Students’ Geometric Thinking 8
In the last 20 years, the perception of learning as internalization of knowledge is criticized and problemized in mathematics education society (Lave & Wenger, 1991; Sfard, 2000; Forman & Ansell, 2001). Lave and Wenger (1991) describe learning as a process of “increasing participation in communities of practices” (p.49). Sfard (2000) also emphasized the new understanding of learning as “Today, rather than speaking about “acquisition of knowledge,” many people prefer to view learning as becoming a participant in a certain discourse” (p.160).
This new perspective in the understanding of learning brings different views to mathematics teaching practice. While the structure of mathematics lessons is organized in the sequence of Initiation- Response-Evaluation (IRE) in the traditional mathematics classrooms, with the reform movement, participation of the students become the centre of the mathematics classrooms (O’ Connor, 1993; Steele, 2001). Initiating topic or problems, starting or enhancing discussions, providing explanations are the role of the teacher in the traditional classrooms but these roles become a part of students’ responsibilities in the reform mathematics classrooms (Forman, 1996).
Turkey also tries to organize their mathematics curriculum according to these reform movements. With the new elementary mathematics curriculum, in addition to developing mathematical concepts, the goal of mathematics education is defined as enhancing students’ problem solving, communication and reasoning abilities. Doing mathematics is no more defined only as remembering basic mathematical facts and rules and following procedures, it also described as solving problems, discussing the ideas and solution strategies, explaining and defending own views, and relating mathematical concepts with other mathematical concepts and disciplines (MEB, 2006).
Parallel to new understanding of learning, reform movements in mathematics education, and new Turkish elementary mathematics curriculum, students’ roles such as developing alternative solution strategies and sharing and discussing these strategies gain great importance in mathematics education. Mathematics teachers are advised to create classroom discourse in which students will be encouraged to use different approaches for solving problems and to justify their thinking. This means that some researches and new mathematics curriculum give so much importance to encourage students to develop alternative problem solving strategies and share them with others. (MEB, 2006; Carpenter, Fennema, Franke, Levi & Empson, 1999; Reid, 1995).
One of the aims of the new mathematics curriculum is that the students stated their mathematical thinking and their implications during the mathematical problem solving process (MEB, 2006). According to new curriculum, the students should have opportunity to solve the problems using different strategies and to explain their thinking related to problem solving to their friends and teacher. Moreover, the students’ should state their own mathematical thinking and implications during the problem solving process and they should develop problem solving strategies in mathematics classrooms (MEB, 2006). Fraivillig, Murphy and Fuson (1999) reported that creating this kind of classrooms requires that teacher has knowledge about students’ mathematical thinking.
One of the most important studies related to children’s mathematical thinking is Cognitively Guided Instruction (CGI). The aim of this study is to help the teachers organize and expand their understanding of children’s thinking and to explore how to use this knowledge to make instructional decisions such as choice of problems, questions to ask children to acquire their understanding. The study was conducted from kindergarten through 3rd grade students. At the beginning of the study, researchers tried to explore students’ problem solving strategies related to content domains addition, subtraction, multiplication and division. The findings from this investigation is that students solve the problems by using direct modeling strategies, counting strategies derived facts strategy and invented algorithms. In order to share their findings with teachers, they conducted workshops. With these workshops, the teachers realized that the students are able to solve the problems using a variety of strategies. After this realization, they started to listen to their students mathematical explanations, tried to elicit those strategies by asking questions, tried to understand children’s thinking and encouraged the use of multiple strategies to solve the problems in their classrooms (Franke, & Kazemi, 2001, Fennema, Carpenter, & Franke, 1992). At the end of the study, the students whose teachers encourage them to solve the questions with different strategies and spend more time for discussing these solutions showed higher performance (Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996).
Similar finding is also observed the study of Hiebert and Wearne (1993). They concluded that when the students solve few problems, spend more time for each problem and explain their alternative solution strategies, they get higher performance. As indicated the new curriculum in Turkey (MEB,2006), the teacher should create a classroom in which students develop different problem solving strategies, share these with their classmates and their teacher and set a high value on different problem solving strategies during the problem solving process. Encouraging the students to solve the problems is important since while they are solving the problems, they have a chance to overview their own understanding and they take notice of their lack of understandings or misunderstandings (Chi & Bassock, 1989, as cited in Webb, Nemer & Ing, 2006). Moreover, Forman and Ansell (2001) stated that if the students develop their own problem solving strategies, their self confidence will be increase and they can build their mathematical informal knowledge.
