• Research in Mathematical Education
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• v.12 no.3
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• pp.215-233
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• 2008

Deep comprehension of basic mathematical notions and concepts is a basic condition of a successful teaching. Some elements of algebraic thinking belong to the elementary school mathematics. The question "What stays the same and what changes?" link arithmetic problems with algebraic conception of variable. We have studied beliefs and comprehensions of future elementary school mathematics teachers on early algebra. Pre-service teachers from three academic pedagogical colleges deal with mathematical problems from the pre-algebra point of view, with the emphasis on changes and invariants. The idea is that the intensive use of non-formal algebra may help learners to construct a better understanding of fundamental ideas of arithmetic on the strong basis of algebraic thinking. In this article the study concerning arithmetic series is described. Considerable number of pre-service teachers moved from formulas to deep comprehension of the subject. Additionally, there are indications of ability to apply the conception of change and invariance in other mathematical and didactical contexts.

• The Mathematical Education
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• v.53 no.1
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• pp.57-73
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• 2014

This study investigated the connections of mathematical thinking of students at the second, fourth, and sixth grades with regard to multiplication, fraction, and proportion, all of which have multiplicative structures. A paper-and-pencil test and subsequent interviews were conducted. The results showed that mathematical thinking including vertical thinking and relational thinking was commonly involved in multiplication, fraction, and proportion. On one hand, the insufficient understanding of preceding concepts had negative impact on learning subsequent concepts. On the other hand, learning the succeeding concepts helped students solve the problems related to the preceding concepts. By analyzing the connections between the preceding concepts and the succeeding concepts, this study provides instructional implications of teaching multiplication, fraction, and proportion.

• The Mathematical Education
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• v.53 no.2
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• pp.163-184
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• 2014

The purpose of this study was to develop the assessment tools using graphing calculators for the assessment of the mathematical process which was emphasized in 2009 reformed mathematics curriculum. In this paper, we presented three sample calculator-based test items for the assessment of students' mathematical process abilities and scoring rubrics for the paper and pencil assessment and assessment based on observation on each item. In order to improve mathematics teachers' understanding of the assessment tools using graphing calculators and to show the procedures of assessment using technological devices, we also drew up assessment guidelines. We hope the results of the study contribute to the promotion of assessment environment encouraging the use of graphing calculators in assessments.

• Research in Mathematical Education
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• v.22 no.1
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• pp.47-82
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• 2019

Pennsylvania is one of the states that adopted the Common Core State Standards for Mathematics (CCSSM) and crafted its own standards (The PA Core State Standards). Pennsylvania teachers are required to have a clear understanding of the PA Core Standards. It is timely and appropriate to study Pennsylvania teachers' beliefs, as the standards have been adopted and implemented for several years since the revision of the PA Core Standards (2014). This study examined how eight western Pennsylvania elementary school teachers' beliefs about teaching and learning mathematics related to the SMP. To this end, I conducted an in-depth interview with each participating teacher. The in-depth interviews featured the teachers' overarching mathematical instructional goals and their productive beliefs. Furthermore, I linked these beliefs with the CCSSM Standards for Mathematical Practice (SMP).

• The Mathematical Education
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• v.44 no.4 s.111
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• pp.547-563
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• 2005

There have been many papers reporting that the axiomatic approach in Abstract Algebra is a big obstacle to overcome for the students who are not trained to think in an abstract way. Therefore an instructor must seek for ways to help students grasp mathematical concepts in Abstract Algebra and select the ones suitable for students. Mathematics faculty and students generally consider Abstract Algebra in general and quotient groups in particular to be one of the most troublesome undergraduate subjects. For, an individual's knowledge of the concept of group should include an understanding of various mathematical properties and constructions including groups consisting of undefined elements and a binary operation satisfying the axioms. Even if one begins with a very concrete group, the transition from the group to one of its quotient changes the nature of the elements and forces a student to deal with elements that are undefined. In fact, we also have found through running abstract algebra courses for several years that students have considerable difficulty in understanding the concept of quotient groups. Based on the above observation, we explore and analyze the nature of students' knowledge about $Z_n$ that is the set of congruence classes modulo n. Applying the genetic decomposition method, we propose a model to lead students to achieve the correct concept of $Z_n$.

• The Mathematical Education
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• v.58 no.4
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• pp.589-607
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• 2019

The study aimed to investigate what high-school students recognize mathematics textbooks and how they use textbooks in their learning mathematics in and out-of mathematics classrooms. For this purpose, we developed a set of interview questions in order to unpack what high-school students thought about mathematics textbooks and how they intended to use the textbooks for their learning mathematics. Eleven high-school students participated in the interview; the interview lasted for about an hour for each student. The data from the interviews were analyzed. The findings from the data analysis suggested as follows: a) the students seemed to consider mathematics textbooks as crucial medium for a mathematics classroom material and thus, they were likely to obliged to use the textbooks for preparing for not only tests and examination conducted regularly in schools but college entrance examination conducted nationwide; b) however, the students appeared to use the textbooks in limited ways in which they looked into the textbooks to prepare for mid-term or final exam only, not for their understanding mathematical contents as a main resource; and c) the students seemed to realize that they rarely have had an opportunity to develop mathematical thinking capabilities and understand mathematical ideas conceptually through the mathematics textbooks.

