Math Progress International 11-14 Answers

[Agalia Valcin]

The study aims to compare and determine 6th-8th graders' efficiencies, strategies and representations of student from different grades (from 6th to 8th) when dealing with problems related to linear and quadratic patterns. Research data was obtained from tests applied to 246 students and clinical interviews implemented with 18 students. It was shown that when grade increases students' efficiencies of generalizing pattern improve in a positive way in all levels. Besides this, as learning levels of 6th and 8th grade students increase, the variety in pattern generalization strategies changed at least in all types of patterns. While students at different learning levels generally used recursive strategy to solve all the problems, the number of the students using explicit strategies is relatively low. The students of high levels use many types of representation in generalization; in contrast, the weak level students prefer mostly numerical representations.

Neste artigo so estudadas as estratgias e representaes usadas por alunos turcos de diferentes sries (da 6 a 8) quando resolvendo problemas que envolvem a busca de padres lineares e quadrticos. Os dados foram coletados a partir da aplicao de testes especficos em 246 alunos e de entrevistas clnicas com 18 desses alunos. Os resultados apontam que a eficcia nos padres de generalizao aumentam positivamente de uma srie escolar para outra, bem como aumenta a variedade das estratgias de generalizao. Enquanto alunos de diferentes momentos da escolarizao usam estratgias recursivas para resolver os problemas relativos busca de padres, mas a quantidade de estudantes que mobilizam estratgias explcitas relativamente pequena. Alunos de sries mais avanadas usam vrios tipos de representaes no processo de generalizar, ao passo que alunos das sries mais iniciais, de modo acentuado, preferem a representao numrica.

In order to educate 21st century people the field of mathematics education is passing through dramatic changes. In that changing process, the approaches on high-level mathematical reasoning such as exploring, conjecturing and generalizing has taken a significant place in basic mathematics instruction (THOMPSON; THOMPSON, 1995). Thus, new standards and curricula have been developed and put into practice around the world. Turkish education system has affected by this process and Turkish elementary mathematics education curriculum has changed following the new expectations stated by Turkish Ministry of Education (TME). Efforts to study, recognize and develop patterns in new curriculum appear as an important learning field, based on the fact that

Mathematics is the science of patterns and harmonies. In other words, mathematics is the science of number, shape, space, size and the relations between them. Discovering the relations that the patterns include, and generalizing them can be helpful for students in developing their skills while perceiving the world around them better. In addition, the patterns represented in different forms and especially expressed as symbolic will make agreat contribution to basic concepts of algebra to be formed [...] (TME, 2005, p.95).

Many mathematics educators have dealt with patterns from different points of view and agreed on the idea that discovering and generalizing patterns are important for learning mathematics. They have also expressed that the study of patterns could improve students' algebraic concepts at early ages, and contribute to that algebraic thinking required in future learning (KENNY; SILVER, 1997; ENGLISH; WARREN, 1998; ORTON; ORTON, 1999; ZAZKIS; LILJEDAHL, 2002; LANNIN, 2003; SMITH, 2003). Lee (1996, p.103) stated that "algebra, and indeed all of mathematics is about generalizing patterns". Armstrong (1995) emphasized that exploring patterns at early gradesimprove children thinking abilities from an algebraic point of view, and signalized the importance of making generalizations in algebra using the patterns. Smith (2003), stressing on the relationship between patterns, functions and algebra, claimed that these three components must be integrated in the curriculum. In addition, many types of patterns - like linear and non-linear (quadratic and geometrical sequence), numerical, pictorial, arithmetical, geometrical and repetitive patterns - can be themes in the studies about pattern generalization including different education levels, from elementary schooling to pre-service school teachers, or from primary to secondary school (ORTON; ORTON, 1999; BISHOP, 2000; LANIN, 2005; RIVERA, 2007; AMIT; NERIA, 2008). In literature, great numbers of different types of patterns are studied using different strategies in order to generalize them. In order to determine such strategies, first of all, pattern types must be classified systematically. In this respect, Both Ley's and Feifie's studies about classifying different types of patterns are of huge importance. Ley (2005) puts forward linear patterns in five different formats as visual, geometric, table, number sequence and word problem. Feifie (2005), as seen in Figure 1, defines three different types of patterns and focuses on their five different formats.

The study of pattern generalization in school mathematics has been the focus of research conducted over the last years. In these studies, while focusing generalizing pattern types, it has been suggested that it should be focused on the detailed studies on strategies used in generalizing patterns. Stacey (1989), in his study about generalizing linear pattern problems with students in different educational levels, claims that they are more competent to find the near term than the far term in both shape and sequence of numbers. Orton and Orton (1999) defines patterns as a kind of approach leading to algebra and claims that students aged between 10-13 are much more able in generalizing linear patterns in sequences like "1, 4, 7, ..." than generalizing non-linear ones, and they are also able to find near and far terms in linear patterns easily when compared with quadratic patterns. Feifei (2005), studying the ability of 8th grade students in problems including linear, quadratic and geometric sequences, finds out that these students are much more able to deal with linear pattern problems than with quadratic ones, and are also more able to deal with quadratic pattern problems than with geometric sequencespattern ones.

