Students’ understanding of the structure of deductive proof

Abstract

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.

Geometry Education, Including the Use of New Technologies: A Survey of Recent Research

“…It appears that the fundamental issue of understanding the need for axioms and for accepting some statements as definitions to avoid circularity has been largely under-researched in the mathematics education community (though see Fujita, Jones, & Miyazaki, 2011;Miyazaki, Fujita, & Jones, 2017). Another under-researched area seems to be exploring the existence of a mathematical choice between defining (and classifying) the quadrilaterals hierarchically or in partitions (compare de Villiers 1994; Usiskin et al 2008).…”

Designing a Web-based Learning Support System for Flow-chart Proving in School Geometry

“…We have provided more detailed accounts of these theoretical and pedagogical aspects in Miyazaki et al ( , 2017 and in . Here, we only briefly explain each of them and then summarize the design principles that informed our web-based learning system.…”

“…The RCGP, however, does not pay detailed attention to the level of 'chaining elements'. Therefore, in this paper, we use the following levels of learner understanding of proof structure initially elaborated by Miyazaki et al ( , 2017: pre-, partial-and holistic-structural levels. These levels are described in Table 1 and the overall Fig.…”

“…However, if they choose ∠ABO = ∠ACO as an assumption, they might fall into logical circularity. This latter case can reveal learners' uncertainty about the structure of proofs; see the final section below (and also Miyazaki et al 2017).…”

“…Through the improvement, learners can understand that they should not use conclusions as assumptions. This understanding is essential for the Partial-structural Relational (b) hypothetical syllogism) sub-level (Miyazaki et al 2017). In fact, both J1 and J2, after they experienced LEVELS 2, 3 and 4 and received various types of feedback from the system, came to know that in a proof they cannot use the same singular proposition as both assumption and conclusion.…”