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

• Communications of Mathematical Education
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• v.25 no.2
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• pp.451-472
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• 2011

In this paper we studied problem solving related with geometric interpretation of algebraic expressions. We analyzed algebraic expressions, related these expressions with geometric interpretation. By using geometric interpretation we could find new approaches to solving mathematical problems. We suggested new problem solving methods related with geometric interpretation of algebraic expressions.

• The Mathematical Education
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• v.58 no.1
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• pp.79-99
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• 2019

This study is a study to collect information about 'Limitations of functions' related learning. Especially, this study was conducted on three students who can find answers by algebraic procedure in the process of extreme problem solving. Students have had the experience of converting from their algebraic procedures to graphical expressions. This shows how they reflect on their algebraic procedures. This study is a study that observes these parts. To accomplish this, twelfth were teaching experiment in three high school students. And we analyzed the contents related to the research topic of this study. Through this, students showed the difference of expressions in the method of finding limits by using algebraic interpretation methods and graphs. In addition, we examined the connectivity of the limitations of functions problem solving process of functions using algebraic procedures and graphs in the process of converting algebraic expressions to graph expressions. This study is a study of how students construct limit concepts. As in this study, it is meaningful to accumulate practical information about students' limit conceptual composition. We hope that this study will help students to study limit concept development process for students who have no limit learning experience in the future.

• Journal of Educational Research in Mathematics
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• v.8 no.2
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• pp.439-452
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• 1998

In this Study, I concerned the learning-teaching of the use of letters in algebra. Our Study can be summarized as follows; First, I tried to establish the theoretical Foundation necessary for the learning-teaching of the use of letters in literal expressions. Second, I made a course of study that leads to the understanding of the meaning and the use f literals in algebraic expressions. Third, Based on this course of study, I held classes on First-grade students in middle school and I carried on an investigation their understanding of the meaning and the use of literals in algebraic expressions. Finally, I made an analysis of findings in this investigation and identified student's a better understanding of the meaning and the use of literals in algebraic expressions.

• School Mathematics
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• v.6 no.4
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• pp.411-433
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• 2004

Many researchers have reported that 7th graders have severe difficulties in using letters and algebraic expressions. This study investigated the way Microsoft Excel contributes to student's understanding of letters and algebraic expressions. For six hours through two weeks, four 7th grade students experienced various activities with Excel after school and both before and after the experimentation, the interviews to check their understanding was conducted. The results were as follows; First, after the experimentation, students used various letters to express formulas and recognized that letters represent not only some objects but also changing objects. Also they accepted that same objects could be represented by different letters and different objects could be represented by the same letters. Second, Excel improved students' abilities to discriminate variables and invariables in the problem and to find mathematical relationships among variables. And with Excel students could divide the whole calculation procedure into several steps in order to handle it more easily. Also, Excel made immediate numerical feedback possible and it made students express the calculation in a more formalized way than a paper and pencil environment did.

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

• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.6 no.4
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• pp.325-336
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• 1994

A numerical simulation of velocity and temperature fields and Nusselt number distributions is performed by using the algebraic stress model (ASM) for the velocity profiles and low Reynolds number ${\kappa}-{\varepsilon}$ model and the algebraic heat flux model(AHFM) for turbulent heat transfer in a $180^{\circ}$ bend with a constant wall heat flux. In the low Reynolds number ${\kappa}-{\varepsilon}$ model, turbulent Prandtl number is modified by considering the streamline curvature effect and the non-equilibrium effect between turbulent kinetic energy production and dissipation rate. Every heat flux term presented in the transport equation of turbulent heat flux is reduced to algebraic expressions in a way similar to algebraic stress model. Also. in the wall region, low Reynods number algebraic heat flux model(AHFM) is applied.

• Journal of Educational Research in Mathematics
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• v.24 no.3
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• pp.359-374
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• 2014

Students' difficulties and errors in relation to mathematical proofs are worth while to say one of the dilemmas in mathematics education. The potential elements of their difficulty are scattered over the process of proving in geometry as well as algebra. This study aims to investigate whether middle school students understand the context of algebraic proof including literal expressions. We applied 24 third-grade middle school students a test item which shows a proof including a literal expression and missing the conclusion. Over the half of them responded wrong answers based on only the literal expression without considering its context. Three of them were interviewed individually to show their thinking. As a result, we could find some characteristics of their thinking including the perspective on proof as checking the validity of algebraic expression and the gap between proving and understanding of proof etc. From these, we also discussed about several didactical implications.

• School Mathematics
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• v.5 no.1
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• pp.59-76
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• 2003

Algebraic signs are important on learning and problem solving of algebra. This study investigated the contents of letters and expressions in textbooks by syntactics, semantics and pragmatics, and considered the introduction and extension processes of algebraic signs didactically. We also categorized the signs, and looked into textbook problems in view of semiotic. The result is that textbook is constructed in syntactics and semantics. Finally, the assessment of 7th grade students' competence in syntactics, semantics, syntactics+- semantics, pragmatics, and problem solving shows that students' ability in syntactics and pragmatics Is a predictive variable for algebraic problem solving.

• Education of Primary School Mathematics
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• v.19 no.3
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• pp.223-247
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• 2016

Given the importance of developing algebraic thinking from early grades, this study investigated an overall performance and main characteristics of algebraic thinking from a total of 197 third grade students. The national elementary mathematics curriculum in Korea does not emphasize directly essential elements of algebraic thinking but indicates indirectly some of them. This study compared our students' performance related to algebraic thinking with results of Blanton et al. (2015) which reported considerable progress of algebraic thinking by emphasizing it through a regular curriculum. The results of this study showed that Korean students solved many items correctly as compatible to Blanton et al. (2015). However, our students tended to use 'computational' strategies rather than 'structural' ones in the process of solving items related to equation. When it comes to making algebraic expressions, they tended to assign a particular value to the unknown quantity followed by the equal sign. This paper is expected to explore the algebraic thinking by elementary school students and to provide implications of how to promote students' algebraic thinking.