Not only mathematical thinking, but also geometrical thinking has very crucial role for developing mathematical thinking since National Council of Teachers of Mathematics in USA (2000) stated that “geometry offers an aspect of mathematical thinking that is different from, but connected to, the world of numbers” (p.97). While students are engaging in shapes, structures and transformations, they understand geometry and also mathematics since these concepts also help them improve their number skills.
There are some studies which dealt with children’s thinking but a few of them examine children’s geometrical thinking especially two dimensional and three dimensional geometry. One of the most important studies related to geometrical thinking is van Hiele Theory. The theory categorizes children’s geometrical thinking in a hierarchical structure and there are five hierarchical levels (van Hiele, 1986). According to these levels, initially students recognize the shapes as a whole (Level 0), then they discover the properties of figures and recognize the relationship between the figures and their properties (level 1 and 2). Lastly the students differentiate axioms, definitions and theorems and they prove the theorems (level 3 and 4) (Fuys, Geddes, & Tischler, 1988).
Besides, there are some other studies which examined geometrical thinking in different point of view. For example, the study of Ng (1998) is related to students’ understanding in area and volume at grade 4 and 5. But, Battista and Clements (1996) and Ben-Chaim (1985) investigated students’ geometric thinking by describing students’ solution strategies and errors in 3-D cube arrays at grades 3, 4 and 5. On the other hand, Chang (1992) carried out a study to understand spatial and geometric reasoning abilities of college students. Besides of these studies, Seçil (2000), Olkun (2001), Olkun, Toluk (2004), Özbellek (2003) and Okur (2006) have been conducted studies in Turkey. Generally, the studies are about students’ geometric problem solving strategies (Seçil, 2000), the reason of failure in geometry and ways of solution (Okur, 2006), the misconceptions and missing understandings of the students related to the subject angles at grade 6 and 7 (Özbellek, 2003). In addition to these, studies has been done to investigate the difficulties of students related to calculating the volume of solids which are formed by the unit cubes (Olkun, 2001), number and geometry concepts and the effects of using materials on students’ geometric thinking (Olkun & Toluk, 2004).
When the studies are examined which has been done in Turkey, the number of studies related to spatial ability is limited. Spatial ability is described as “the ability to perceive the essential relationships among the elements of a given visual situation and the ability to mentally manipulate one or two elements and is logically related to learning geometry” (as cited in Moses, 1977, p.18). Some researchers claimed that it has an important role for mathematics education since spatial skills contribute an important way to the learning of mathematics (Fennema & Sherman, 1978; Smith, 1964) and Anderson (2000) claimed that mathematical thinking or mathematical ability is strongly related with spatial ability. On the other hand, Moses (1977) and Battista (1990) found that geometric problem solving and achievement are positively correlated with spatial ability. So, developing students’ spatial ability will have benefit to improve students’ geometrical and also mathematical thinking and it may foster students’ interest in mathematics.
Since spatial ability and geometric thinking are basis of mathematics achievement, then one of the problems for researchers may be to investigate students’ geometric thinking (NCTM, 2000; Anderson, 2000; Fennema& Sherman, 1978; Smith, 1964). For this reason, generally this study will focus on students’ geometrical thinking. Particularly, it deals with how students think in three-dimensional and two-dimensional geometry, their solution strategies in order to solve three-dimensional and two-dimensional geometry problems, the difficulties which they confront with while they are solving them and the misconceptions related to geometry. Also, whether or not the students use their mathematics knowledge or daily life experiences while solving geometry questions are the main questions for this study.
The purpose of this study is to assess and describe students’ geometric thinking. Particularly, its purpose is to explain how the students approach to three-dimensional geometry, how they solve the questions related to three-dimensional geometry, what kind of solution strategies they develop, and what kind of difficulties they are confronted with when they are solving three-dimensional geometry problems. Also, the other purpose is to analyze how students associate their mathematics knowledge and daily life experience with geometry.
