Reasoning-and-proving in geometry in school mathematics textbooks in Japan

Fujita, Taro; Jones, Keith

doi:10.1016/j.ijer.2013.09.014

Cited by 36 publications

(20 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In doing so, we focus on the domain of geometry because it is a key area in which proofs and proving are learned and used in secondary school mathematics. While there have been numerous studies on proof and proving in secondary school geometry covering students' capabilities regarding proofs and their perspectives on proving in classes (Herbst & Brach, 2006;Hoyles & Healy, 2007;Senk, 1985), teachers' conceptions of proof (Knuth, 2002a(Knuth, , 2002b, textbook analysis (Fujita & Jones, 2014;Otten, Gilbertson, Males, & Clark, 2014), task design (Cirillo & Herbst, 2012;Komatsu, in press), and classroom-based research, including investigation of student-teacher interactions and teaching interventions to enhance student learning (Martin, McCrone, Bower, & Dindyal, 2005;Miyazaki, Fujita, & Jones, 2015), only a few studies have related to proof validation, mostly focusing on whether students can discern the invalidity of circular arguments (in which conclusions are used as suppositions). For instance, McCrone and Martin (2004) surveyed 18 American high school students and showed that only 22% of the students correctly judged a circular argument invalid.…”

Section: Research On Proof Validationmentioning

confidence: 99%

Proof validation and modification in secondary school geometry

Komatsu

Jones

Ikeda³

et al. 2017

The Journal of Mathematical Behavior

Self Cite

View full text Add to dashboard Cite

Proof validation is important in school mathematics because it can provide a basis upon which to critique mathematical arguments. While there has been some previous research on proof validation, the need for studies with school students is pressing. For this paper, we focus on proof validation and modification during secondary school geometry. For that purpose, we employ Lakatos' notion of local counterexample that rejects a specific step in a proof. By using Toulmin's framework to analyze data from a task-based questionnaire completed by 32 ninth-grade students in a class in Japan, we identify what attempts the students made in producing local counterexamples to their proofs and modifying their proofs to deal with local counterexamples. We found that student difficulties related to producing diagrams that satisfied the condition of the set proof problem and to generating acceptable warrants for claims. The classroom use of tasks that entail student discovery of local counterexamples may help to improve students' learning of proof and proving. first category encompasses studies in which different types of mathematical argument (visual, inductive, generic, or deductive) are presented to students and the students are asked to evaluate whether the arguments are personally convincing and whether the arguments can be considered to constitute proofs (e.g., Healy & Hoyles, 2000; Martin & Harel, 1989; Segal, 1999). A second category relates to students' understanding of given correct proofs. Based on models of proof comprehension (Mejia-Ramos, Fuller, Weber, Rhoads, & Samkoff, 2012; Yang & Lin, 2008), studies in this category seek to identify effective proof comprehension strategies and examine the effectiveness of specific training to improve students' proof comprehension (Hodds, Alcock, & Inglis, 2014; Samkoff & Weber, 2015). A third category involves proof validation; here, researchers show students purported deductive proofs and ask them to determine whether the proofs are valid or invalid (

show abstract

Section: Research On Proof Validationmentioning

confidence: 99%

Proof validation and modification in secondary school geometry

Komatsu

Jones

Ikeda³

et al. 2017

The Journal of Mathematical Behavior

Self Cite

View full text Add to dashboard Cite

show abstract

“…Although there is no official teaching sequence prescribed in the Japanese national BCourse of Study^, a progression can be found in the seven authorised textbooks and in the practice of many schools (Fujita & Jones, 2014). Building on informal proofs in earlier grades, students in Grade 8 are introduced to deductive proof through studying properties of angles and lines, triangles, and quadrilaterals; here they learn the structure of deductive proofs and how to construct the proofs, and then explore and prove properties of triangles and properties of quadrilaterals.…”

Section: Students' Understanding Of Deductive Proofs In Japanese Secomentioning

confidence: 99%

Students’ understanding of the structure of deductive proof

2016

Self Cite

View full text Add to dashboard Cite

While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students' understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Prestructural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.

show abstract

“…Also, standards-based textbooks offer more reasoning and proving opportunities, which was also confirmed by Davis (2012). Fujita and Jones (2013) examined Japanese geometry textbooks on reasoning and proving opportunities and showed that a larger portion of exercises, compared to American textbooks, are devoted to reasoning and proving. However, this finding may be skewed because their study analyzed chapters that explicitly address proofs, triangle congruence, and parallelogram as argued by Cai * Korean high school geometry topics are three dimensional coordinates and figures and vectors.…”

Section: Textbook Studies On Reasoning and Provingmentioning

confidence: 69%

“…Several studies have examined reasoning and proving opportunities in various textbooks (Davis, 2010;Fujita & Jones, 2013;Stylianides, 2009;Thompson et al, 2012). Stylianides (2009) found in his analysis of the Connected Mathematics series that about 40% of tasks are related to reasoning and proving and rationale.…”

Section: Textbook Studies On Reasoning and Provingmentioning

confidence: 99%

Reasoning and Proving Opportunities in Textbooks: A Comparative Analysis

Hong

Choi²

2018

International Journal of Research in Education and Science

View full text Add to dashboard Cite

In this study, we analyzed and compared reasoning and proving opportunities in geometry lessons from American standard-based textbooks and Korean textbooks to understand how these textbooks provide student opportunities to engage in reasoning and proving activities. Overall, around 40% of exercise problems in Core Plus Mathematics Project (CPMP) ask for reasoning and proving activities while 20% of exercise problems in Korean textbooks are about reasoning and proving activities. One of the major findings is that Korean and CPMP students have different opportunities in learning geometry. It may be interesting to investigate geometric reasoning and proving performance of Korean and CPMP students to see how textbooks influence their learning.

show abstract

Reasoning-and-proving in geometry in school mathematics textbooks in Japan

Cited by 36 publications

References 8 publications

Proof validation and modification in secondary school geometry

Proof validation and modification in secondary school geometry

Students’ understanding of the structure of deductive proof

Reasoning and Proving Opportunities in Textbooks: A Comparative Analysis

Contact Info

Product

Resources

About