24 SES 09, Studies in Geometry
Three-dimensional polycubical shapes are shapes constructed from unit cubes and many middle school children have difficulty to understand how to represent them in two-dimensional environments (Fujita, Kondo, Kumakura & Kunimune, 2017; Pittalis & Christou, 2013; Widder & Gorsky, 2013). Misconceptions including overgeneralisation and overspecialisation (e.g., Ulusoy & Cakiroglu, 2017), rote learning or rule and cue following (e.g., Altun, 2013; Bingolbali & Ozmantar, 2012), and insufficient and inappropriate geometry instruction/teaching (e.g., Ding & Jones, 2006; Lipowsky et al., 2009) are claimed, by the researchers from various contexts, to be possible reasons affecting poor student-performance in learning three-dimensional shapes. For example, Lipowsky et al. (2009) have noted that the effects of teacher characteristics and geometry instruction on students’ learning become more important than they are assumed in the studies. Likewise, Bingolbali and Ozmantar (2012) have asserted that geometry teachers tend to tell students what to do from how to solve problems to what to note and praise them for following the rules set by them, and these are not encouraging for students to create new methods to solve or create new ideas. The present study has focussed on observing whether and how those reasons affect middle school students’ understanding of polycubical shapes in a specific public school context. We examined whether and how the factors in the literature affect the seventh grade middle school students’ learning of three-dimensional geometry. The research also tried to give an explanation to students’ and teachers’ approach to the topic with the support from classroom observations, and find context-specific factors which might not be investigated earlier. Thus, the research question answers “What are the context-specific factors affecting Turkish middle school students’ learning of two-dimensional representations of three-dimensional shapes constructed from unit cubes in middle school classes?”.
Two-dimensional representation of three-dimensional polycubical shapes is a part of many European curriculums including English, French, German, Polish and Turkish national mathematics curriculums. Many researchers reported that it is important for students to work with three-dimensional shapes and translate them into two-dimensional forms (e.g., Battista, 1998; Pittalis & Christou, 2013; Widder & Gorsky, 2013). For example, Battista et al. (1998) have argued that activities with three-dimensional shapes that support three-dimensional thinking are vital for students’ construction and use of mathematical concepts. It is also claimed that gaining three-dimensional thinking skills brings many advantages to middle school students including increasing the ability to visualise (Messner & Horman, 2003), spatial awareness and the specialization in mathematics and specifically in geometry (Uttal & Cohen, 2012).
Unfortunately, despite its prevalence and importance, students’ poor performance in this topic has been reported by many researchers for over 30 years (e.g., Battista et al., 1998; Nardi & Steward, 2003; Pittalis & Christou, 2013). To illustrate, according to Clements and Battista’s (1992) review, Usiskin (1987) reported that middle school students’ performance dealing with two and three-dimensional shapes was poor, and this prevents them taking geometry classes in high school as those classes are elective. Likewise, Brown et al. (1988) and Kouba et al. (1988) reported students’ low academic achievement in shapes and their properties, and their visualisation. Findings of more recent studies are barely different to those of the 1980s. Students’ difficulty in three-dimensional geometry, its causes and possible solutions are still the subjects of much active research (e.g. Altun, 2013). Moreover, similar to the research findings, some national and international test results also have brought to light that middle school students are not achieving goals of national and local programmes with regard to three-dimensional geometry (MoNE, 2016; OECD, 2017).
This study is the initial part of a design-based research. The aim was to observe four naturally occurring cases (teachers) to discover the problems related to students’ learning of three-dimensional polycubical shapes more detailly. The first phase of a design-based research can be thought as the analysis of the problem both in the naturally occurring place and in the literature (Bakker & van Eerde, 2014; Herrington, McKenney, Reeves, & Oliver, 2007). In the current case, it is important to pre-determine the possible factors which might affect students’ progression and the process of students’ learning. The next step will be designing geometry lessons according to the determined factors and therefore the needs of the students in order to improve their performance on two-dimensional representations of three-dimensional polycubical shapes. The study was conducted in two public middle schools where two mathematics teachers in each agreed to be a part of the study and invited the researchers to observe their students while they were teaching two-dimensional representations of three-dimensional polycubical shapes during their regular lessons. There were 25 to 30 middle school students, who were 12- to 13-years old, in each of four classrooms. Middle school students were observed for four lesson hours during their regular mathematics lessons when they were learning the topic. Each lesson observation was 40 minutes. 16 lessons of four classes, being 4 lessons in each classroom, were observed as a non-participant observer, sitting at the back side of the classroom and taking field notes in five-minute intervals. A sketch of each classroom was drawn prior to the field notes. The observation protocol had two parts (a descriptive part and a reflective part) and was used to structure the field notes. Copies of the materials (presentations, activity sheets and books) used by the teachers and the students in the class were also collected as additional data. The data were organised according to the emerging themes which were explained in the findings section as three major reasons for students’ poor performance in two-dimensional representations of polycubical shapes.
