• Journal for History of Mathematics
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• v.17 no.1
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• pp.61-68
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• 2004

This paper aims at investigating the algebraic solution of cubic and quartic equation and eliciting the didactical meanings of them. First, I examine the event which relates to the equation in the history of mathematics and investigate the algebraic solution of cubic and quartic equation. And then I elicit the didactical suggestions which are required of teachers and students when they investigate the algebraic solution of cubic and quartic equation. In general, the investigation of these solutions is the valuable task which requires the algebraic intuition and technique for students and certificates expert knowledge for teachers.

• Communications of Mathematical Education
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• v.29 no.4
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• pp.661-685
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• 2015

This study is to search for a program of professional development of mathematics teachers on the viewpoint of content knowledge of mathematics. To do this, we select algebraic subject as content knowledge for solution of polynomials and develop material for group study based on selected subject. We supply the developed material to teachers and discuss the possibility of application and the acceptability of it. For discussion, we collect data through tests and questionnaire. Through analysing the data, we obtain the positive result.

• 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.

• Research in Mathematical Education
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• v.15 no.4
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• pp.311-325
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• 2011

In this paper I propose that teaching students the most efficient method of problem solving may curtail students' creativity. Instead it is important to arm students with a variety of problem solving heuristics. It is the students' responsibility to decide which heuristic will solve the problem. The chosen heuristic is the one which is meaningful to the students.

• Journal of the Korean School Mathematics Society
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• v.8 no.3
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• pp.367-382
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• 2005

In this paper, we deal with various contents relating to the group concept in school mathematics and teaching of algebraic structures indirectly by combining these contents. First, we consider structure of knowledge based on Bruner, and apply these discussions to the teaching of algebraic structure in school algebra. As a result of these analysis, we can verify that the essence of algebraic structure is group concept. So we investigate the previous researches about group concept: Piaget, Freudenthal, Dubinsky. In our school, the contents relating to the group concept have been taught from elementary level indirectly. Tn elementary school, the commutative law and associative law is implicitly taught in the number contexts. And in middle school, various linear equations are taught by the properties of equality which include group concept. But these algebraic contents is not related to the high school. Though we deal with identity and inverse in the binary operations in high school mathematics, we don't relate this algebraic topics with the previous learned contents. In this paper, we discussed algebraic structure focusing to the group concept to obtain a connectivity among school algebra. In conclusion, the group concept can take role in relating these algebraic contents and teaching the algebraic structures in school algebra.

• The Mathematical Education
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• v.58 no.1
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• pp.1-20
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• 2019

As the importance of algebraic thinking in elementary school has been emphasized, the links between fraction knowledge and algebraic thinking have been highlighted. In this study, we analyzed the solution methods and characteristics of thinking by fifth graders who have not yet learned fraction division when they solved 'reverse fraction problems' (Pearn & Stephens, 2018). In doing so, the contexts of problems were extended from the prior study to include the following cases: (a) the partial quantity with a natural number is discrete or continuous; (b) the partial quantity is a natural number or a fraction; (c) the equivalent fraction of partial quantity is a proper fraction or an improper fraction; and (d) the diagram is presented or not. The analytic framework was elaborated to look closely at students' solution methods according to the different contexts of problems. The most prevalent method students used was a multiplicative method by which students divided the partial quantity by the numerator of the given fraction and then multiplied it by the denominator. Some students were able to use a multiplicative method regardless of the given problem contexts. The results of this study showed that students were able to understand equivalence, transform using equivalence, and use generalizable methods. This study is expected to highlight the close connection between fraction and algebraic thinking, and to suggest implications for developing algebraic thinking when to deal with fraction operations.

• The Mathematical Education
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• v.42 no.5
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• pp.697-706
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• 2003

In this paper we consider the equality sign as a mathematical concept and investigate its meaning, errors made by students, and subject matter knowledge of mathematics teacher in view of The Model of Mathematic al Concept Analysis, arithmetic-algebraic thinking, and some examples. The equality sign = is a symbol most frequently used in school mathematics. But its meanings vary accor ding to situations where it is used, say, objects placed on both sides, and involve not only ordinary meanings but also mathematical ideas. The Model of Mathematical Concept Analysis in school mathematics consists of Ordinary meaning, Mathematical idea, Representation, and their relationships. To understand a mathematical concept means to understand its ordinary meanings, mathematical ideas immanent in it, its various representations, and their relationships. Like other concepts in school mathematics, the equality sign should be also understood and analysed in vie w of a mathematical concept.

• Research in Mathematical Education
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• v.14 no.1
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• pp.33-50
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• 2010

The mathematical thinking which transforms important mathematical content and developed into mathematical structure is a vital process in building up mathematical ability as mathematical knowledge based on structure. Such process based on students' recognition of mathematical concept. Developing mathematical thinking into mathematical structure happens when different cognitive units are connected and compressed to form schema of solution, which could happen through some guided problems. The effort of arithmetic approach in problem solving did not necessarily provide students the structure schema of solution. The using of equation to solve the problem is based on the schema of building equation, and is not necessary recognizing the structure of the solution, as the recognition of structure may be lost in the process of simplification of algebraic expressions, leaving only the final numeric answer of the problem.

• Research in Mathematical Education
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• v.25 no.3
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• pp.189-199
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• 2022

A debate about the importance of geometry courses has existed for years. The questions have revolved around its significance to students and teachers alike. This study looks to determine whether a teacher taking a college-level geometry course has a positive relationship with their students' algebraic reasoning skills. Using data from the High School Longitudinal Study 2009 (HSLS09: Ingels et al., 2011, 2014), it was determined that 9th-grade teachers who took a college-level geometry course had a significant positive association with their students' 11th-grade algebraic reasoning scores. This study suggests that teachers who take geometry during college have a lasting effect on their students. The implications of these findings and how they may affect higher education are discussed.

• Research in Mathematical Education
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• v.25 no.1
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• pp.65-89
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• 2022

Despite its recognition in the field of mathematics education and mathematics, students' understanding about proof and performance on proof tasks have been far from promising. Research has documented that teachers tend to accept empirical arguments as proofs. In this study, an online survey was administered to examine how Korean secondary mathematic teachers make judgements on mathematical arguments varied along representations. The results indicate that, when asked to judge how convincing to their students the given arguments would be, the teachers tended to consider how likely students understand the given arguments and this surfaces as a controversial matter with the algebraic argument being both most and least convincing for their students. The teachers' judgements on the algebraic argument were shown to have statistically significant difference with respect to convincingness to them, convincingness to their students, and validity as mathematical proof.