• Education of Primary School Mathematics
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• v.10 no.2
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• pp.125-139
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• 2007

The purpose of this study seeks entry into a method to make the use of educational aids popular. To achieve it, it is observed that instructions applying worksheets to make an activation of use of educational aids have influences on mathematical achievement and mathematical disposition and attitude. All variables exception with the frequence of use of educational aids are controlled in both experimental group and comparative group. According to the result, there is no significant difference of mathematical achievement in pre t-test between two groups, while experimental group get 10 points higher than comparative group in average (t=0.519, p<0.01). On the other hand, within intra-experimental group the influences of use of educational aids on mathematical achievement is positive without the achievement levels of students. The difference dependent on the levels of student is sought by ANCOVA using prescores as a covariance, and it appears in the significance level of 5%(F=4.885, p<0.05), and the effect is more in the lower level of students than in the middle and high level.

• Journal of Educational Research in Mathematics
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• v.11 no.2
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• pp.239-256
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• 2001

The notion of metaphor has been increasingly popular in research of mathematics education. In particular, metaphor becomes a useful unit for analysis to provide a profound insight into mathematical reasoning and problem solving. In this context, this paper takes metaphor as an analytic unit to examine the relationship between objectivity and subjectivity in mathematical reasoning. Specifically, the discourse analysis focuses on the code switching between literal language and metaphor in mathematical discourse. It is shown that the linguistic code switching is parallel with the switching between two different kinds of mathematical knowledge, that is, factual knowledge and mathematical imagination, which constitute objectivity and subjectivity in mathematical reasoning. Furthermore, the pattern of the linguistic code switching reveals the dialectical relationship between the two poles of mathematical reasoning. Based on the understanding of the dialectical relationship, this paper provides some educational implications. First, the code-switching highlights diverse aspects of mathematics learning. Learning mathematics is concerned with developing not only technicality but also mathematical creativity. Second, the dialectical relationship between objectivity and subjectivity suggests that teaching and teaming mathematics is socioculturally constructed. Indeed, it is shown that not all metaphors are mathematically appropriated. They should be consistent with the cultural model of a mathematical concept under discussion. In general, this sociocultural perspective on mathematical metaphor highlights the sociocultural organization of teaching and loaming mathematics and provides a theoretical viewpoint to understand epistemological diversities in mathematics classroom.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.3
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• pp.885-905
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• 2010

This study questions that mathematical evaluations strive to memorize fragmentary knowledge and have an objective test. To solve these problems on mathematical education We did descriptive test. Through the descriptive test, students think and express their ideas freely using mathematical terms. We want to know if that procedure is correct or not, and, if they understand what was being presented. We studied this because We want to analyze where and what kinds of faults they committed, and be able to correct an error so as to establish a correct mathematical concept. The result from this study can be summarized as the following; First, the mistakes students make when solving the descriptive tests can be divided into six things: error of question understanding, error of concept principle, error of data using, error of solving procedure, error of recording procedure, and solving procedure omissions. Second, students had difficulty with the part of the descriptive test that used logical thinking defined by mathematical terms. Third, errors pattern varied as did students' ability level. For high level students, there were a lot of cases of the solving procedure being correct, but simple calculations were not correct. There were also some mistakes due to some students' lack of concept understanding. For middle level students, they couldn't understand questions well, and they analyzed questions arbitrarily. They also have a tendency to solve questions using a wrong strategy with data that only they can understand. Low level students generally had difficulty understanding questions. Even when they understood questions, they couldn't derive the answers because they have a shortage of related knowledge as well as low enthusiasm on the subject.

• Journal of Educational Research in Mathematics
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• v.10 no.1
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• pp.85-102
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• 2000

The concept of variables is central to mathematics teaching and learning in junior and senior high school. Understanding the concept provides the basis for the transition from arithmetic to algebra and necessary for the meaningful use of all advanced mathematics. Despite the importance of the concept, however, much has been written in the last decade concerning students' difficulties with the concept. This Thesis is based on research to investigate the hypothesis that LOGO programming will contribute to 6th grader' learning of variables. The aim of the research were to; .investigate practice on pupils' understanding of variables before the activity with a computer; .identify functions of LOGO programming in pupils' using and understanding of variable symbols, variable domain and the relationship between two variable dependent expressions during the activity using a computer; .investigate the influence of pupils' mathematical belief on understanding and using variables. The research consisted predominantly of a case study of 6 pupils' discourse and activities concerning variable during their abnormal lessons and interviews with researcher. The data collected for this study included video recordings of the pupils'work with their spoken language.