ERG SES G 09, Mathematics and Education
Tessellation is defined as “tiling of the plane with a repeating unit consisting one or more shapes through transformations so that there is no gap between the shapes and no overlapping of shapes” (Van de Walle, Karp, & Williams, 2007, p.422). Tessellation has important role in mathematics education since it helps students to develop understanding of mathematical concepts, properties of geometric shapes and connections between mathematical concepts. Tessellation activities in middle school also have many implications in terms of students’ engagements in geometry (Callingham, 2004). In addition to that, it also helps students to develop understanding of connection between mathematics and real world. For instance, students mostly face with tessellations in their daily lives such as walls used to surround their school and the tiles covering the floors (Ward, 2003).
Tessellations provide constructivist approach for learning geometry (Ward, 2003). Considering its importance, tessellations are included in mathematics education programs in many countries including Australia, Finland, Canada and Turkey. For instance, students create tessellations by translating, reflecting and rotating the given figure and make reasoning on which shapes would tessellate or not in Australian classrooms. In the same way, 4th, 5th and 6th grade students in Finland are provided with environments in which they work on tessellations.
When the literature is reviewed, it is seen that there are limited studies in the content of tessellations. One of these studies was conducted by Callingham (2004) regarding the primary students’ understanding of tessellations. She used van Hiele model as a basis for analysis and revealed that students are in the visualization, analysis and abstraction levels. In the same way, Kılıç, Köse, Tanışlı and Özdaş (2007) conducted a study with fifth graders and observed that students’ geometric thinking levels in tessellation can be categorized under visualization and analysis levels. Both of these studies were conducted to determine students’ geometric levels in the content of tessellations. In both studies, any intervention or activities were not implemented to enhance students’ thinking levels. Therefore, these studies do not suggest significant method to improve students’ geometric thinking levels.
In addition to the studies above, in the literature there are limited studies regarding teachers’ knowledge in the content of tessellations. Teachers have a crucial role in the classroom as a mediator between students and knowledge through designing the instruction in a way that students can be involved in learning mathematics (Thompson, 1992). For that reason, as being the future teachers, pre-service teachers should work and practice with tessellations to develop proficiencies in this subject in their teacher training course. In this sense, investigation of prospective teachers’ geometric thinking would give valuable information regarding their understanding on tessellation and reveal the points that they struggle with.
Considering this importance, this study aimed to investigate the effect of activity-based instruction on prospective teachers’ geometric thinking levels through van Hiele theory in the content of tessellations. The following research question is investigated in the current study.
What is the effect of activity-based instruction on prospective mathematics teachers’ geometric thinking levels in the concept of tessellations?
Current study is the first stage of a larger study that examined prospective middle school mathematics teachers’ geometric thinking levels through activity-based instruction. In this paper, we presented the effect of activity-based instruction on prospective mathematics teachers’ geometric thinking levels. In order to determine the effect of activity-based instruction, one-group pretest-posttest design in which measures or observations related to a single group are recorded before and after the treatment (Fraenkel & Wallen, 2006) was used. Data were collected from the prospective mathematics teachers in a public university in Ankara, Turkey. There was a classroom with 19 prospective teachers (18 females and 1 male) in which instruction and activities related to tessellations were implemented. In the selected classroom, participants took 4 hr activity-based instruction in each week. During the instructions, participants were involved in a constructivist learning environment so that they prepare activities and present them in the classroom environment and discuss the implementation of the related topic in the real classroom environment in the light of the middle school mathematics curriculum. Therefore, in the current study, the instruction was designed through enriched activities to teach tessellation concept and evaluate its effect on prospective teachers’ geometric thinking levels. Before the interventions started, prospective mathematics teachers were administered Tessellation Test (TT) developed by the researchers as pretest. After that, the intervention was implemented as lecture of the Geometric Thinking and Geometric Concepts unit in Van De Walle, Karp, & Bay-Williams’s (2013) book. In the second week, participants worked on tessellation activity so that they analyzed and constructed given tessellation with papers by using geometric transformations. In the third week, they used technology to apply geometric transformations to create both the unit and whole tessellation. After these three weeks, TT was administered as posttest in the fourth week. Students’ responses to the test was analyzed in the basis of five-levels of van Hiele theory of geometric thought (visualization, analysis, abstraction, deduction and rigor). In order to investigate the effect of activity-based instruction on prospective mathematics teachers’ geometric thinking levels in the concept of tessellations, quantitative analysis techniques were applied. Both descriptive and inferential statistics were conducted through SPSS program. Since the study was conducted with small sample size, to decide on the difference between thinking levels of prospective teachers in pretest and posttest, the Wilcoxon Signed Rank Test, nonparametric alternative to the repeated measures t-test (Pallant, 2001) was used.
The aim of the current study was to decide on the effect of activity-based instruction on prospective mathematics teachers’ geometric thinking levels in the concept of tessellations. According to the descriptive statistics, before the intervention, four participants were in Level 0, twelve of them were in Level 1, two of them were in Level 2 and one of them was in Level 3. On the other hand, after the intervention, it was seen that only one of the participants was in Level 0 while six of them were in Level 1, eleven of them were in Level 2 and one of them was in Level 3. Therefore, only one student stayed in level 0, while two students moved from level 0 to level 1, one moved from level 0 to level 2, four students in level 1 stayed in the same level, while 8 students moved from level 1 to level 2. Students in level 2 and 3 stayed in the same level after the instruction. Moreover, the result of Wilcoxon Signed Rank Test indicated a statistically significant increase in prospective mathematics teachers’ geometric thinking levels, z= –3.21, p< .001, with a large effect size (r= .52). The median level on the Tessellation Test increased from level 1 (Md= Level 1) to level 2 after the treatment (Md= Level 2). In conclusion, it can be deduced that the activity-based instruction was effective mostly for the ones who were in level 0 and 1. Moreover, prospective mathematics teachers were classified at most in level 2, although they are expected to handle these topics and have higher levels of thinking in geometry as being teachers of future. In this regard, teacher education programs can be revised considering the importance of tessellation in enhancing prospective teachers’ spatial abilities and efficiency in geometry.
Callingham, R. (2004). Primary Students’ Understanding of Tessellation: An Initial Exploration. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 183-190). Fraenkel, J. R., & Wallen, N. E. (2006). How to design and evaluate research in education (6th ed.). Boston: McGraw Hill. Kılıç, Ç., Köse, N. Y., Tanışlı, D., & Özdaş, A. (2007). Determining the Fifth Grade Students’ van Hiele Geometric Thinking Levels in Tessellation. Elementary Education Online, 6(1), 11-23. Pallant, J. (2001). SPSS Survival Manual: A Step by Step Guide to Data Analysis Using SPSS for Windows (Versions 12). Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. Handbook of Research on Mathematics Teaching and Learning, 127–146. Retrieved from http://psycnet.apa.org/psycinfo/1992-97586-007 Van de Walle, J. A., Karp, K. S., & Williams, J. M. B. (2013). Elementary and middle school mathematics. Teaching developmentally (8th ed.). Boston, MA: Pearson Education, Inc. Ward, R. A. (2003). Teaching tessellations to preservice teachers using TesselMania! Deluxe: A Vygotskian approach. Information Technology in Childhood Education Annual, 2003(1), 69-78.
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