Ali Eraslan

The notion of reducing abstraction in quadratic functions

One possible approach students can cope with abstract algebra concepts is reducing abstraction. This notion occurs when learners are unable to adopt mental strategies as they deal with abstraction level of a given task. To make these concepts mentally accessible for themselves, learners unconsciously reduce the level of the abstraction of the concepts (O. Hazzan and R. Zazkis, Reducing abstraction: the case of school mathematics, Educ. Stud. Math. 58 (2005), pp. 101–119). Reducing abstraction as a theoretical framework has been used for explaining students’ thought process in areas of advance mathematical topics in collegiate mathematics. By analysing two tenth-grade algebra honour students’ ways of thinking on quadratic functions, this article shows how the notion of reducing abstraction can be used for analysing students’ mental processes in secondary school mathematics and provides insights into students learning about graphs as well as reports two different strategies on translation tasks as an act of reducing the level of abstraction: one provides an effective method, while the other becomes misleading.

The notion of compartmentalization: the case of Richard

One of the important phenomena observed in the learning of mathematics is compartmentalization. This phenomenon occurs when a learner has two different, potentially contradictory schemes in his or her cognitive structure; in a typical case, a student deals with the same mathematical concept in an inconsistent or incoherent way, or activates a less relevant scheme to the situation in his or her mind. This research reports a case of compartmentalization phenomenon in which Richard, a tenth-grade algebra honours student, was interviewed on quadratic tasks. Results show evidence for the existence of compartmentalization related to the vertex of a parabola. The article concludes with some suggested educational implications.

REVEALING PRIMARY SCHOOL CHILDREN'S MATHEMATICAL MODELING POTENTIAL: HAIRDRESSER SALON PROBLEM

To be able to cope with complex systems, it is important for mathematics education that children have experience with situations in which they gradually engage the real problem with complex systems and interdisciplinary complex systems at starting from the primary school. In doing this, mathematical modeling consisting of cycles of model eliciting and relating model with real-life is one of the available approaches. Therefore, the purpose of this study was to reveal the models of fourth-grade primaryschool children, who have no prior experience with modeling problems in their classroom. This research study was conducted during the 2016-2017 academic year, in a university-foundation primary school in a large city along the Black Sea Region of Turkey. Participants were a total of 20 children in one of the 4 grades. The children in groups of four were assigned a modeling activity, the Hairdresser Salon Problem, to work on for two class-hours as a group. Modeling processes of each group and presentations at the end were video and audio-taped. The mathematical thoughts, developed models, written responses of the primary-school children were qualitatively analyzed in the light of the modeling cycle developed by Blum and Ferri [1]. The results showed that children were able to concert symbolic data into numerical data, take into account of different parameters (hair cutting and washing cost, waiting time, distance from home, hair washing time, hair cutting time, hair styling time and customer satisfaction) together and make mathematical calculations in the modeling process. At the end, they presented and explained their models if they were appropriate.

Fourth-Grade Primary School Students' Thought Processes and Challenges Encountered during the Butter Beans Problem.

While the need to access, use, and create knowledge has been continuously increasing in the 21st century, one of the main targets in education is to have a qualified work force be a citizen of the world, emphasizing world citizenship over individualism (Ministry of National Education [MoNE], 2013). Therefore, educating individuals to have skills like construction, hypothesis, identification, description, verification, prediction, manipulation, analytical thinking, and teamwork and who can effectively deal with problems and creatively develop solutions have become important educational goals (English & Watters, 2004). In this respect, mathematics education has greater importance in educating problem-solving individuals with analytical and creative-thinking skills. In line with these developments, Turkish curriculum was reshaped in 2005 to train individuals for these skills. Instead of emphasizing a step-by-step approach, memorization, or learning rules, the main focus of the current primary mathematics curriculum is to train individuals to use mathematics in their daily lives, to have problem-solving skills, to share their thoughts as a team, to have self-confidence in mathematics, and to develop a positive attitude towards mathematics (MoNE, 2009).When considering the goals of mathematics education, it has become critically important that students understand and be able to explain mathematical concepts, test hypotheses, and analyze relationships, as well as learn how to reconstruct existing knowledge (Thomas & Hart, 2010). Behaviors related to mathematics appear in all levels of educational programs, from pre-school to higher education, with adaptations according to ones level of development. Lesh and Zawojewsky (2007) stated that memorizing mathematical processes and then applying these methods to similar problems is not enough. They emphasized the need for students to face complex and multifaceted problem situations and gain experience this way, thus allowing them to develop new skills and mathematical thinking to prepare them for their future life after school. At this point, primary education is an important period for developing these skills (English & Watters, 2004). Mathematical models and modeling approaches can be utilized to analyze complex problems that represent real-life situations students can actively participate in (Sriraman & Lesh, 2006). Therefore, model-eliciting activities that bring about situations where students can create solutions to problems that involve mathematical modeling should be used as early as primary school, allowing them to face complex, real-life situations like this at an early age (English, 2006b). However, model and modeling in Turkeys new primary-school mathematics education program make reference to solid materials such as cubes, cones, algebra tiles, pattern blocks, fraction sets, and ten-based blocks to help students easily understand abstract mathematical concepts (MoNE, 2009). The only parts that emphasize higher-level mathematical thinking in the new program are project and performance assignments, which are rarely used effectively by teachers.Research studies conducted in primary schools have revealed that modeling activities enable students to express, test, revise, and change their thoughts several times (Eraslan, 2011a); improve the use of mathematical language, the ability to work in groups, social interactions, reading data from tables, and successful dealings with graphics (Watters, English, & Mahoney, 2004); enhance meta-cognition and critical thinking skills (English & Watters, 2004); contribute to overcoming some of young childrens conceptual shortcomings (English & Watters, 2004); and discover the fundamental ideas and processes of problems, determine the priorities of basic elements according to interrelationships, and make mathematical calculations to transform qualitative data into quantitative (English, 2007). On the other hand, some students were observed to have difficulties interpreting and understanding a variety of representational formats of the presented data (English & Watters, 2004), converting data into different representational formats (English, 2012a), introducing created models systematically (English, 2003), and determining proper parameters (e. …