Developing Students Ability To Write Mathematical Proof By Polya Method

Both writing and reading a proof is equally not easy. Some mathematicians attested that students found difficulties in mathematical proving. Mathematics and mathematics education experts like Jones (1997, 2001), Weber (2001), and Smith (2006) found that difficulty in proof writing is due to: lack of theorem and concept understanding, lack of proving ability, and there is a teaching-learning process that unites with the subject. So, it is truly needed that class of writing proof in order to help to generate students’ ability to do mathematical proving. Polya method is going to be that purpose. Keywords: mathematics proof, Polya metho

The Informal Logic of Mathematical Proof

Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is that a more nuanced understanding of mathematical proof and discovery may be achieved by paying attention to the aspects of mathematical argumentation which can be captured by informal, rather than formal, logic. Two accounts of argumentation are considered: the pioneering work of Stephen Toulmin [The uses of argument, Cambridge University Press, 1958] and the more recent studies of Douglas Walton, [e.g. The new dialectic: Conversational contexts of argument, University of Toronto Press, 1998]. The focus of both of these approaches has largely been restricted to natural language argumentation. However, Walton's method in particular provides a fruitful analysis of mathematical proof. He offers a contextual account of argumentational strategies, distinguishing a variety of different types of dialogue in which arguments may occur. This analysis represents many different fallacious or otherwise illicit arguments as the deployment of strategies which are sometimes admissible in contexts in which they are inadmissible. I argue that mathematical proofs are deployed in a greater variety of types of dialogue than has commonly been assumed. I proceed to show that many of the important philosophical and pedagogical problems of mathematical proof arise from a failure to make explicit the type of dialogue in which the proof is introduced.Comment: 14 pages, 1 figure, 3 tables. Forthcoming in Perspectives on Mathematical Practices: Proceedings of the Brussels PMP2002 Conference (Logic, Epistemology and the Unity of the Sciences Series), J. P. Van Bendegem & B. Van Kerkhove, edd. (Dordrecht: Kluwer, 2004

Improving legibility of natural deduction proofs is not trivial

In formal proof checking environments such as Mizar it is not merely the validity of mathematical formulas that is evaluated in the process of adoption to the body of accepted formalizations, but also the readability of the proofs that witness validity. As in case of computer programs, such proof scripts may sometimes be more and sometimes be less readable. To better understand the notion of readability of formal proofs, and to assess and improve their readability, we propose in this paper a method of improving proof readability based on Behaghel's First Law of sentence structure. Our method maximizes the number of local references to the directly preceding statement in a proof linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page

The Tur\'{a}n number and probabilistic combinatorics

In this short expository article, we describe a mathematical tool called the probabilistic method, and illustrate its elegance and beauty through proving a few well-known results. Particularly, we give an unconventional probabilistic proof of a classical theorem concerning the Tur\'{a}n number $T(n,k,l)$. Surprisingly, this proof cannot be found in existing literature.Comment: 5 pages; to appear in Amer. Math. Monthly 201

Analisis Kemampuan Membaca Bukti Matematis Pada Mata Kuliah Statistika Matematika

Mathematical Statistics is one of the course that are considered difficult, so students need mathematical skills to learn it. One of the skills required to learn that course is skill to read mathematical proof. The aims of this research is analyzing of the skill to read mathematical proof in Mathematical Statistics Course. The method used is qualitative. The results of the analysis of skill to read mathematical proof in Mathematical Statistics courses at one of the private universities in East Jakarta is seen that students are still difficulties in checking the truth and write the concepts used in each step of proof. For any prerequisite course, they still do not understand. This is one of the factors that make a student's skill to read mathematical proof is not good. Based on this research, it appears that the skill to read mathematical proof of students on Mathematical Statistics course is still not good

UNDERGRADUATE STUDENTS’ PROOF CONSTRUCTION ABILITY IN ABSTRACT ALGEBRA

The opinion of mathematics education expert toward the necessity of introducing mathematical proof to be thought at all levels was increased. Number of mathematics teacher in America conducted intensive discussion about whether mathematics proof should be included or excluded in mathematics curriculum. Teachers agree on the importance of proof and on the necessity for students to develop the skills needed to construct proofs. However many students of all levels of education face serious difficulties with constructing mathematical proof. Whereas, the limitedness on proving ability would influence on learning other advanced mathematics such as real analysis, abstract algebra, and others. That condition would hamper the development of students’ reasoning and others mathematical thinking abilities. The objective of developing proof methodology was to improve students’ ability on understanding mathematical proof, and proof constructing of mathematical statements. Some approaches had been developed, among them was concept of generic proof. Generic proof method of example level was explained of a concepts in general based on a specific example or case. The purpose of this paper is to categorizing and describing the different types of processes that undergraduate students use to construct proofs. This study involved 87 undergraduate students and two kinds instruments those proof reading test and a proof construction test. Keywords: mathematical proof, geometr

Premise Selection for Mathematics by Corpus Analysis and Kernel Methods

Smart premise selection is essential when using automated reasoning as a tool for large-theory formal proof development. A good method for premise selection in complex mathematical libraries is the application of machine learning to large corpora of proofs. This work develops learning-based premise selection in two ways. First, a newly available minimal dependency analysis of existing high-level formal mathematical proofs is used to build a large knowledge base of proof dependencies, providing precise data for ATP-based re-verification and for training premise selection algorithms. Second, a new machine learning algorithm for premise selection based on kernel methods is proposed and implemented. To evaluate the impact of both techniques, a benchmark consisting of 2078 large-theory mathematical problems is constructed,extending the older MPTP Challenge benchmark. The combined effect of the techniques results in a 50% improvement on the benchmark over the Vampire/SInE state-of-the-art system for automated reasoning in large theories.Comment: 26 page