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

Miyazaki, Mikio; Fujita, Taro; Jones, Keith

doi:10.1007/s11858-015-0712-5

Cited by 28 publications

(22 citation statements)

References 17 publications

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“…In the first learning phase, constructing flow-chart proofs in an open problem situation helped to scaffold students' understanding of the structure of deductive proofs (Miyazaki, Fujita, & Jones, 2015). Since students at this early stage of learning about a deductive proof might see it as a rather meaningless set of symbols about the properties of geometric shapes, through the follow-chart proving we provided a visualisation of the structure of the proofs.…”

Section: A Sequence Of Lessons To Introduce the Structure Of Deductivmentioning

confidence: 99%

Students’ understanding of the structure of deductive proof

2016

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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.

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Section: A Sequence Of Lessons To Introduce the Structure Of Deductivmentioning

confidence: 99%

Students’ understanding of the structure of deductive proof

2016

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“…The conclusion of the study by [10] also showed that the use of Flow Proof can overcome students' difficulties in performing proof. Similarly to research by [11], the use of Flow Proof with open problems encouraged student's development toward understanding the structure of proof by providing assumptions with conclusions.…”

Section: Fig 4 Students Learning Resultsmentioning

confidence: 99%

Students’Ability in Analyzing by Using a Flow Proof to Solve Problems in Real-Analysis Lecture

Murni¹,

Subhan²

2018

Proceedings of the 2nd International Conference on Mathematics and Mathematics Education 2018 (ICM2E 2018)

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Real analysis is one of subjects that help the students to experience critical thinking processes. Students must understand mathematical definitions and theorems in order to solve problems and perform a process of analytical thinking, so that their analysis will be developed to the given problems. The research that has been done showed that students' ability in analyzing can be improved by using Flow Proof in a preliminary analysis, but not their ability in narrating the proof yet.

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“…Hasil penelitian menunjukkan bahwa ketika siswa diberikan suatu masalah dalam Geometri dan siswa diminta untuk menuliskan premis serta pernyataan yang akan dibuktikan, begitu juga dengan membangun diagram Flow Proof dan Two-Column Proof, terlihat bahwa siswa dapat menyelesaikan pembuktian, 2) penelitian Miya zaki, Fujita, Jones, dan Iwanaga (2017). Hasil penelitian menunjukkan bahwa penggunaan Flow Proof pada pelajaran Geometry dapat mengatasi kesulitan siswa dalam melakukan pembuktian, dan 3) penelitian Miyazaki, Fujita, dan Jones (2015). Hasil penelitian menunjukkan bahwa penggunaan Flow Proof dengan problem terbuka mendorong perkembangan siswa terhadap pemahaman struktur bukti dengan memberikan berbagai hubungan asumsi dengan kesimpulannya.…”

Section: Pendahuluanunclassified

Penggunaan Flow Proof pada Perkuliahan Analisis Real untuk Meningkatkan Kemampuan Mahasiswa dalam Menganalisis Pembuktian

Helma

2019

jep

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Real Analysis is the subjects to help the students in critical thought. Students must have the definitions and theorems to solve the problems and perform a process which is often called the preliminary analysis. After that, students construct the proof based on the preliminary analysis. Based on the characteristics of the problem in Real Analysis, the solution given to the problem is using flow proof in composing preliminary analysis The research shows that the student's ability in analyzing can be improved by using flow proof This is an open access article distributed under the Creative Commons 4.0 Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

Abstract: Abstract

Cited by 28 publications

References 17 publications

Students’ understanding of the structure of deductive proof

Students’ understanding of the structure of deductive proof

Students’Ability in Analyzing by Using a Flow Proof to Solve Problems in Real-Analysis Lecture

Penggunaan Flow Proof pada Perkuliahan Analisis Real untuk Meningkatkan Kemampuan Mahasiswa dalam Menganalisis Pembuktian

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