Different Types of Algebraic Thinking: an Empirical Study Focusing on Middle School Students
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Different Types of Algebraic Thinking: an Empirical Study Focusing on Middle School Students Demetra Pitta-Pantazi 1
& Maria
Chimoni 1 & Constantinos Christou 1
Received: 4 December 2018 / Accepted: 18 July 2019/ # Ministry of Science and Technology, Taiwan 2019
Abstract Central in the frameworks that describe algebra from K-12 is the idea that algebraic thinking is not a single construct, but consists of several algebraic thinking strands. Validation studies exploring this idea are relatively scarce. This study used structural equation modeling techniques to analyze data of middle school students’ performance on tasks that correspond to four algebraic thinking strands: (i) Generalized Arithmetic, (ii) Functional Thinking, (iii) Modeling Languages, and (iv) Algebraic Proof. The study also examined the role that cognitive abilities play in students’ algebraic thinking. Results emerging from confirmatory factor analysis showed that the proposed model adequately explains students’ algebraic thinking. Additionally, results emerging form latent path analysis showed that students are first able to solve Functional Thinking tasks and only when this is achieved, they proceed to solve Generalized Arithmetic tasks, then Modeling Languages tasks, and finally Algebraic Proof tasks. Lastly, the quantitative analyses indicated that students’ cognitive abilities (analogical, serial, and spatial reasoning) predict students’ algebraic thinking abilities. Keywords Algebraic proof . Algebraic thinking . Cognitive abilities . Functional thinking .
Generalized arithmetic
Introduction During the twentieth century, a well-established principle among various national curricula was that arithmetic should precede algebra (Carraher & Schliemann, 2007). Only at the beginning of the twenty-first century researchers and educators started highlighting the problems arising from this compartmentalization of arithmetic and algebra and the need for a more comprehensive and longitudinal approach starting from
* Demetra Pitta-Pantazi [email protected]
1
Department of Education, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
D. Pitta-Pantazi et al.
elementary school (Kaput, 2008). Specifically, the Principles and Standards of School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000) suggested spreading algebra across K-12, an action which highlights that algebra should be viewed as a way of thinking, rather than as a discrete topic (Kaput, 2008). A number of approaches have been pursued to describe the multidimensional nature of algebraic thinking (Kaput, 1998; Kieran, 1996). Kaput (1998) suggested a comprehensive theoretical framework that involves four algebraic thinking strands; (i) Generalized Arithmetic, (ii) Functional Thinking, (iii) Modeling Languages, and (iv) Abstract Algebra, of which Algebraic Proof is a main component. To date, there are no validation studies that empirically tested the hypothesis that students’ algebraic thinking could be described through these four strands nor are there any stu
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