n. asghary

The principles and standards of school mathematics (NCTM) in the branch of algebra propose standards that consider the development of students' understanding of algebraic symbolization and especially the understanding of the variable concept as one of the basic needs of students. Functional thinking is also the highway of algebraic thinking and teachers should consider it as the heart and soul of math education. Figural patterns have characteristics that are favorable for starting the generalization and development of functional thinking and can be used in school mathematics. Also, attention to mathematical structure should be an important part of the teaching and learning mathematics. The structure of a mathematical pattern is the way a pattern is organized and is often expressed as a generalization. The purpose of this research was to investigate the performance of students in building the concept of figural pattern generalization based on APOS theory (action-process-object and schema) and to improve the stages using the states of awareness of the structure, and will help teachers and students to have a more accurate evaluation of the figural pattern generalization and better identify the problems and improve their stages of understanding.

The data collection method was quantitative-qualitative. The data collection tool included a researcher-designed test and a semi-structured interview. The statistical population included 493 seventh grade students of public schools in Malekan city (Azerbaijan-e-sharghi). According to Cochran's sample size formula, 220 male and female seventh-grade students were selected and participated in the researcher-designed test. The validity of the test was checked and confirmed by three mathematics teachers and four experienced teachers. The reliability and internal consistency of the questions were confirmed by finding Cronbach's alpha coefficient and alpha as 0.69.

Students' mental structures were identified according to APOS theory in the figural pattern generalization. The highest percentage of correct response was at the action stage and the lowest percentage was at the object stage. The stages of APOS were hierarchical and from simple to complex, and the results of this research confirmed the characteristics of this theory. At the action stage, about 18% of the students were unsuccessful, and according to the first attentional state (holing wholes) looking at the figural pattern was introduced as a tool to improve the understanding of these students. About 60% were unsuccessful at the process stage, which the discerning details and recognizing relationships as second and third attentional states helped students to reach the process stage. About 88% of the students failed to reach the object that the fourth attentional state (perceiving properties) as auxiliary tool was introduced. Reasoning on the basis of identified properties as the fifth attentional state was used to upgrade students to the schema stage in APOS theory.

This research provided a framework for measuring and evaluating students in the figural pattern generalization that teachers can use in better teaching of this concept. At each stage, solutions can be provided according to the student s' stage to improve their understanding of generalization to higher stage. This research showed the power of APOS theory in compatibility with other constructionist theories. Adapting the APOS theory with the theory of structural awareness and benefiting from this adaptation in order to improve the stage of understanding of the seventh-grade students in figural pattern generalization was considered as one of the innovative aspects of this research.

In this paper the differential quadrature method is implemented to find numerical solution of two and three-dimensional telegraphic equations with Dirichlet and Neumann's boundary values. This technique is according to exponential cubic B-spline functions. So, a modification on the exponential cubic B- spline is applied in order to use as a basis function in the DQ method. Therefore, the Telegraph equation (TE) is altered to a system of ordinary differential equations (ODEs). The optimized form of Runge-Kutta scheme has been implemented by four-stage and three-order strong stability preserving (SSPRK43) to solve the resulting system of ODEs. We examined the correctness and applicability of this method by four examples of the TE.

Proportional reasoning is regarded as the gate of functional thinking and is a potent means for developing algebraic thinking which necessitates students’ understanding of the functional relationship between measure spaces. The ability to recognize and identify the structural similarity and multiple comparisons in the proportional reasoning process is the base of algebra and advanced mathematics. The concept of proportion and the necessities of developing proportional thinking are cognitively complex and its teaching demands concept-oriented approaches. Studing the quality of teachers' perceptions draws the perspective of the method and development of conceptual structures among the students. The present study focused on determining the extent of recognizing non-proportional situations and also the kind of selected strategies to solve proportional verbal issues in the teaching activity. Considering the importance of the context of this problem, the study focused on four semantic types of the problems in this field. Considering the pedagogical thinking of teachers in solving proportional problems provides the discussion on the obstacles of using the proportional reasoning among different semantic types .

