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RESEARCH PRODUCT

An Alternative Paradigm to Evaluate the Acquisition of the van Hiele Levels

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This article presents an alternative way of analyzing the van Hiele level of students’ geometrical reasoning. We evaluate the students’ answers, taking into account the van Hiele level they reflect and their mathematical accuracy. This gives us a description of how accomplished the students are in applying the procedures associated with each of the van Hiele levels and allows us to determine the students’ degree of acquisition of the van Hiele levels. In this way we obtain a clearer picture of the students’ geometrical reasoning than with the traditional assignment of one van Hiele level to the learners. An example of the application of this method is provided: We describe a test that evaluated students’ ability to reason in three-dimensi onal geometry, some responses of students (9 eighth-grade pupils and 41 future primary school teachers), and the classification of their responses using our method. Approximately 25 years ago, the van Hieles proposed a model of the development of geometric thinking that identified five differentiated levels of thinking, ordered so that the students moved sequentially from one level of thinking to the next as their capability increased (van Hiele, 1957, 1986; van Hiele-Geldof, 1957). In the last 10 years there has been a growing interest in the van Hieles’ model of the development of geometric thinking (Fuys, Geddes, & Tischler, 1988; Gutierrez & Jaime, 1989; Hoffer, 1983; Senk, 1985). An important focus of research has been on ways to determine students’ level of thinking. The main goal of this article is to present an alternative method to evaluate the students’ van Hiele level of reasoning, thus offering a way of identifying those students who are in transition between levels. The way the student’s level of reasoning is ascertained plays an important role in research related to the van Hiele model. Most researchers have determined a student’s van Hiele level for a topic following assessment criteria based on the number of right answers to a written test (Gutierrez & Jaime, 1987; Mayberry, 1983; Usiskin, 1982) or on the thinking level shown by the student in each activity during an interview (Burger & Shaughnessy, 1986; Fuys et al., 1988). In both cases, the respective criteria have assigned each student to one van Hiele level. Although most students show a dominant level of thinking when answering open-ended questions, a large number of them clearly reflect in their answers the presence of other levels, and there are some students whose answers show two consecutive dominant levels of reasoning simultaneously (Usiskin, 1982; Burger & Shaughnessy, 1986; Fuys et al., 1988). Burger and Shaughnessy and Fuys et al. suggested that these students were in transition between two levels, but their approaches to the problem have been different. Burger and Shaughnessy sought a consensus in the evaluators’ opinions; Fuys et al. assigned a student to Level 1-2 to indicate that the student clearly used both Levels 1 and 2 of reasoning for an activity.