Students’ understanding of the structure of deductive proof

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: Pre-structural, 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.     

Learning to Teach Science: The Curriculum of Student Teaching in Hiroshima "Attached" Schools

Journal of Science Teacher Education -     

Levels of Proof in Lower Secondary School Mathematics

The purpose of this study is to establish levels from an inductive proof to an algebraic demonstration in lower secondary school mathematics. I propose that we can establish six levels of proof in lower secondary school mathematics as steps from an inductive proof to an algebraic demonstration on the basis of three axes (contents of proof, representation of proof, and students' thinking). To reach this conclusion, I firstly examine the meaning of demonstration in lower secondary school mathematics and proof in lower secondary school mathematics, and show the relationships between them. Secondly, I set out four basic levels of proof, as seen from two aspects (contents and representation of proof). Thirdly, I subdivide them into six levels from the third aspect of students' thinking. Finally, I illustrate my discussion with a 7th grader's activities.This revised version was published online in September 2005 with corrections to the Cover Date.     

Cognitive Incoherence of Students Regarding the Establishment of Universality of Propositions through Experimentation/Measurement

The purpose of this research is to construct a conceptual framework for use in capturing students’ unstable perception regarding the establishment of the universality of propositions through experimentation/measurement in school geometry. As a conceptual framework, this research uses a quadrangular–pyramid model comprised of five representative aspects: idealism/teleology, pessimism, optimism, actualism, and na?ve pre-established harmony. In constructing the framework, I decided on the following viewpoints: (1) the students take as a criterion of the universality of propositions whether the proposition and the result of experimentation/measurement match or not; (2) the students assume that the result will always match the proposition; and (3) the students consider improving the method of experimentation/measurement based on whether the result matches or not. To illustrate the phenomenon of students’ unstable perception, I focus on the intermediate aspects between actualism and na?ve pre-established harmony, in which the cognitive incoherence of students is conspicuous in that it is easy-to-manifest. I discovered two types of unstable perception through analyzing two 8th graders’ responses to questionnaires: (1) one student understood asymptotic accessibility to the match by improving the measurement, although the student was skeptical about reaching the final stage; and (2) another student understood the necessity of the match in order to guarantee her geometrical activity, although she was confused when confronted with the result in that it did not match, even when measured with high-precision equipment.