The study attempt to answer the following questions:
- How do 4th, 5th, 6th, 7th and 8th grade elementary students’ solve the questions related to three-dimensional geometry problems?
- What kind of solution strategies do 4th, 5th, 6th, 7th and 8th elementary students develop in order to solve three-dimensional geometry problems?
- What kind of difficulties do 4th, 5th, 6th, 7th and 8th elementary students face with while they are solving three-dimensional geometry problems?
- How do 4th, 5th, 6th, 7th and 8th elementary students associate their mathematics knowledge and daily life experience with geometry problems?
Most of the countries have changed their educational program in order to make learning be more meaningful (NCTM, 2000; MEB, 2006). The development of Turkish curriculum from 2003 to up till now can be assessed the part of the international educational reform. Particularly, the aim of the changes in elementary mathematics education is to make the students give meaning to learning by concretizing in their mind and to make the learning be more meaningful (MEB, 2006). In order to make learning more meaningful, knowing how the students think is critically important. For this reason, this study will investigate students’ mathematical thinking especially geometrical thinking since geometry provides opportunity to encourage students’ mathematical thinking (NCTM,2006).
The result of the international exams such as Trends in International Mathematics and Science Study (TIMSS) and the Programme for International Student Assessment (PISA) and national exams Secondary School Entrance Exam “OrtaöÄŸretim KurumlarÄ± ÖÄŸrenci Seçme SÄ±navÄ± (OKS)” show that the success of Turkish students’ in mathematics and especially in geometry is too low. Ministry of National Education in Turkey stated that although international average is 487 at TIMSS-1999, Turkish students’ mathematics average is 429. Moreover, they are 31st country among 38 countries. When the sub topics are analyzed, geometry has least average (EARGED, 2003). The similar result can be seen the Programme for International Student Assessment (PISA). According to result of PISA-2003, Turkish students are 28th county among 40 countries and Turkish students’ mathematics average is 423 but the international average is 489. When geometry average is considered, it is not different from the result of TIMSS-1999 since international geometry average is 486 but the average of Turkey is 417 ((EARGED, 2005). As it can be realized from result of both TIMSS-1999 and PISA-2003, Turkish students’ average is significantly lower than the international average.
Since in order to get higher mathematical performance, being aware of children’s mathematical thinking has crucial role (Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996). For this reason, knowing students’ geometric thinking, their solution strategies and their difficulties related to geometry problems will help to explore some of the reasons of Turkish students’ low geometry performance in international assessment, Trends in International Mathematics and Science Study (TIMSS) and Program for International Student Assessment (PISA), and in national assessment, Secondary School Entrance Exam “OrtaöÄŸretim KurumlarÄ± ÖÄŸrenci Seçme SÄ±navÄ± (OKS).”
As a result, when geometry and being aware of students’ problems solving strategies and their difficulties when they are solving geometry problems has important roles on mathematics achievement are taken into consideration, studies related to geometry and students’ geometric thinking are needed. Besides, Turkish students’ performance in international assessments is considered; it is not difficult to realize that there should be more studies related to geometry. For these reasons, the study will assist in Turkish education literature.
Significance of the Study
Teachers’ knowledge about children’s mathematical thinking effect their instructional method. They teach the subjects in the way of children’s thinking and they encourage students to think over the problems and to develop solution strategies. With such instructional method, classes are more successful (Fennema, Carpenter, & Franke, 1992).