In addition to this, researches about generalizing different pattern problems - from primary school to university - put generalizing pattern problems in the center of the issue. In these studies, researchers define many strategies to deal with generalization in problems related to patterns such as recursive, common ratio, counting, additive, explicit or non-explicit, whole-subject, linear, guess-and-check, trial-and-error and contextual strategies, for instance (STACEY, 1989; ORTON; ORTON, 1999; BECKER; RIVERA, 2004, 2005; AMIT; NERI, 2008). Besides, all these studies, especially the one in which Stacey (1989) defines strategies in generalizing pattern problems,give foundation to many other studies.For instance,in a study about linear patterns, Stacey (1989) put forward some strategies about linear patterns including near and far generalization for 9-13 years students. The main contribution of her research was the classification of students' strategies when solving contextualized linear generalization tasks, whether or not leading to correct answers. Strategies found were counting, whole-object, difference and linear. Based on this study, author concludes that a significant number of students used incorrectly a direct proportion method when trying to generalize. Another study based on Stacey's work was held by Garcia-Cruz and Martion (1997). They developed a study to understand the processes of generalization of secondary school students. Their categorization of the methods used by these students was based on Stacey's work. According to these researchers, there are three types of strategies: visual, numeric and mixed (numeric and visual). If the drawing had a fundamental role in finding the pattern, it would be assumed as a visual strategy. However, if the numerical sequence had a fundamental role for finding the pattern, then the strategy would be assumed as numeric. The students who used mixed strategies especially focused on the numerical sequence, but for confirming the validity of the solution they usually draw some sketch. This provides the setting for students that use visual strategies for checking the validity of the reasoning and for students that use numerical strategies. Moreover, Rivera and Becker (2005) claim that students mostly use numeric strategies, and describe three types of generalization: numerical, figural and pragmatic. Students that use numerical generalizationimplemented trial-and-error with little sense about what factors in the linear pattern mean. Students using figural generalization concentrated on relations between numbers in the sequence and could define variables under the scope of a functional relationship. Students that work based on a pragmatic generalization used numerical and figural strategies together in order to define sequences of numbers based on both properties and relationships. Ley (2005) describedthe three differentsolutionstrategies forlinearpattern problems: recursive, whole-object and explicit. According to Ley (2005), a recursive strategy involves the use of the previous term in the sequence to find the next term. Both, children and adults, commonly try to solve the problem finding the difference between two terms and then adding that result to the last term in order to determine the next number in the sequence. The whole-object strategy involves, often mistakenly, the proportional reasoning to solve question related to pattern. An explicit strategy involves generalizing the relationship between the two variables so that any value can be determined. This is the first step in a gradual progression towards expressing functions using formulas and equation. When using an explicit strategy, the rule is invariant and applicable to the near and the far terms, and therefore is conducive to create a general rule and reach the nth term. Lanin (2005) analyses strategies of 6th grade students that, in generalizing pattern problems, use a particular approach. Lanin's framework includes explicit and non-explicit strategies (non-explicit strategies are counting and recursive; while explicit strategies are whole-object, guessing and checking, contextual). In a more recent study, Amit and Neri (2008) focus on the strategies of middle school students (12-14 aged) in generalizing linear and non-linear pattern problems, and conclude that capable students have high mathematical abilities for pattern problems resembling generalization. They also declare that students use recursive method for local generalizationsand functional method for global generalization, in those cases that problems include both types of patterns. In addition, they state that students use, in general, additive and global strategies, and they emphasize that most of the students have preferred multiplication strategies to additive ones. Besides, most of the studies examining strategies of generalization of patterns study, generally, linear patterns. Only few of these studies examine non-linear patterns (KREBS, 2003; EBERSBACH; WILKENING, 2007; AMIT; NERIA, 2008). As an example, we can point out that according to the studies related to non-linear patterns (KREBS, 2003; EBERSBACH; WILKENING, 2007), additive strategies are mostly common and it has been stated that there has been a great tendency towards linearity, although the patterns are obviously non-linear. Moreover, while additive (expansive) strategies used in linear pattern problems are much more employed then a global generalization - because of the constant and obvious difference between each two successive patterns - , in non-linear patterns it can be said that this approach involving the use of visual approaches can prevent students from understanding the global structure (KREBS, 2003; RIVERA, 2007; AMIT; NERIA, 2008).

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