• Journal of Educational Research in Mathematics
• /
• v.17 no.4
• /
• pp.453-475
• /
• 2007

School algebra starts with introducing algebraic expressions which have been one of the cognitive obstacles to the students in the transfer from arithmetic to algebra. In the recent studies on the teaching school algebra, algebraic thinking is getting much more attention together with algebraic expressions. In this paper, we examined the processes of the transfer from arithmetic to algebra and ways for teaching early algebra through algebraic thinking factors. Issues about algebraic thinking have continued since 1980's. But the theoretic foundations for algebraic thinking have not been founded in the previous studies. In this paper, we analyzed the algebraic thinking in school algebra from historico-genetic, epistemological, and symbolic-linguistic points of view, and identified algebraic thinking factors, i.e. the principle of permanence of formal laws, the concept of variable, quantitative reasoning, algebraic interpretation - constructing algebraic expressions, trans formational reasoning - changing algebraic expressions, operational senses - operating algebraic expressions, substitution, etc. We also identified these algebraic thinking factors through analyzing mathematics textbooks of elementary and middle school, and showed the middle school students' low achievement relating to these factors through the algebraic thinking ability test. Based upon these analyses, we argued that the readiness for algebra learning should be made through the processes including algebraic thinking factors in the elementary school and that the transfer from arithmetic to algebra should be accomplished naturally through the pre-algebra course. And we searched for alternative ways to improve algebra curriculums, emphasizing algebraic thinking factors. In summary, we identified the problems of school algebra relating to the transfer from arithmetic to algebra with the problem of teaching algebraic thinking and analyzed the algebraic thinking factors of school algebra, and searched for alternative ways for improving the transfer from arithmetic to algebra and the teaching of early algebra.

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

• Journal of Elementary Mathematics Education in Korea
• /
• v.24 no.1
• /
• pp.89-108
• /
• 2020

Since arithmetic is the foundation of school mathematics, it needs to be taught meaningfully in the direction of improving arithmetical thinking levels of students beyond the fluency of computing skills. Therefore, in this study, the arithmetical thinking levels of 100 students in 5th grade were analyzed by applying the arithmetical thinking level test. As a result, 82 students were at 1st level and 15 students were at 2nd level of the arithmetical thinking. I analyzed the characteristics of arithmetical thinking and types of errors and misconceptions made by the students, and derived some didactical implications for arithmetic education in elementary school mathematics.

• Journal of Educational Research in Mathematics
• /
• v.27 no.4
• /
• pp.765-789
• /
• 2017

Arithmetic is the basis of school mathematics and in fact, number and operation in elementary school curriculum is the most basic and essential domain. Even though there has been a consensus that arithmetic should be taught more meaningfully beyond the emphasis of calculation skills and teachers should emphasize the aspect of the arithmetical thinking, it is difficult to find studies which focus on the arithmetical thinking itself. So this research aims to explore the meaning of the arithmetical thinking and extract the arithmetical thinking factors. In order to solve the research problems, we reviewed and analyzed the literatures and then conducted Delphi survey to extract arithmetical thinking factors. From the results of this research, we found the meaning of arithmetical thinking and the arithmetical thinking factors. Especially, the arithmetical thinking consists of 18 factors. It is important to pay attention to students' arithmetical thinking because there are various factors of the arithmetical thinking. It is necessary to identify the aspects of arithmetical thinking reflected in school mathematics based on the meaning of arithmetical thinking and its factors. Based on this, it is possible to find effective teaching and learning methods of arithmetic focusing on the arithmetical thinking.

• Journal of Educational Research in Mathematics
• /
• v.12 no.4
• /
• pp.457-472
• /
• 2002

In this paper, we deal with the algebraic thinking from the perspective of ‘process’ and ‘object’ aspects. Generally, mathematical concepts have come from the concrete process. We consider the origin of algebra as the arithmetic calculations. Also, the concept of school arithmetic is beginning from actions or procedures. However, in order to develop the alge- braic thinking and to apply this thinking, we have to see the history of algebraic thinking, and find this duality. Next we investigate various researches relating to the ‘process-object duality’. Theses studies suppose that the concept formation and thinking process should be stared from the process-object duality. Finally, we reinterprete many difficulties in algebra - equals sign, variables, algebraic expressions, and linear equations, the principle of permanence of form- from the perspective of the process-object duality.

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

• East Asian mathematical journal
• /
• v.29 no.4
• /
• pp.355-392
• /
• 2013

Generally school algebra is to start with introducing variables and algebraic expressions, which have major cognitive obstacles to students in the transfer from arithmetic to algebra. But the recent studies in the teaching school algebra argue the algebraic thinking from an early algebraic point of view. We compare the Korean elementary mathematics textbooks with Americans from this perspective. First, we discuss the history of school algebra in the school curriculum. And Second, we investigate the recent studies in relation to early algebra. We clarify the goals of this study(the importance of early algebra in the elementary school) through these discussions. Next we examine closely the number sense in the arithmetic and the symbol sense in the algebra. And we conclude that the operation sense can connect these senses within early algebra using the algebraic thinking. Finally, we compare the elementary mathematics books between Korean and American according to the components of the operation sense. In this comparative study, we identify a possibility of teaching algebraic thinking in the elementary mathematics and early algebra can be introduced to the elementary mathematics textbooks from aspects of the operation sense.

• Journal of Educational Research in Mathematics
• /
• v.26 no.3
• /
• pp.489-508
• /
• 2016

Number and operation is the most basic and crucial part in elementary mathematics but is also well known as a part that students have lots of difficulties. A lot of researches have been done in various ways to solve this problem but it can't be solved fundamentally by emphasizing calculation method and skill. So we need to go over it in terms of relevant arithmetical thinking. This study aims to diagnose the cause of an underachiever's difficulties about arithmetic and finds a prescription for her by analyzing her level of arithmetical thinking based on Guberman(2014) and understanding about arithmetic. To achieve this goal, we chose an 6th grader who's having a hard time particularly in number and operation among mathematics strands and conducted a case study carrying out arithmetical thinking level tests on two separate occasions and analyzing her responses. As a result of analyzing data, her arithmetical thinking corresponded to Guberman's first level and it is also turned out that student is suffering from some arithmetic concepts. We suggest several implications for teaching of arithmetic at elementary school in terms of the development of arithmetical thinking based on analysis result and discussion about it.

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

• East Asian mathematical journal
• /
• v.38 no.2
• /
• pp.257-275
• /
• 2022

Recently, there are studies the argument that arithmetic rules established by the four fundamental arithmetic operations, in other words, commutative laws, associative laws, distributive laws, should be explicitly described in mathematics textbooks and the curriculum. These rules are currently implicitly presented or omitted from textbooks, but they contain important principles that foster mathematical thinking. This study aims to evaluate the current level of understanding of these computation rules and provide implications for the curriculum and textbook writing. To this end, the correct answer ratio of the five arithmetic rules for 1-4 grades 398 in five elementary schools was investigated and the type of error was analyzed and presented, and the subject to learn these rules and the points to be noted in teaching and learning were also presented. These results will help to clarify the achievement criteria and learning contents of the calculation rules, which were implicitly presented in existing national textbooks, in a new 2022 revised curriculum.