Difficulties in Learning and Teaching Numbers:A Literature Review on the Obstacles and Misconceptions of Learners and Instructors

This paper attempts to summarize a number of research studies on numbers.The purpose of this study was to investigate and identify the obstacles encountered by students when they are dealing with number reasoning(whole numbers,integers,and rational numbers)and the difficulties faced by pre-service teachers in teaching arithmetic,including their misconceptions and weaknesses when they teach arithmetic and operations.There are two main sections in this paper:students’cognitive obstacles for number reasoning,and pre-service teachers’misconceptions of arithmetic.With the summarized misconceptions and obstacles of both,students and teachers,this paper provides efficient and effective thinking strategies that may help both,learners and instructors overcome obstacles,revise misconceptions,and strengthen their understanding,in order to develop proficiency in number reasoning and arithmetic operations.

The Generalization and Proof of “Square Root of 2 Is Not a Rational Number” on the Integral Domain

In this paper, the traditional proof of “square root of 2 is not a rational number” has been reviewed, and then the theory has been generalized to “if n is not a square, square root of n is not a rational number”. And then some conceptions of ring, integral domain, ideal, quotient ring in Advanced algebra, have been introduced. Integers can be regarded as an integral domain, the rational numbers can be regard as a fractional domain. Evens and odds are principal ideals in integral domain. The operations on evens and odds are operations on quotient ring. After introducing “the minimalist form” in fraction ring. The paper proves the main conclusion: in a integral domain, multiplicative subset S produces a fraction ring S−1R, and n is not a square element in R, then to every element a∈R, a2≠n.

Ranking method for the reciprocal judgment matrix based on the unascertained three-valued judgments

The ranking problem is studied when the pairwise comparisons values are unoertain in the analytic hierarchy process （AHP）. The method of constructing the judgment matrix is presented when the pairwise comparisons values are denoted by the unascertained three-valued reciprocal scales. By turning the reciprocal judgment matrix into attribute judgment matrix, the method to check the consistency of the pairwise comparisons judgment matrix and the calculation method of weighting coefficients are given. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

Multi-agent Blind Model and Its Application to Regional Eco-environmental Quality Assessment

Through denoting each expert as an agent and viewing a multiple criteria decision-making as a synthesis problem of aggregating experts＇ ratings, a multi-agent blind model （MABM） is developed for regional eco-environmental quality assessment. In this model, the ratings of the evaluated object under an index, given by expert group, are first utilized to construct a series of blind numbers. In general, each index will correspond to different blind numbers. On the basis of aggregating index weights, the rank score in the form of a blind number is obtained for the evaluated object. Then, by means of calculating expected value of the above blind number, its rank score is further converted into a crisp value. By way of comparing the expected value with classification standards, eco-environmental quality of the evaluated sample could he identified successfully in the end. As a case, the MABM is used to evaluate the eco-environmental quality of Chaohu Lake basin. Study result shows that the MABM is a useful model for regional eco-environmental quality assessment.

Periodic Orbits Around Kerr Sen Black Holes

We investigate periodic orbits and zoom-whirl behaviors around a Kerr Sen black hole with a rational number q in terms of three integers(z,w,v),from which one can immediately read off the number of leaves(or zooms),the ordering of the leaves,and the number of whirls.The characteristic of zoom-whirl periodic orbits is the precession of multi-leaf orbits in the strong-field regime.This feature is analogous to the counterpart in the Kerr space-time.Finally,we analyze the impact of the charge parameter b on the zoom-whirl periodic orbits.Compared to the periodic orbits around the Kerr black hole,it is found that typically lower energies are required for the same orbits in the Kerr Sen black hole.