP), and (c) a combined approach (DIA+IMP). Controls (CONT) received no metacognitive instruction. Participants were 115 9th-grade students in an Israeli junior high school studying linear functions and graph representations. Results indicated that students using the combined approach outperformed their peers in all measures of mathematical reasoning and metacognitive knowledge. The IMP students outperformed the DIA and CONT students in problem-solving skills and metacognitive strategy use, and the DIA students outperformed their peers in reducing conceptual errors. The authors discuss the practical and theoretical implications of these metacognitive approaches.

Effects of multimedia environments on kindergarten children’s mathematical achievements and style of learning

The purpose of the study is two‐fold: (a) to investigate the effects of learning mathematics with multimedia embedded in different styles of learning (cooperative learning versus individual learning) in kindergarten on students’ mathematical achievements; (b) to examine students’ preference for style of learning with computers in kindergarten. Participants were 116 students (girls and boys) who studied in kindergarten classes. One group was exposed to multimedia embedded in cooperative learning (CL), the second group was exposed to multimedia embedded in individual learning (IL) and the control group (C) was not exposed to multimedia. Findings indicated that the CL and IL students significantly outperformed the C group in mathematical achievement. The IL students further improved their mathematical skills at the higher level, while the CL students further increased their positive attitude about cooperative learning. Theoretical and practical implications are discussed. Les effets des environnements multimedia sur les résultats en mathématiques des enfants de l’école maternelle et sur leur style d’apprentissage Le but de la présente étude est double: (a) il s’agit d’étudier à l’école maternelle les effets de l’apprentissage des mathématiques avec des multimedia intégrés dans différents styles d’apprentissage (l’apprentissage coopératif opposé à l’apprentissage individuel), sur les résultats qu’obtiennent les élèves en mathématiques. Et (b) Il s’agit d’examiner quel est le style d’apprentissage sur ordinateur que préfèrent les élèves de maternelle. Il y avait 116 participants, élèves (garçons et filles) de maternelle. L’un des groupes a été exposé aux multimedia intégrés dans l’apprentissage coopératif (CL), le deuxième groupe a été exposé aux multimedia intégrés dans l’apprentissage individuel (IL) et le groupe de contrôle (C) n’a pas été exposé aux multimedia. Les observations ont montré que les élèves de CL et IL avaient des résultats en mathématiques nettement supérieurs à ceux du groupe de contrôle (C). Les élèves en apprentissage individuel ont amélioré leurs compétences mathématiques au niveau le plus élevé tandis que les élèves en apprentissage coopératif ont renforcé leur attitude favorable vis‐à‐vis de l’apprentissage coopératif. L’article examine les implications théoriques et pratiques. Auswirkungen von Multimediaumgebungen auf die Mathematikleistungen und den Lernstil von Kindergartenkindern Diese Studie verfolgt zwei Ziele: (a) die Auswirkungen auf das Mathematiklernen mit Multimediaunterstützung in verschiedenen Lernformen (kooperatives Lernen vs. Einzellernen) im Kindergarten auf die Mathematikleistungen von Kindern zu untersuchen und (b) herauszufinden, welche Form des Lernens mit Computern die Kinder präferieren. 116 Kindergartenkinder (Mädchen und Jungen) nahmen an der Studie teil. Eine Teilgruppe nutzte Multimedia beim kooperativen Lernen (CL), die zweite beim individuellen Lernen (IL) und die Kontrollgruppe (C) arbeitete ohne Multimediaunterstützung. Die Ergebnisse wiesen darauf hin, dass die Schüler der CL und IL Gruppen bei mathematischen Leistungen die Schüler der Kontrollgruppe signifikant übertrafen. Die Teilnehmer der Einzellerner‐Gruppe verbesserten außerdem ihre Fachkenntnisse auf höherem Niveau, während die Teilnehmer aus der kooperativ lernenden Gruppe außerdem ihre positive Einstellung zum gemeinsamen Lernen weiter verstärkten. Theoretische und praktische Folgerungen werden diskutiert.

