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`<\ l| X Yt (4X ܭYD >0 t ȵ<\ t t ĳt 䲔 D LX. tL (4X Y ܭYD >DŴ t |T D tLŴp ɔ\ `D X\ (4X ܭYD >D LŔ (4X lp<\ XՔ t ĳt p tǔ |D ̹ܴp ĳD .
8mD ` LŔ ĳ] l´ 8mՀ0 XՔ t <\ an, n+c, an+c, an^(2), n^(2)+c,an^(2)+bn+c <\ Xp YX } @ X (4 8mD ` D H\. tǔ \ YX % ĳ ĳt t. \ t D 8 X Jĳ] XՔ ĳ ɔ` t.;In the algebra curriculum of each country such as USA, UK, and Australia, the Pattern Generalization learning where the learner takes the lead in creating the basis of thinking breaking away from the traditional algebraic learning method is emphasized(Kim Sung-joon). As a pattern provides a chance to find a certain rule in the surrounding environment and express it mathematically, it can be a good subject that can recognize the value and usefulness of mathematics. Also, it can be a good tool to improve the mathematical thinking skills such as inductive inference and deductive inference(Kim Soo-hwan et al. 2009). In spite of such an advantage of pattern, it has not yet settled down as a core subject in the mathematics curriculum, and, in most of the cases, pattern learning is regarded as an activity of finding simple sequence or regularity or a recreation activity(Zazkis and Liljedahl, 2002). For this reason, it is required to analyze the students' level and actual state of Pattern Generalization to present the implications of Pattern Generalization for algebraic education. Accordingly, in this study, we intend to look into the high rank second year middle school students' stage and strategy of Pattern Generalization. In accordance with such a necessity and an objective of study, we have established the following research issues in this study:
1. What is the high rank second year middle school students' stage of Pattern Generalization depending on the Forms of Expressions?
a. What is the stage of Pattern Generalization depending on the Forms of Expressions?
b. What is the justification type of the stage of Pattern Generalization?
2. What is the high rank second year middle school students' strategy of Pattern Generalization depending on the Forms of Expressions?
a. What is the strategy and the success rate of Pattern Generalization depending on the Forms of Expressions and the success rate?
b. What change has taken place when a student who has failed in Pattern Generalization is induced to approach it with another strategy?
In this study, in order to grasp the Pattern Generalization of second year middle school students depending on the Forms of Expressions, preliminary tests were conducted twice. Based on the result, a study was carried out through convenience sampling of 82 second year students of C Middle School located in Seoul who belong to the high rank group26). The objects of the study were the 78 students excluding the 4 students whose response to the test questions was insufficient. In order to produce the questionnaire to be used for this study, the visual patterns which appeared in preceding studies were analyzed first. The attributes of pattern were classified into Form of Expression, Shape of Pattern, Material of Pattern, Linearity, Increase/Decrease, and other , and Form of Expression was adopted as the major variable of this study. Accordingly, efforts were made to control the remaining variables. The Forms of Expressions used here were an, n+c, an+c, an^(2), n^(2)+c, and an^(2)+bn+c . In order to look into the stage of Pattern Generalization depending on the Forms of Expressions, correction and supplementation were carried out based on the stages of Pattern Generalization of Stacy(1989) and Kim Sung-joon(2003), and, in order to look into the justification type at the stage of Pattern Generalization, the justification types of Balacheff(1987) and Tall(1995) were reorganized. In this study, the justification types were classified into Empirical Justification through Examples, Inductive Empirical Justification, Visual Logical Justification, and Logical Justification throug< h Induction Principle. Also, in order to investigate the strategy of Pattern Generalization, correction and supplementation were carried out on the basis of the generalization strategy of Lannin(2003). In order to enhance the reliability in solving the research issues, another teacher participated in this study in addition to this researcher as a coder. Also, in order to carry out a deeper analysis on the basis of the written test, individual interviews with the students selected among those who participated in this test were conducted. In order to observe the learning environment of the students and to closely observe the thoughts of the students at the same time, the context of the interview was recorded and filmed. The video files and recorded contents were copied and used for analysis.
The result of analyzing the stage of Pattern Generalization is as follows:
First, as a result of analyzing the stage of Pattern Generalization, it could be seen that the high rank second year middle school students are distributed much over the Starting Stage and Disambiguation Stage . The students at the Starting Stage are the students who are unable to well obtain the number of patterns of 10th stage though they can well draw the figure of the next stage of the given pattern, and the students at the Disambiguation stage are the students who can solve the expression of Pattern Generalization. The fact that most of the students are distributed over the Starting Stage and the Disambiguation Stage like this can be interpreted to mean that though they can easily derive the expression of generalization if they once find the mathematical rule of the pattern, it is not easy for them to move from the Starting Stage to the next stage if they fail to find the rule . Accordingly, it can be seen that finding the mathematical rule of the pattern is important to the extent that it determines the success and fail in deriving the generalized expression.
Second, when we look into the distribution of the students who have succeeded in generalization by question, the frequency of success drops as we move from Question No. 1 to Question No. 6. That is to say, it means that the question becomes more difficult as we mover from an, to n+c, an+c, an^(2), n^(2)+c, and an^(2)+bn+c.
Third, it can be seen that Empirical Justification through Examples and Visual Logical Justification methods are most widely used among the justification types. As
Question No. 1 and Question No. 2 were of simple linear expression, the obtained expressions were justified by the method of easily checking whether the expression obtained was right by substituting the relevant part of the obtained expression with an appropriate value. As Questions No. 4 to 6 were of quadratic expression, the obtained expressions were justified by grasping the structure of the given pattern rather than substituting the relevant part of the obtained expression with an appropriate value. This means that they checked whether their expressions were right or not, and attempted verification, though not strict. Accordingly, it was found that, as to the questions for which the rule could be easily found, a tendency of justifying them by substituting the relevant parts with one or two values was shown, and, as to the questions for which the rule could not be easily found, the justification was carried out using a relatively logical method.
The result of looking into the strategy of pattern generation by question is as follows:
First, Contextual Strategy showed the highest frequency, and was followed by Recursive Strategy. That is to say, these two strategies were thought to be quite attractive strategies to the students in solving the pattern questions. Also, among these strategies, as Contextual Strategy has shown a very high correct answer rate, it can be regarded as an effective strategy, though there may be a difference
depending on the question.
Second, we need to pay attention to Counting Strategy and Guess and Check Strategy. These two strategies showed higher frequencies of failure than success. First, the st<udents who use Counting Strategy are presumed to be mostly the students who remain at the starting stage. These students attempted to find the rule of the pattern by drawing a picture only to fail. Next, the reason why many students who used Guess and Check Strategy failed in establishing a general expression was thought to be because the students who used this strategy only tried to find the expression using only their intuition without considering why the expressions they found were like that. That is to say, though they can find the expressions through trials and errors, they fail more frequently.
In conclusion, it is found through this study that it is helpful for the students to approach the questions with different strategies to find the rule of the pattern. As finding the mathematical rule of the pattern plays an important role in deriving the generalized expression at this time, it will be helpful to structurally approach the pattern when finding the rule of the pattern, and this will also help in creating the general expression.
As it is better to present easier questions first when presenting the questions, it is
recommended to present them in the order of an, n+c, an+c, an^(2), n^(2)+c and an^(2)+bn+c and to present the pattern questions of the level a little higher than that of the student. This will be also helpful for improvement in the thinking skills of students. Also, it is important to develop the habit of re-checking the result of calculation to prevent mistakes.~http://dspace.ewha.ac.kr/handle/2015.oak/212006;
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