ERG SES D 11, Mathematics and Education
Geometry which is considered as one of the major concepts in mathematics and mathematics education (Clements & Battista, 1992; NCTM, 2000) involves three cognitive processes; construction, reasoning, and visualization (Duval, 1998). To comprehend these processes and the connections among them is necessary for being proficient in geometry (Duval, 1998). Throughout the development of geometry, it was stated that geometric constructions have an important role (Stupel, Oxman, & Sigler, 2014) and it is one of the earliest concepts of mathematics (Kuzle, 2013). Geometric construction was described as “a problem situation in which it is required that a desired figure be drawn with the aid of specified instruments (such as the straightedge and compasses) and using specific given data” (Albrecht, 1952, p.5). According to Lim (1997), geometric construction involves carrying out procedural steps to construct geometric entities such as perpendicular line, parallel line, and angle bisectors using geometrical instruments. Some researchers described geometrical constructions by mentioning the use of compass and straightedge (e.g., Axler & Ribet, 2005; Djorić & Janičić, 2004).
How students construct geometric figures is directly connected to the development of their geometric reasoning (Köse, Tanışlı, Erdoğan, & Ada, 2012). Thus, using different tools such as compass, straightedge, and dynamic geometry programs might cause some differences in students’ reasoning process in geometric constructions. For example, Geometer’s Sketchpad which is a dynamic geometry program involves tools in the construction menu such as ‘midpoint’, ‘intersection’, and ‘angle bisector’. If students want to find the bisector of a given angle in Geometer’s Sketchpad, they can select the angle and the tool angel bisector in the construction menu and then they automatically get the angle bisector. However, if students want to find the angle bisector via using compass and straightedge, they have to find create strategies, use basic construction rules, and think geometrically to find the appropriate steps for the construction.
According to Axler and Ribet (2005), to understand Euclidean geometry, it is necessary to know about the nature and scope of compass and straightedge constructions. While studying about constructions, students develop their process of logical thinking, make conclusions, and reach conjectures about the given constructions (Djorić & Janičić, 2004). Moreover, they are leaded to participate in a deeper thinking process and developed their knowledge of geometry (Stupel et al., 2014). Since geometric construction problems require students to suggest accurate conclusions with an appropriate language and constructive consideration, they can be used in the present education system and constitute an important field for training students (Djorić & Janičić, 2004). In this manner, prospective middle school mathematics teachers should know about geometric constructions via using tools since they will be mathematics teachers in the future. Moreover, although studies regarding reasoning and visualization components of geometry (Duval, 1998) were seen in the literature, the number of studies related to construction particularly with compass and straightedge is limited (Karakuş, 2014). In this respect, the purposes of the study are to investigate prospective middle school mathematics teachers’ competencies in using compass and straightedge and the difficulties that they encountered during construction process based on collective argumentation. In the current research collective argumentation refers to “multiple people working together to establish a claim” (Conner, Singletary, Smith, Wagner, & Francisco, 2014, p.184). Based on these purposes, research questions are stated as follows.
1. To what extent are prospective middle school mathematics teachers successful in geometric constructions via using compass and straightedge?
2. What are the difficulties that prospective middle school mathematics encountered during geometric constructions via using compass and straightedge?
To address research questions, case study was utilized in the present study. According to Creswell (2007), in case study, researchers explore a case or cases over time by collecting in-depth data via various sources of information such as interviews, observations, and documents. The purposive sampling in which researchers select potential participants to supply data (Fraenkel, & Wallen, 2005) was used to determine the participants. Since the present study requires conducting deep investigation of the construction process of participants, the accessibility of them is an important issue. By accepting this as a criterion, prospective middle school mathematics teachers in a state university in Ankara were selected as the participants. Moreover, it was expected that prospective teachers give more time and effort for geometric constructions when they are asked within the context of an undergraduate course. Then, it was decided to collect data in a geometry related elective course. Thus, seven junior (3rd year) prospective middle school teachers who took the elective course are selected as the participants. Geometric constructions explained in the book of Alexander and Koeberlein (2011) were used to collect data. Specifically, eight geometric construction used in the present study were stated as follows: construction of a line segment congruent to a given line segment, construction of an angle congruent to a given angle, construction of the midpoint of a line segment, construction of angle bisector of a given angle, construction of perpendicular line to a given line at a point on that line, construction of perpendicular line to a given line form a point not on the given line, construction of parallel line to a given line, and construction of tangent to a circle at a point on the circle. In the first week of the elective course, how compass and straightedge are used was explained. Then, prospective teachers were asked to do constructions via using compass and straightedge by themselves. After a period, they were encouraged to share their ideas with others. At the end of each construction, valid construction methods were aimed to reach through a collective argumentation process. Video recordings, documents, and field notes were used as data sources. Firstly, prospective teachers’ documents and their construction process in the classroom discussion were analyzed to answer the first research question. Then, their difficulties in the construction process were analyzed by creating codes and themes in order to answer the second research question.
According to the findings, prospective teachers were able to suggest at least one valid method for each construction at the end of the classroom discussion. In some constructions, they also suggested more than one method. This finding is inconsistent with the study of Öçal and Şimşek (2017) which stated that in-service mathematics teachers were not successful in basic geometric constructions. The reason of this situation might be related to classroom environment in the present study. In the classroom discussion, prospective teachers explained their ideas and decided together whether the suggested method provides a valid construction or not. Since they discuss about possible methods throughout the constructions, they had opportunity to use the new idea in the following constructions. On the other hand, the difficulties students encountered in geometric construction could be categorized as not using compass and straightedge properly, not noticing the difference between drawing and construction, not using notation in geometry, not explaining how they constructed step by step, getting confused in the construction process because of unnecessary drawings, not remembering the geometry concepts, and using more complex constructions or ideas to do basic constructions. Some of these difficulties were also mentioned in the study of Karakuş (2014). When prospective teachers use the tools inappropriately, it might lead them to misinterpret their actions in construction process. Moreover, it was seen that prospective teachers hesitated in deciding whether they conducted a valid construction or not. Since the terms drawing and construction in geometry have different meanings (Albrecht, 1952), they might be confused with these terms. Since prospective teachers reached various methods in some constructions via using compass-straightedge through collective argumentation process, a similar setting might be useful for middle school students while studying about construction. In further studies, follow-up interviews with prospective teachers might be conducted to reach in-depth explorations.
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