The findings show three major reasons for low student-performance in the observed context. The first reason for this is “passive learning”. It is observed that even mathematics teachers integrated concrete manipulatives (unit cubes and linking cubes) into their classes, they mostly choose to use those manipulatives themselves to illustrate polycubical shapes instead of offering students an opportunity construct polycubical shapes and explore the topic. Many researchers argue that hands-on activities with manipulatives might be more helpful for students’ learning of three-dimensional shapes (Fujita et al., 2017); thus, we planned to give students an opportunity to explore the topic with the help of linking cubes in the future lessons. The second reason for students’ poor performance appears to be the reliance on “technology as a drill rather than a learning tool”. The technology was used in all of the lessons in different ways. While some of the teachers showed videos from the Ministry of Education’s moodle in the end of the lessons to summarise what they have taught, others solved questions on the smart board, questions which were prepared by them on office tools prior to the lessons to repeat the subject they have already taught. The final underlying reason affecting students’ success has found to be “exam-focused instruction”. It was observed that almost all of the observed lessons included past ministry exam questions, and mathematics teachers emphasised the importance of the topic using those questions. We believe that giving real-life examples and relating the topic to the students’ daily practices might be more helpful for students than a sole emphasis on ministry exam questions. The future lesson plans we will design for the next iteration of this study will include such real-life examples in order to help students give more importance and value to learn the topic.
Altun, M. (2013). Ortaokullarda Matematik Ogretimi. Bursa: Aktuel. Bakker, A., & van Eerde, H. (2014). An introduction to design-based research with an example from statistics education. In Doing qualitative research: Methodology and methods in mathematics education (pp. 429–466). Berlin: Springer. Battista, M., Clements, D., Arnoff, J., Battista, K., & Borrow, C. (1998). Students’ Spatial Structuring of 2D Arrays of Squares. Journal for Research in Mathematics Education, 29(5), 503–532. Bingolbali, E., & Ozmantar, M. F. (Eds.). (2012). Ilkogretimde Karsilasilan Matematiksel Zorluklar ve Cozum Onerileri (3rd ed.). Ankara: Pegem Akademi. Clements, D., & Battista, M. (1992). Geometry and Spatial Reasoning. In Handbook of Research on Mathematics Teaching and Learning (pp. 420–464). New York: Macmillan. Ding, L., & Jones, K. (2006). Teaching Geometry in Lower Secondary School in Shangai, China. Proceedings of the British Society for Research into Learning Mathematics, 26(1), 41–46. Fujita T., Kondo Y., Kumamura H., & Kunimune S. (2017). Students’ geometric thinking with cube representations: Assessment framework and empirical evidence. Journal of Mathematical Behavior, 46, 96-111. Herrington, J., McKenney, S., Reeves, T., & Oliver, R. (2007). Design-based Research and doctoral students: Guidelines for preparing a dissertation proposal. In Ed-Media (pp. 4089–4097). Chesapeake: AACE. Lipowsky, F., Rakoczy, K., Pauli, C., Drollinger-Vetter, B., Klieme, E., & Reusser, K. (2009). Quality of geometry instruction and its short-term impact on students’ understanding of the Pythagorean Theorem. Learning and Instruction, 19(1), 527–537. Messner, J., & Horman, M. (2003). Using Advanced Visualization Tools to Improve Construction Education. In International Conference on Construction Applications of Virtual Reality. MoNE. (2016). PISA 2015 Turkish National Report. Ankara: MoNE. Nardi, E., & Steward, S. (2003). Is Mathematics T.I.R.E.D? A Profile of Quiet Disaffection in the Secondary Mathematics Classroom. British Educational Research Journal, 29(3), 345–366. OECD. (2017). Mathematics performance (PISA) (indicator). Pittalis, M., & Christou, C. (2013). Coding and decoding representations of 3D shapes. The Journal of Mathematical Behavior, 32(3), 673-689. Ulusoy, F., & Cakiroglu, E. (2017). Middle School Students’ Types of Identification for Parallelogram: Underspecification and Overgeneralization. AIBU Faculty of Education Journal, 17(1), 457–475. Uttal, D. & Cohen, C. (2012). Spatial Thinking and STEM Education: When, Why, and How? In Psychology of Learning and Motivation (pp. 147–181). Widder, M., & Gorsky, P. (2013). How students solve problems in spatial geometry while using a software application for visualizing 3d geometric objects. Journal of Computers in Mathematics and Science Teaching, 32(1), 89–120.
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