Algebra includes the relationships between quantities, the use of symbols, the modeling of phenomena, and the mathematical expression of change (Carraher, Martinez & Schliemann, 2008). In order to learn algebra by understanding its content, it is necessary to have algebraic thinking skills, which is one of the types of mathematical reasoning. Driscoll (1999, 2001) interpreted algebraic thinking as thinking about quantitative situations that support clarifying relationships between variables, based on Cuoco, Goldenberg, and Mark (1996)'s useful ways of thinking about mathematical content which they defined as habits of mind. Driscoll (1999) put forward a theoretical framework for the habits that students should acquire in order to develop algebraic thinking skills, by claiming that when the student learns symbols, they will take an important step in expressing generalizations, revealing algebraic structures, forming relationships, and formulating mathematical situations. Driscoll (1999) conceptualized habit of algebraic thought as Building Rules to Represent Functions and Abstracting from Computation as habits of mind, which are taken place the umbrella term of the Doing-Undoing.

Doing-Undoing:This algebraic habit of mind is an umbrella term for the other two habits. Students should be able to both conclude an operation related to algebra and reach the starting point by working backwards from the result of an operation which they found the result. Thanks to this mental habit, students not only focus on reaching the result, but also think about the process.

Building Rules to Represent Functions: This mental habit includes recognizing and analyzing patterns; investigating and representing relationships; making generalizations beyond specific examples; analyzing how processes or relationships have changed; and looking for evidence of how and why rules and procedures work (Magiera, van den Kieboom & Moyer, 2013). The sub-themes of this habit are; organizing information, predicting patterns, chucking the information, different representations, describing a rule, describing change, justifying a rule.

Abstracting from Computation:It is the capacity to think about calculations regardless of the numbers used. Abstraction is important for this habit of mind. Abstraction is the process of extracting mathematical objects and relations based on generalization (Lew, 2004). The sub-themes of this habit are; computational shortcuts, calculating without computing, generalizing beyond examples, equivalent expressions, symbolic expressions, justifying shortcuts.

Magiera et. al. (2013) investigated algebraic habits of mind 18 elementary school pre-service teacher with problems. Magiera et. al. (2017) examined pre-service teachers' habits of building rules to represent functions in the scope of the algebra problem. Strand & Mills (2017) stated that the studies in the literature examined pre-service teachers' algebraic thoughts within the scope of problem solving. Therefore, in the studies in the literature, it is seen that problems are used as a tool to examine the pre-service teachers' algebraic thoughts. Also, Kieran et al. (2016) mentioned the importance of problems in developing algebraic thinking. In this context, unlike other studies, this study will examine how pre-service teachers' algebraic habits of mind differ in well-structured and ill-structured problems. Simon (1973) stated that students' solutions to ill-structured problems differ from their solutions to well-structured problems. In addition, Webb & Mastergeorge (2003) examined the differences in the solution strategies of student groups solving ill-structured problems and well-structured problems. Kim & Cho (2016) examined how pre-service teachers' motivations affect their problem solving processes in ill-structured problems. In this study, it is aimed to compare the algebraic thinking styles used by pre-service elementary mathematics teachers in the process of solving a well-structured and ill-structured algebra problem. For this purpose, "What is the difference between the algebraic habits of mind that teacher candidates use in solving a well-structured and ill-structured algebra problem?" an answer to the research question was sought.

This study aims to compare the algebraic thinking styles used by pre-service elementary mathematics teachers in solving a well-structured and ill-structured algebra problem. The descriptive survey model was used as a method in the study. The sample of the research consists of 62 pre-service teachers who took the "Algebra Teaching" course in the elementary mathematics education teaching program of a state university in Turkey. As a data collection tool in the research, the researcher used algebra problem named "Crossing the River" which Driscoll proposed to reveal the algebraic thinking habits of the mind of pre-service teachers. Half of the sample group was presented with well-structured version of the problem and the other half with ill-structured version. In the study, the data were analyzed descriptively. In descriptive analysis, the data obtained are summarized and interpreted under predetermined themes, categories or codes (Robson, 2009). Such analyzes are made to describe profiles of people, events or situations. Descriptive studies require extensive prior knowledge of the situation or event described. In this context, well-structured problem and ill-structured problem were analyzed using the algebraic habits of mind framework in the study of Driscoll (1999). In this framework, “Doing-Undoing, Building Rules to Represent Functions and Abstracting from Computation” are categorized according to the characteristics of the algebraic habits of the mind.

It has been seen that the way that asking the question is effective in developing algebraic habits of mind for solving the question. Hence, in the well-structured algebra problem, predicting patterns, chucking the information, different representations, describing a rule, describing change, justifying a rule, equivalent expressions, symbolic expressions and calculating without computing habits have come to prominence. In the ill-structured algebra problem, organizing information, predicting patterns, chucking the information, different representations, describing a rule, describing change, justifying a rule, generalizing beyond examples, equivalent expressions and symbolic expressions have come to the fore. From this point of view, structuring the problem in line with the habit desired to be acquired by the student is effective in directing the student to use the expected habits or strategy at the end of the process. In this problem, starting from arithmetic, finding the desired result, that is, creating the rule, may be a suitable method for the initial stage. It is important to choose the appropriate problem for the transition to algebra. For example, in a classical arithmetic problem, the student asks, “How many times do 2 children and 8 adults cross the river?” can solve the problem by using only arithmetic without trying to generalize or create rules without going into a thinking process. You can find it here without seeing the rule or pattern. In the well-structured algebra problem, it has naturally become a necessity for the pre-service teacher to describing a rule while they try to calculate the number of trips of 8 adults and 2 children one by one. At this point, it is important to choose the problems that will enable the students, pre-service teachers or teachers evolve the algebraic thinking habits of the mind in the desired direction.

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