Critical review of geometry in current textbooks in lower secondary schools in Japan and the UK

This paper reports on an initial analysis of current best-selling textbooks for lower secondary schools in Japan and the UK (specifically Scotland) using an analytic framework derived from the study of the textbooks in the “Trends in International Mathematics and Science Study” (TIMSS). Our analysis indicates that, following the specification of the mathematics curriculum in these countries, Japanese textbooks set out to develop students’ deductive reasoning skills through the explicit teaching of proof in geometry, whereas comparative UK textbooks tend, at this level, to concentrate on finding angles, measurement, drawing, and so on, coupled with a modicum of opportunities for conjecturing and inductive reasoning

Lower secondary school students’ understanding of algebraic proof

Secondary school students are known to face a range of difficulties in learning about proof and proving in mathematics. This paper reports on a study designed to address the issue of students’ cognitive needs for conviction and verification in algebraic statements. Through an analysis of data from 418 students (206 from Grade 8, and 212 from Grade 9), we report on how students might be able to ‘construct’ a formal proof, yet they may not fully appreciate the significance of such formal proof. The students may believe that formal proof is a valid argument, while, at the same time, they also resort to experimental verification as an acceptable way of ‘ensuring’ universality and generality of algebraic statement

Flow-chart proofs with open problems as scaffolds for learning about geometrical proofs

AcceptedArticleCopyright © FIZ Karlsruhe 2015The final publication is available at Springer via http://dx.doi.org/10.1007/s11858-015-0712-5Recent research on the scaffolding of instruction has widened the use of the term to include forms of support for learners provided by, amongst other things, artefacts and computer-based learning environments. This paper tackles the important and under-researched issue of how mathematics lessons in lower secondary school can be designed to scaffold students’ initial understanding of geometrical proofs. In order to scaffold the process of understanding the structure of introductory proofs, we show how flow-chart proofs with multiple solutions in ‘open problem’ situations are a useful form of scaffold. We do this by identifying the ‘scaffolding functions’ of flow-chart proofs with open problems through analysis of classroom-based data from a class of Grade 8 students (aged 13-14 years old) and quantitative data from three classes. We found that using flow-chart proofs with open problems supported the students’ development of a structural understanding of proof by giving them a range of opportunities to connect proof assumptions with conclusions. The implication is that such scaffolds are useful to enrich students’ understanding of introductory mathematical proofs.Grant-in-Aid for Scientific Research, Ministry of Education, Culture, Sports, Science, and Technology, Japan

Combining scaffolding for content and scaffolding for dialogue to support conceptual break throughs in understanding probability

The final publication is available at Springer via http://dx.doi.org/10.1007/s11858-015-0720-5In this paper, we explore the relationship between scaffolding, dialogue and conceptual breakthroughs, using data from a design-based research study into the development of understanding of probability in 10-12 year old students. Our aim in the study was to gain insight into how the combination of the scaffolding of content using technology and scaffolding for dialogue in the expectation that this would facilitate conceptual breakthroughs. We analyse video-recordings and transcripts of pairs and triads talking together around TinkerPlots software with worksheets and teacher interventions, focusing on moments of conceptual breakthrough. The dialogue scaffolding promoted both dialogue moves specific to the context of probability and dialogue in itself. This paper focuses on an episode of learning that occurred within dialogues (framed and supported by the scaffolding. We present this as support for our claim that combining scaffolding for content with scaffolding for dialogue can be effective. This finding contributes to our understanding of both scaffolding and dialogic teaching in mathematics education by suggesting that scaffolding can be used effectively to prepare for conceptual development through dialogue.7th European Community Framework Programme - Marie Curie Intra European Fellowshi

Functions of Open Flow-chart Proving in introductory lessons of formal proving

ArticleLiljedahl, P., Oesterle, S., Nicol, C., & Allan, D. (Eds.). (2014). Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 4):225-232. Vancouver, Canada: PME.conference pape

Supporting students to overcome circular arguments in secondary school mathematics : the use of the flowchart proof learning platform

ArticleProceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education．Ankara, Turkey, 2011-7-10/15, 2011, 2:353-360．conference pape

Introducing the structure of proof in lower secondary school geometry: a learning progression based on flow-chart proving

ArticleProceedings of the 12th International Congress on Mathematical Education (ICME-12)．COEX, Seoul, Korea, 2012-7-8/15, 2012, 2858-2867．conference pape

Proof and proving in current classroom materials

Research across many countries reports that teaching the key ideas of proof and proving to all students is not an easy task. This paper reports on the session of the BSRLM Geometry Working Group which examined current classroom material from the UK with the intention of uncovering the ‘opportunities for proof’ in geometry that are provided by such material. To carry out such an analysis three analytical frameworks are compared. Two of the analytical frameworks, while placing proof and proving in a wider context of learners’ mathematics, may not fully uncover the detail of proof and proving. The third analytical framework, while permitting a detailed analysis of explicit proof and proving, may not fully account for textbooks that devote most space to discussions of proof and proving and/or contain problems that implicitly provoke proof. This comparison reveals some of the complexity of textbook analysis and suggests that further work is needed on a suitable analytical framework