# An Analysis of Middle School Mathematics Textbooks from the Perspective of Fostering Algebraic Thinking through Generalization *

## Article excerpt

Although the concept of algebraic thinking is considered related to algebra, it is seen to have a wider meaning than algebra and also to have many definitions emphasizing its different aspects. Algebraic thinking can be said to be a subset of mathematical thinking and to use many basic skills such as reasoning, representation, functional thinking, and generalization (Bednarz, Kieran, & Lee, 1996; Driscoll & Moyer, 2001; Kaput, 2000; Mason, 1996). Of these, generalization is prominent in terms of its central role in mathematics.

While Mason (1996) defined generalization as the heart of mathematics, Kaput (2008) defined the focus of algebraic thinking as a complex process of symbolization that serves the aim of generalization. Polya (1957) referred to generalization as the center of mathematical activities and as a basis for developing mathematical knowledge. Many studies have emphasized the importance of generalization in developing students' thinking processes (Blanton, 2008; Common Core State Standards for Mathematics [CCSSM], 2010; Mason, Johnston-Wilder, & Graham, 2005; National Council of Teachers of Mathematics [NCTM], 2000; Van de Walle, Karp, & Bay-Williams, 2012).

CCSSM, which is regarded as the backbone of educational reform in the United States, and NCTM, the leading organization in mathematics education, advocate that mathematics exercises that include generalizations improve mathematics education. Concordantly, Turkey's mathematics curriculum, which was revamped in 2006 and 2013, has emphasized the importance of providing students with generalization skills. Also when students are trying to improve their problem-solving skills, generalizations are provided as a necessary solution. In addition, an indicator that should be taken into consideration for providing students with reasoning skills is stated in the curriculum as something that "makes logical generalizations and inferences" (Milli Eğitim Bakanlığı [MEB], 2006, 2013).

In short, generalization is a fundamental cognitive function in the thinking process (Dumitraşcu, 2015) and thus has a critical role in teaching and learning (Kaput, Carraher, & Blanton, 2008). In the process of mathematics teaching and learning, students need high-level thinking that emphasizes generalization; otherwise learning difficulties are inevitable. Indeed, many studies on students' ability to generalize in every grade in mathematics confirm this view (Akkan & Çakıroğlu, 2012; Bishop, 1997; Çayır & Akyüz, 2015; Gray, Loud, & Sokolowski, 2005; Haldar, 2014; Kaput & Blanton, 2000; Lee & Lee, 2015; MacGregor & Stacey, 1997; Özdemir, Dikici, & Kültür, 2015). For instance, some studies have determined students to have difficulty with: formulating and writing mathematical thinking when generalizing (Kaput & Blanton, 2000), expressing simple relationships using algebraic notations (Bishop, 1997), generalizing arithmetic (Haldar, 2014), and generalizing patterns (Akkan & Çakıroğlu, 2012; Çayır & Akyüz, 2015; Lee & Lee, 2015; Özdemir et al., 2015). For all these reasons, generalizations have been identified as the critical point of this research focused on algebraic thinking.

Textbooks, by guiding teachers and providing resources for learning, are a rather important teaching material. The quality of a textbook largely guides teaching activities and contributes to how students learn subject (Demirel & Kıroğlu, 2005; Güzel & Adıbelli, 2011). Textbooks are also tools for implementing curricula (Duman, Karakaya, Çakmak, Eray, & Özkan, 2001). While a textbook does not fully reflect what happens in the classroom, it does however show the instructional objectives that can influence students' mathematical knowledge (Dumitraşcu, 2015). All these significant points have led studies to analyze textbooks under various subject areas in the mathematics education literature (Ashcraft & Christy, 1995; Bakılan-Mutu, 2008; Freeman & Porter, 1989; Jitendra, Deatline-Buchman, & Sczesniak, 2005; Kerpiç & Bozkurt, 2011; Tanışlı & Köse, 2011; Taşdemir, 2011; Yeniterzi & Işıksal-Bostan, 2015). …

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