Metacognitive failure in constructing proof and how to scaffold it

This research aims to describe the students’ metacognitive failure in constructing proof and the scaffolding support. The participants of this qualitative case study were eight preservice mathematics teachers of six-semester, State University of Malang. We carried out a test about proof construction problems in Abstract Algebra. Then we verified the data using triangulation of constant comparative method from a test and a task-based interview with the stimulated recall. The results indicated two groups of students in proving strategy. Group I performed “appropriate” syntactic strategy and Group II vice versa. Blindness was experienced by the subject that does not recognize errors detection or the ambiguity of the proof. Mirage occurred when the subject recognizes an error detection on the proper strategy or application of a theorem, then is unable to verify the truth of his work. Misdirection appeared when the subject recognizes a lack of progress, then uses an incomplete or irrelevant concept. Vandalism emerged with no progress or detection of errors of the strategy then the subject performs some irrelevant steps to the issue or uses a misconception. Practically, the teachers can use these results for learning innovations in scaffolding-based proof courses. The scaffolding might need some development and application in supporting students to overcome difficulty in proving mathematical sentences.

Sources of Students’ Difficulties with Proof By Contradiction

The literature on proof by contradiction (PBC) is nearly unanimous in claiming that this proof technique is “more difficult” for students than direct proof, and offers multiple hypotheses as to why this might be the case. To examine this claim and to evaluate some of the hypotheses, we analyzed student work on proof construction problems from homework and examinations in a university “Introduction to Proof” course taught by one of the authors. We also conducted stimulated-recall interviews with student volunteers probing their thought processes while solving these problems, and their views about PBC in general. Our results suggest that the knowledge resources students bring to bear on proof problems, and how these resources are activated, explain more of their “difficulties” than does the logical structure of the proof technique, at least for this population of students.

Preliminary study of an ancient earthquake-proof construction technique monitoring by an innovative structural health monitoring system

<p class="Abstract">The historical and cultural heritage analysis of the Italian territory is of primary importance, because this territory is one of the richest in the world and can enrich our knowledge in different field. In fact, in the field of structural engineering, a new discovery was made in Calabria, in the south of Italy. By investigating various architectural treatises related to earthquake-proof constructions, new knowledge was ganed by analyzing buildings made with fictile tubules bricks. One of them is an unprecedented anti-seismic construction widespread in southern Calabria and patented by Pasquale Frezza.</p><p class="Abstract">In order to avoid the collapses of these structures, in this work, an innovative method to monitor and obtain the mechanical properties of these structures in real time, minimizing measurement uncertainty is proposed.</p>

Supporting Mathematical Argumentation and Proof Skills: Comparing the Effectiveness of a Sequential and a Concurrent Instructional Approach to Support Resource-Based Cognitive Skills

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as ‘resource-based,’ as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students’ work on proof construction tasks: (i) a sequential approach focusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) a concurrent approach focusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students.

Enhanced Realizability Interpretation for Program Extraction

This thesis presents Intuitionistic Fixed Point Logic (IFP), a schema for formal systems aimed to work with program extraction from proofs. IFP in its basic form allows proof construction based on natural deduction inference rules, extended by induction and coinduction. The corresponding system RIFP (IFP with realiz-ers) enables transforming logical proofs into programs utilizing the enhanced re-alizability interpretation. The theoretical research is put into practice in PRAWF1, a Haskell-based proof assistant for program extraction.

Conception and development of inductive reasoning and mathematical induction in the context of written argumentations

Nowadays, mathematical reasoning and making proof have taken importance for all students from the grade level of elementary education to university. More specifically, mathematical induction (MI) is a kind of proof and reasoning strategy taking place nearly all grade levels. Moreover, teachers are important factors affecting student learning and they can acquire necessary knowledge and skills developmentally in their teacher education programs. This paper makes contributions to domain of research by investigating the development of PMT’s conception of MI in the context of written argumentations encouraging MI. In other words, the purpose of this multiple case study is to explore how PMT’s conception of mathematical induction develop through their written argumentations. These cases show that there exist improvements in PMT’s written argumentations, conception of MI and proof construction activities related to MI. In other words, the more organized and structured they produced written argumentation, the more successfully they use and make mathematical induction.

Efficient verification of parallel matrix multiplication in public cloud: the MapReduce case

With the advent of cloud-based parallel processing techniques, services such as MapReduce have been considered by many businesses and researchers for different applications of big data computation including matrix multiplication, which has drawn much attention in recent years. However, securing the computation result integrity in such systems is an important challenge, since public clouds can be vulnerable against the misbehavior of their owners (especially for economic purposes) and external attackers. In this paper, we propose an efficient approach using Merkle tree structure to verify the computation results of matrix multiplication in MapReduce systems while enduring an acceptable overhead, which makes it suitable in terms of scalability. Using the Merkle tree structure, we record fine-grained computation results in the tree nodes to make strong commitments for workers; they submit a commitment value to the verifier which is then used to challenge their computation results’ integrity using elected input data as verification samples. Evaluation outcomes show significant improvements comparing with the state-of-the-art technique; in case of 300*300 matrices, 73% reduction in generated proof size, 61% reduction in the proof construction time, and 95% reduction in the verification time.

DEDUCTIVE OR INDUCTIVE? PROSPECTIVE TEACHERS’ PREFERENCE OF PROOF METHOD ON AN INTERMEDIATE PROOF TASK

The emerging of formal mathematical proof is an essential component in advanced undergraduate mathematics courses. Several colleges have transformed mathematics courses by facilitating undergraduate students to understand formal mathematical language and axiomatic structure. Nevertheless, college students face difficulties when they transition to proof construction in mathematics courses. Therefore, this descriptive-explorative study explores prospective teachers' mathematical proof in the second semester of their studies. There were 240 pre-service mathematics teachers at a state university in Surabaya, Indonesia, determined using the conventional method. Their responses were analyzed using a combination of Miyazaki and Moore methods. This method classified reasoning types (i.e., deductive and inductive) and types of difficulties experienced during the proving. The results conveyed that 62.5% of prospective teachers tended to prefer deductive reasoning, while the rest used inductive reasoning. Only 15.83% of the responses were identified as correct answers, while the other answers included errors on a proof construction. Another result portrayed that most prospective teachers (27.5%) experienced difficulties in using definitions for constructing proofs. This study suggested that the analytical framework of the Miyazaki-Moore method can be employed as a tool to help teachers identify students' proof reasoning types and difficulties in constructing the mathematical proof.