The study was done by descriptive survey method. The statistical population included 180 teachers of primary schools and mathematics teachers of the first level of the secondary schools, and prospective teachers who participated in the study voluntarily. The research instrument was a researecher-developed test containing 17 problems comprised of 3 nonproportional situations of additive problem types and 14 direct proportional problems, presented in the missing-value type which were either researcher-devised or selected from reliable research sources. The content validity of the test was confirmed by professors in the field of mathematics and testing and psychometrics. The collected data were analyzed using inferential and descriptive statistics.

The results of the first study revealed that the primary school teachers and the prospective teachers were faced with some difficulties in recognizing non-proportional statements. It seems that the superficial characteristics of verbal problem including having a structure similar to the proportional problems of the type of missing value and also the multiplicative nature of numerical structure are involved in determining the situation as a proportional structure. In studying the the strategies of solving the proportional problems in the teaching activity, the responses of the participants were analyzed using the descrtiptive method based on 9 types of problem-solving strategies. The results of the analyses showed that all of the first levele of the secondary school teachers and the prospective teachers of both of these levels, at least in one of their first two priorities in teaching these problems, applied algorithmic proportional strategies or algebraic equation formulation while being slightly influenced by semantic types. Teachers of the primary schools had little desire to use the algorithmic proportional strategies. On the contrary, as compared to other teachers, they had a higher preference for using functional and numerical proportional reasoning. However, they did not prefer to use proportional functional reasoning in their activities. On the other hand, the first two priorities of the primary school teachers were not included in any semantic types, utilization of more complex proportional reasonings, and scale factor.

The results emphasize the necessity of the development of pedagogical content knowledge in this field in order to develop the application of the strategies of functional proportional reasoning and appropriate representations by teachers which are aimed at providing more desirable conditions for students’ proportional reasoning development. Unexpected behaviors of prospective teachers in this study emphasize creating higher sensitivity to the consequences of delaying the emergence of students’ relative thinking in the instructional plans of teacher training courses.

The figural patterns have a unique capacity to enhance functional thinking. The patterns generalization in school mathematics is considered as a way to promote functional thinking. Variable is one of the concepts in patterns generalization. Paying attention to figural patterns provides an opportunity for students to understand the meaning of variable and how to use it. Reference is also a central concept in patterns generalization. The number of variables is one of the characteristics that has been proposed in the pattern generalization tasks, but all the research has been related to one variable, linear and quadratic patterns. The aim of this study was to identifying the prior schemas in generalization of two-variable figural patterns. As regard to the concept of two variables, understanding three-dimensional space is a prerequisite for understanding and generalizing two-variable patterns. In these patterns, instead of one independent variable, there are two independent variables that change simultaneously and affect the dependent variable. Understanding these patterns requires the development of the R2 space scheme to R3 space, which is not a cognitively complex step and does not require the reconstruction of the existing scheme.

The present research is part of a broad research which is done using quantitative-qualitative (mixed) research method. The research framework is APOS theory and based on the use of ACE (Activities, Class discussions and Exercises) teaching cycles. This research was conducted in three steps. In the first step, initial genetic decomposition for generalization of two-variable figural patterns was designed using the background, self-concept analysis and researchers’ experiences. It includes the prior schemas for generalization. In the second step, from the total 493 students of Malekan city (in East Azerbaijan) as the statistical population of research, a sample of 220, 7th grade students were selected based on the Cochran formula for determination of sample size. Then, a test that includes 7 tasks was designed based on APOS framework. The validity of the test was confirmed by three experts in mathematics education and four experienced teachers. Internal consistency of questions was estimated with Cronbach’s alpha and reported to be 0.69. Students responded the test at 90 minutes. The third step of research began with 19 students, with permission from the education and training office of Malekan, and school principals and parents of students. This step is done in three cycles.

Using the analysis of students' responses to this test based on the APOS framework and doing three cycles of the research were conducted with the teaching method of Activity-Class Discussion-Exercise (ACE) with 19 students; genetic decomposition was finalized in this way, and defects of students in reference schema, R3 schema and variables schema as prior schemas in generalization of two-variable figural patterns were identified and encoded. Most of students had a good understanding of working with two variables. However in the context of generalization of two-variable figural pattern revealed many difficulties at the naming of variables,  and using independent and dependent variables in  proper position

Identifying the mental constructs of students in generalizing patterns eases better teaching and learning.

