• School Mathematics
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• v.14 no.4
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• pp.445-468
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• 2012

This study is tried in order to link informal arithmetic reasoning to formal algebraic reasoning. In this study, we investigated elementary school student's non-formal algebraic reasoning used in algebraic problem solving. The result of we investigated algebraic reasoning of 839 students from grade 1 to 6 in two schools, Korea, we could recognize that they used various arithmetic reasoning and pre-formal algebraic reasoning which is the other than that is proposed in the text book in word problem solving related to the linear systems of equation. Reasoning strategies were diverse depending on structure of meaning and operational of problems. And we analyzed the cause of failure of reasoning in algebraic problem solving. Especially, 'quantitative reasoning', 'proportional reasoning' are turned into 'non-formal method of substitution' and 'non-formal method of addition and subtraction'. We discussed possibilities that we are able to connect these pre-formal algebraic reasoning to formal algebraic reasoning.

• East Asian mathematical journal
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• v.31 no.2
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• pp.145-165
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• 2015

The purpose of this study is to suggest how to go about teaching and learning secondary school algebra by analyzing problem-solving ability and problem-solving process through algebraic reasoning. In doing this, 393 students' data were thoroughly analyzed after setting up the exam questions and analytic standards. As with the test conducted with technical school students, the students scored low achievement in the algebraic reasoning test and even worse the majority tried to answer the questions by substituting arbitrary numbers. The students with high problem-solving abilities tended to utilize conceptual strategies as well as procedural strategies, whereas those with low problem-solving abilities were more keen on utilizing procedural strategies. All the subject groups mentioned above frequently utilized equations in solving the questions, and when that utilization failed they were left with the unanswered questions. When solving algebraic reasoning questions, students need to be guided to utilize both strategies based on the questions.

• School Mathematics
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• v.13 no.4
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• pp.581-598
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• 2011

Given the growing agreement that algebra should be taught in the early stage of the curriculum, considerable studies have been conducted with regard to early algebra in the elementary school. However, there has been lack of research on how to organize mathematic lessons to develop of algebraic reasoning ability of the elementary school students. This research attempted to gain specific and practical information on effective algebraic teaching and learning in the elementary school. An exploratory qualitative case study was conducted to the fourth graders. This paper focused on the associative law of the multiplication. This paper showed what kinds of activities a teacher may organize following three steps: (a) focus on the properties of numbers and operations in specific situations, (b) discovery of the properties of numbers and operations with many examples, and (c) generalization of the properties of numbers and operations in arbitrary situations. Given the steps, this paper included an analysis on how the students developed their algebraic reasoning. This study provides implications on the important factors that lead to the development of algebraic reasoning ability for elementary students.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.4
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• pp.655-676
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• 2016

The purpose of this study is to examine the effects of teaching on functional thinking, one of the algebraic thinking in sixth grade students level. For this study, we developed functional thinking based-teaching through analyzing mathematical curriculum and preceding research, which consisted of 12 classes, and we investigated the effects of teaching through quantitative and qualitative analysis. In the results of this study, functional thinking based-teaching was statistically proven to be more effective in improving algebraic reasoning skills and lower elements which is an algebraic reasoning as generalized arithmetic and functional thinking, compared to traditional textbook-centered lessons. In addition, the functional thinking based-teaching gave a positive impact on the functional thinking level. Thus functional thinking based-teaching provides guidance on the implications for teaching and learning methods and study of the functional thinking in the future, because of the significant impact on the mathematics learning in six grade students.

• 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.

• 제어로봇시스템학회:학술대회논문집
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• 1993.10a
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• pp.1058-1062
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• 1993

A fuzzy reasoning method is proposed for the implementation of control systems based on non-fuzzy microprocessors. The essence of the proposed method is to search the local active miles instead of the global rule base. Thus the reasoning is conveniently performed on a master cell as a fuzzy accelerating kernel, which is transformed from an active fuzzy cell. The interpolative reasoning is simplified via adopting the algebraic product of fulfillment for the conditional connective AND and the weighted average for the rule sentence connective ALSO.

• Journal of the Korean Institute of Telematics and Electronics B
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• v.33B no.12
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• pp.84-94
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• 1996

In this work, a fuzzy petri net model for modeling a general form of fuzzy production system which consists of chaining fuzzy production rules and so requires multistage reasoning process is presented. For the obtained fuzzy petri net model, the net will be transformed into some matrices, and also be systematically led to an algebraic form of a state equation. Since it is fond that the approximate reasoning process in fuzzy systems corresponds to the dynamic behavior of the fuzzy petri net, it is further shown that the multistage reasoning process can be carried out by executing the state equation.

• The Mathematical Education
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• v.58 no.2
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• pp.161-185
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• 2019

This study aims to examine pre- and in-service mathematics teachers' reasoning and how they justify their reasoning. For this purpose, we developed a set of mathematical tasks that are based on mathematical contents for middle grade students and conducted the survey to pre- and in-service teachers in Korea. Twenty-five pre-service teachers and 8 in-service teachers participated in the survey. The findings from the data analysis suggested as follows: a) the pre- and in-service mathematics teachers seemed to be very dependent of the manipulation of algebraic expressions so that they attempt to justify only by means of procedures such as known algorithms, rules, facts, etc., rather than trying to find out a mathematical structure in the first instance, b) the proof that teachers produced did not satisfy the generality when they attempted to justify using by other ways than the algebraic manipulation, c) the teachers appeared to rely on using formulas for finding patters and justifying their reasoning, d) a considerable number of the teachers seemed to stay at level 2 in terms of the proof production level, and e) more than 3/4 of the participating teachers appeared to have difficulty in mathematical reasoning and proof production particularly when faced completely new mathematical tasks.

• 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.

• Journal of Elementary Mathematics Education in Korea
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• v.19 no.4
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• pp.457-484
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• 2015

This study aims to investigate an approach to teach proportional reasoning in elementary mathematics class by analyzing the proportional strategies the students use to solve the proportional reasoning tasks and their percentages of correct answers. For this research 174 sixth graders are examined. The instrument test consists of various questions types in reference to the previous study; the proportional reasoning tasks are divided into algebraic-geometric, quantitative-qualitative and missing value-comparisons tasks. Comparing the percentages of correct answers according to the task types, the algebraic tasks are higher than the geometric tasks, quantitative tasks are higher than the qualitative tasks, and missing value tasks are higher than the comparisons tasks. As to the strategies that students employed, the percentage of using the informal strategy such as factor strategy and unit rate strategy is relatively higher than that of using the formal strategy, even after learning the cross product strategy. As an insightful approach for teaching proportional reasoning, based on the study results, it is suggested to teach the informal strategy explicitly instead of the informal strategy, reinforce the qualitative reasoning while combining the qualitative with the quantitative reasoning, and balance the various task types in the mathematics classroom.