ELEMENTARY SCHOOL FIFTH GRADE STUDENTS ’ THINKING PROCESS IN LINEAR AND QUADRATIC PATTERNS Dilek TANISLI Aynur ÖZDAS Anadolu University,Turkey

22 INTRODUCTION Algebra is one of the most important but also the most diffucult subject in mathematics. Because the doing algebraic operations requires the power of abstracting. Although algebra is a word that has not commonly been heard in grades 3-5 classrooms, the mathematical investigations and conversations of students in these grades frequently include elements of algebraic reasoning.

33 To eliminate the difficulties in algebra, it is necessary to conduct the studies providing the development of the algebraic thinking at the first step in primary school. One of these studies is the pattern studies which are the beginning of abstract and variable concepts.

44 In order to think algebraically students must be able to identify, extend, and generalize patterns to understand quantitative relationships. In other words, algebraic thinking includes the ability to analyze and recognize patterns, to represent the quantitative relationships between the patterns, and to generalize these quantitative relationships. These experiences also assist students to develop the function concept.

55 In many countries, pattern is one of the important parts of elementary programs. Patterns have been discussed for the elementary mathematics program in Turkey since To find out student’s strategies to construct patterns and their cognitive processes in this construction, developing students’ algebraic thinking and function concept is important.

66 Purpose of the Study The main purpose of this study, is part of my Ph. D. dissertation, is to determine the thinking process linear and quadratic shape patterns for elementary fifth grade students, which have the different mathematics success levels (high, middle, low). Within the framework of this aim, the following research questions are addressed:

77 Research Questions In linear and quadratic shape patterns, elementary fifth grade students; How do they define the rule of pattern? How do they extend to the near and next step to pattern? How do they create the requiring pattern?

88 METHOD Participants The study was conducted with 12 elementary fifth grade students, which have the different mathematics success levels (high, middle, low). While selecting the participants of the study out of purposeful sampling techniques, the criterion sampling technique was used.

99 Data Collection and Analyze The data of the study was collected from task- based interviews. All interviews were video-recorded. In addition to the video protocols, written transcripts of the students’ verbal responses as well as their paper-and-pencil activities were used in the analysis.

10 Two linear, and two quadratic shape patterns were asked students in interviews. Each student was interviewed for once during 15-minute. For analysis, the classification which includes “data reduction”, “data display” and “drawing conclusion and verification” were used.

11 DISCUSSION AND CONCLUSION While finding the rule of linear and quadratic shape patterns, the students adopted two approaches namely visual and algebraic (Stacey, 1989; Orton and Orton,1999; Samsan, Linchevski and Olivier, 1999; Krebs, 2005; Becker and Rivera, 2006; Lan Ma, 2007). In visual approach, the students focused on the structure of the shape and in this context, they used recursive, expilicit and other strategies.

12  Within the context of recursive strategies, for both types of pattern, the majority of the students stated how many shapes are required for each new shape (Sasman, Olivier and Linchevski, 1999; Orton and Orton, 1999; Becker and Rivera, 2006; Lan Ma, 2007). As seen in the example, in linear shape pattern the students explained that by adding four white circles to the previous shape, the subsequent shape is obtained.

13  On the other hand, within the context of explicit strategies, some students found the functional relation depending on the structure of the shape (Sasman, Olivier and Linchevski, 1999; Becker and Rivera, 2006; Lan Ma, 2007). For example; in the linear shape pattern given below, the students established the functional relation by associating step number with the number of shape group, constituting three points they selected for each shape x1+1 3x2+1 3x3+1 3x4+1

14  As figured in the example, in the quadratic shape pattern, some of the students constructed the functional relation by associating step numbers with the row number indicated in bold in the shape in each term and square numbers involved in each row (1x1)+1 (2x2)+1 (3x3)+1

15  Within the context of other strategies, some of the students who accepted the visual approach determined how many times the previous shape occur in the subsequent shape (Sasman, Olivier and Linchevski, 1999; Becker and Rivera, 2006).

16 On the other hand, the students adopted the algebraic approach converting the given shape pattern into number pattern firstly (Stacey, 1989; Orton ve Orton,1999, Krebs, 2005; Becker ve Rivera, 2006; Lan Ma, 2007). In this context, the students used recursive, explicit and other strategies.

17  Within the recursive strategies, the strategy to seek the differences between the terms was of the first priority that most of the students focused on. For example, as seen in the following, the students stated that the difference between the terms is 4. [f(n+1)=f(n)+4]

18 Apart from this, the example indicates that there were some students who could find the functional relation by establishing the relation between the terms and the term order (4x1)+1 (4x2)+1 (4x3)+1

19  On the other hand, the students who could establish functional relation at both pattern type could generalize it by verbalizing this relation. For example; there were some expressions such as “ One more of four times of the step number ”.  In addition, more students established functional relations in linear shapes in comparison to in quadratic shapes.

20  Under algebraic approach, within the context of other strategies, the students used the strategies related to a feature of members of a number set, in the number, quadratic and linear patterns they obtained. For example; as it can be seen in the following, some students expressed that in the number patterns they obtained, each number is odd.

21 It was observed that while extending the linear or quadratic shape patterns to a near step the majority of the students used recursive strategies under visual and algebraic approach, on the other hand while extending them to a next step, they focused on explicit strategies.

22 However, some of the students, who could not find any functional relation while extending the linear or quadratic shape patterns to a next step, tend to use “whole-object” strategy which was defined by Stacey (1989) and handled within other strategies and these students obtained wrong results. For example; if n=k.m, then f(n)=k.f(m) After finding f(5)=21, calculates f(50) as n=5.10=50 f(50)=10.f(5) = = 210

23 Programın Etkilerine İlişkin Görüşler  As for forming linear or quadratic shape patterns, the majority of the students could form the linear shape patterns, very few students could form the quadratic shape patterns. For example; linear shape pattern; 1. 2.

24 Quadratic shape pattern;

25 Programın Etkilerine İlişkin Görüşler  On the other hand, some of the students who could not form the linear shape patterns formed the quadratic shape patterns and in the same way, the majority of the students who could not form the quadratic shape patterns formed the linear shape patterns.  In addition to these, there were some students who formed different pattern types apart from these two patterns. Thus, it was observed that while forming the patterns, the students did not take into account of the features of the desired pattern type.

26 Programın Etkilerine İlişkin Görüşler  Moreover, the results also revealed that for both pattern types, there is no relation between the students ’ success levels and the selection of strategy for finding rule, extending the pattern to near and next step and forming the pattern.

27 ELEMENTARY SCHOOL FIFTH GRADE STUDENTS ’ THINKING PROCESS IN LINEAR AND QUADRATIC PATTERNS Dilek TANISLI Aynur ÖZDAS Anadolu University,Turkey