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In mathematics teaching, teachers need to know how students understand the mathematic concepts to help them improve their students' understanding (Schiffer, 2001; Steinberg, Empson, & Carpenter, 2004). This approach is based on listening to and learning from students (Crespo, 2000; Jacobs, Lamb, & Philipp, 2010). Therefore, teacher must make a decision based on student thinking. Identifying strategies that students might use in solving problems allowed teachers to interpret the aspects that students understand or not understand. This allowed teachers to pose problems that take into account students' thinking characteristics. According to the relevant role of mathematical thinking of students in teaching mathematics, some mathematics teacher programs set goals namely the development of teachers' ability to interpret students' mathematical thinking (Eisenhardt, Fisher, Schack, Tassell, & Thomas, 2011).  The research related to the interpretation of students’ mathematical thinking has not become a major study in mathematics education research in Indonesia. Interpretation of students' mathematical thinking is important as it becomes the basic for teachers or pre-service teachers to make informed learning decisions. This learning decision-making takes into account the understanding of each student so as to meet the individual needs of the students. Pre-service teachers need to understand students' thinking to be able to manage problem-solving situations in the classroom. Understanding student thinking can be seen from how they identify key aspects of mathematics. The development of the ability to interpret student thinking can enable teachers to make informed learning decisions, for example, the selection and design of mathematical tasks in solving problems (Chamberlin, 2005). The need for pre-service teachers to base their decisions on students' understanding underscores the importance of characterizing and understanding such skills (Hiebert, Morris, Berk, & Jansen, 2007). This justified the need to focus research attention on how pre-service teachers identify and interpret students' mathematical thingking in different domains (Hines & McMahon, 2005; Lobato, Orrill, Druken, & Jacobson, 2011). Based on the exposure, the researcher perceived the need to do research about the characteristics of teacher candidate interpretation about students' mathematical thinking in solving the problem. Therefore, this study aimed to describe the characteristics of the interpretation of a pre-service teacher about students' mathematical thinking in the domain of inverse proportion.

 

The subject of this research is the 3rd year student of Mathematics Education Program of Muhammadiyah University of Purworejo who can attend to student's strategies in solving Building Construction Problem (BCP). Researchers as the main instrument plan, execute, collect and analyze data, make conclusions and make the reports.  Researchers assigned tasks on the subject to complete the Task of Interpretation of Student Mathematical Thinking (ToIoSMT) by think aloud. The process was recorded with a handy cam-video recorder to obtain the data of visual and verbal description in interpretation.  From the implementation, the researcher obtained written data and think aloud data of the subject. Interview was conducted unstructured to obtain an explanation of what does not appear in subjects’ work or think aloud. Qualitative analysis was used to obtain interpretive characteristics for describing, comparing work, comparing knowledge, and inferring analysis models.

 

This study result the characteristics of interpretation for the four groups of analysis models it appears that these characteristics are relatively similar: 1) paying attention to operations performed or allegedly performed by students; 2) paying attention to the use of the concept of reverse comparison of values ​​as well as the mistakes made by students; and 3) assessing what students understand or do not understand directly or indirectly.  Interpretation of incomplete student work or not writing the details of the operation of inverse proportion the emphasis on the expected operation of the student as well as the assumption that the student understands inverse proportion. Interpretation of student work that uses reasoning comparisons of allegations about operations performed by students that students only operate information that is known only. Interpretation of the students' work that details the settlement of the BCP has paid much attention to the details of the settlement process. Besides giving attention to the concept used is the concept of inverse proportion.

interpretation, pre-service teacher, students’ mathematical thinking