Student and Teacher Perspectives on IMPROVE Self-Regulation Prompts in Web-Based Learning

This chapter describes the results of eight controlled experimentations examining different conditions for implementation of the IMPROVE self-questioning prompts (Kramarski & Mevarech, 2003; Mevarech & Kramarski, 1997) in web-based learning environments from two perspectives, first for students’ learning in the classroom, and second for preservice teachers’ learning during their professional preparation. The IMPROVE method aims to support key aspects of self-regulation targeting learning processes. In evaluating the effect of the IMPROVE prompts, we focused our efforts on assessing progress at high levels of conceptual understanding in the learning domain, referring to mathematical or scientific reasoning among students and teachers alike and also referring to designing traditional and technology-based lessons among the teachers. Thus, we assessed whether learners performed well not only on immediate posttests with items similar to training, but also on tests measuring near and far transfer. In addition, we assessed acquisition of self-regulated learning (SRL) that included offline aptitude questionnaires and online process measures during real-time forum discussions. In this chapter we critically discuss the findings and raise directions for practical implications and future inquiry.

New Perspectives on Integrating Self-Regulated Learning at School

CorrespondenceshouldbeaddressedtoBrachaKramarski;bracha.kramarski@biu.ac.ilReceived20February2013;Accepted20February2013Copyright©2013BrachaKramarskietal. This is an open access article distributed under the Creative Commons AttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.

Online interactions in a mathematical classroom

This study compares the effects of forum discussion with metacognitive guidance versus forum discussion without metacognitive guidance (Forum) on solving real life mathematical tasks and self‐regulated learning. Participants were 43 Grade 7 students (boys and girls) who practiced online problem‐solving in two classes. The metacognitive guidance was based on the IMPROVE metacognitive questioning suggested by Mevarech and Kramarski. The method utilized a series of four self‐addressed metacognitive questions: comprehension questions; connection questions; strategic questions; reflection questions. It was found that students who were exposed to forum discussion with metacognitive guidance (Forum+Meta) outperformed students that were not exposed to metacognitive guidance (Forum discussion) on real life mathematical tasks and self‐regulated learning. It was found that the Forum+Meta students were more positive than the Forum students about their interest in online problem‐solving, mathematical engagement and online communication. Les interactions en ligne dans la classe de Mathématiques Cette étude compare les effets des discussions en Forum avec guidage métacognitif (Forum+Meta) par rapport aux discussions en Forum sans guidage métacognitif (Forum) dans la résolution mathématique de l’apprentissage fondé sur des tâches de la vie réelle et auto‐régulé. Ont participé à l’expérience 43 élèves de 7e année (5e), garçons et filles, qui ont pratiqué la résolution de problèmes en ligne en deux classes. Le guidage métacognitif était basé sur le questionnaire métacognitif IMPROVE tel qu’il a été proposé par Mevarech et Kramarski (1997, 2003). La méthode a utilisé une série de quatre questions métacognitives auto adressées: des questions de Compréhension; des questions de liaison; des questions stratégiques et des questions de réflexion. On a trouvé que les élèves qui ont été exposés à la discussion Forum+Meta ont obtenu de meilleurs résultats que les élèves qui n’étaient pas exposés au guidage métacognitif (discussion en Forum)pour l’apprentissage mathématique fondé sur des tâches de la vie réelle et auto régulé. On a trouvé que les les élèves Forum+Meta avaient un avis plus positif que les élèves Forum seul en ce qui concerne leur intérêt pour la résolution de problèmes en ligne, l’intérêt pour les mathématiques et la communication en ligne. Online‐Interaktionen im Mathematikunterricht In der Studie werden die Wirkung von Forum‐Diskussion mit metakognitiver Anleitung (Forum + Meta) und Forum‐Diskussion ohne metakognitive Anleitung (Forum) beim Lösen mathematischer Anwendungsaufgaben und selbstgesteuertem Lernen verglichen. Teilnehmer waren 43 Siebtklässler (Jungen und Mädchen), die in zwei Klassen mit Online‐Problem‐Lösen beschäftigt waren. Die metakognitive Anleitung wurde nach der von Mevarech und Kramarski (1997, 2003) vorgeschlagenen Befragungsmethode durchgeführt. Die Methode wertete eine Serie von vier selbstadressierten Fragen aus: Verständnisfragen, Verknüpfungsfragen, Strategiefragen und Besinnungsfragen. Es wurde festgestellt, dass Schüler, die nach der Forum + Meta Methode arbeiten mussten, bessere Ergebnisse bei Anwendungsaufgaben und selbstgesteuertem Lernen erzielten, als die, die keine metakognitive Anleitung erfuhren (Forum Diskussion). Auch in Bezug auf ihr Interesse am Online‐Problemlösen, ihr mathematisches Engagement und die Online‐Kommunikation reagierten die “Forum + Meta–Schüler” positiver als die “Forum‐Schüler”.