By identifying the mental structures of students in generalizing patterns, the path of teaching and learning will be smoother. 

‎Complex-valued harmonic functions that are univalent and‎ ‎sense-preserving in the open unit disk $U$ can be written as form‎ ‎$f =h+bar{g}$‎, ‎where $h$ and $g$ are analytic in $U$‎. ‎In this paper‎, ‎we introduce the class $S_H^1(beta)$‎, ‎where $1<betaleq 2$‎, ‎and‎ ‎consisting of harmonic univalent function $f = h+bar{g}$‎, ‎where $h$ and $g$ are in the form‎ ‎$h(z) = z+sumlimits_{n=2}^infty |a_n|z^n‎$ ‎and ‎‎$‎g(z) =‎sumlimits_{n=2}^infty |b_n|bar z^n$‎ for which‎ ‎$$mathrm{Re}left{z^2(h''(z)+g''(z))‎ +2z(h'(z)+g'(z))-(h(z)+g(z))-(z-1)right}<beta.$$‎ It is shown that the members of this class are convex and starlike‎. ‎We obtain distortions bounds extreme point for functions belonging to this class‎, ‎and we also show this class is closed under‎ convolution and convex combinations‎.

Many international studies have been done in order to trace the intellectual path of students in manipulation with the concept of variables and algebraic expressions, as well as to examine and specify their functional problems. However, despite the importance of these two concepts, comprehensive research has not been conducted in the seventh, eighth and ninth grades in Iran, and research has often focused on how to manipulate and write algebraic expressions. Due to the change in Iran's mathematics curriculum in 2009 and the consequent change in mathematics textbooks, the need for a clear picture of students' understanding of these two concepts is doubled.Algebraic expressions are the important part of Algebra and its deep understanding is necessory for solving algebraic problems. The purpose of this study was to investigate grade 7th, 8th and 9th students' understanding of algebraic expressions.

400 students were selected by multistage cluster sampling from 7th , 8th  and 9th  grade students in Tehran. A researcher-made test was designed and implemented. Out of 400 students, 15 students were selected and semi-structured interviews were conducted in order to clarify and interpret students' perceptions.

The results of the test and interviews showed that most students have a poor structural understanding of algebraic expressions and they have understood them merely procedurally. Increasing academic bases did not almost improve structural understanding, but procedural understanding improves with increasing levels.

The results of this study showed that most students in algebraic expressions (simple and complex) have a purely procedural understanding, which means that they have understood algebraic expressions as a set of algorithms and processes. As it turned out from the interviews, these people, when it is necessary to perform an operation on the algebraic expression and consider the algebraic expression as a whole, only memorize the steps as a parrot because they have no understanding of the whole structure of the algebraic expression; and they focus only on the steps within an algebraic expression. In complex algebraic expressions, compared to simple algebraic expressions, the percentage of students who had a purely procedural understanding was reduced. The results of interviews with a number of students showed that this decrease was not due to an increase in students' structural understanding, but for reasons such as ignoring the distributive action, not understanding algebraic expressions, not understanding the process in complex algebraic expressions.

With an aim to investigate the effect of externally imposed body acceleration on two dimensional,pulsatile blood flow through a stenosed artery is under consideration in this article. The blood flow has been assumed to be non-linear, incompressible and fully developed. The artery is assumed to be an elastic cylindrical tube and the geometry of the stenosis considered as time dependent, and a comparison has been made with the rigid ones. The shape of the stenosis in the arterial lumen is chosen to be axially non-symmetric but radially symmetric in order to improve resemblance to the in-vivo situations. The resulting system of non-linear partial differential equations is numerically solved using the Crank-Nicolson scheme by exploiting the suitably prescribed conditions. The blood flow characteristics such as the velocity profile, the volumetric flow rate and the resistance to flow are obtained and effects of the severity of the stenosis, the body acceleration on these flow characteristics are discussed. The present results are compared with literature and found to be in agreement.