Geometry is one of the sub topic of mathematics (MEB,2006) and it has crucial role in representing and solving problems in other sub topics of mathematics. Besides, geometry has important contribution to develop children’s mathematical thinking. On the other hand, in order to understand geometry, spatial ability is useful tool (NCTM, 2000). Battista et al.(1998), Fennema and Tartre (1985) and Moses (1977) emphasized that there is a relationship between spatial ability and achievement in geometry. Moreover, mathematical thinking and mathematical ability is positively correlated with spatial thinking (Anderson 2000). Since geometry, spatial ability and mathematical thinking are positively correlated, being successful in geometry will get higher mathematics achievement. To increase geometry achievement, the teachers should know students’ geometric thinking. Particularly, how students solve problems, what kind of strategies they develop, and what kind of difficulties they face with while they are solving the problems are important concepts in order to get idea about students’ thinking (Fennema, Carpenter, & Franke, 1992). With this study, the teachers will be informed how children think while they are solving geometry problems especially three-dimensional geometry problems, what kind of strategies they develop to solve them, what kind of difficulties they face with related to geometry problems. Furthermore, university instructors will benefit from this study to have knowledge about children’s geometric thinking and this knowledge may be valuable for them. Since they may inform pre-service teachers about children’s thinking and the importance of knowing children’s thinking while making instructional decisions.
As a result, knowing students’ geometric thinking will benefit to increase their geometry achievement and also mathematical achievement, and consequently, this will help to raise the Turkish students’ success of the international exams
Geometry can be considered as the part of mathematics and it provides opportunities to encourage students’ mathematical thinking. Also, geometry offers students an aspect of mathematical thinking since when students engage in geometry, they become familiar with shape, location and transformation, and they also understand other mathematics topics (NCTM, 2000). Therefore, understanding of students’ geometrical thinking, their geometry problem solving strategies and their difficulties in geometry become the base for their mathematical thinking. Also, since geometry is “a science of space as well as logical structure”, to understand students’ geometrical thinking requires knowledge of spatial ability and cognitive ability (NCTM, 1989, p.48).
This chapter deals with some of the literature in four areas related to the problem of this study. The first section of this chapter is related to the van Hiele theory since van Hiele theory explains the level of children’s geometrical thinking (van Hiele, 1986). The second section of this chapter deals with the research studies related to students’ mathematical and geometrical thinking. The third section is devoted to research studies related to spatial ability. And the last section of this chapter reviews the research related to relationship between spatial ability and mathematics achievement.
Section 1: The van Hiele Theory
The van Hiele theory is related to children’s thinking especially their geometrical thinking since the theory categorizes children’s geometrical thinking in a hierarchical structure (van Hiele, 1986). According to theory of Pierre and Diana van Hiele, students learn the geometry subjects through levels of thought and they stated that the van Hiele Theory provided instructional direction to the learning and teaching of geometry. The van Hiele model has five hierarchical sequences. Van Hiele stated that each level has its own language because in each level, the connection of the terms, definitions, logic and symbol are different. The first level is visual level (level 0) (van Hiele, 1986). In this level, children recognize the figures according to their appearance. They might distinguish one figure to another but they do not consider the geometric properties of the figures. For instance, they do not consider the rectangle as a type of a parallelogram. The second level is descriptive level (level 1). In this level, students recognize the shapes by their properties. For instance, a student might think of a square which has four equal sides, four equal angles and equal diagonals. But they can not make relationships between these properties. For example, they can not grasp that equal diagonal can be deduced from equal sides and equal angles. The third level is theoretical level (level 3). The students can recognize the relationship between the figures and the properties. They discover properties of various shapes. For instance, some of the properties of the square satisfy the definition of the rectangle and they conclude that every square is a rectangle. The fourth level is formal logic level (level 4). The students realize the differences between axioms, definitions and theorems. Also, they prove the theorems and make relationships between the theorems. The fifth level is rigor level (level 4). In this level, students establish the theorems in different postulation systems (Fuys, Geddes, & Tischler, 1988).
As a result, the levels give information about students’ geometric thinking to the researchers and mathematics teachers. Mathematics teachers may guess whether the geometry problem will be solved by students or not and at which grade they will solve them.
Section 2: Children’ thinking
The van Hiele theory explains the students’ thinking level in geometry. The levels are important but how students think is as important as their thinking level. To ascertain how students think related to mathematics and especially geometry, a number of studies have been conducted (Carpenter, Fennema, & Franke, 1996; Chang, 1992; Battista, & Clements, 1995; Özbellek, 2003; Olkun, 2005; Ng, 1998; Okur, 2006). Some of these studies are related to mathematical thinking and some of them geometrical thinking. Carpenter et al. (1999) and Olkun (2005) studied children’s mathematical thinking and Chang (1992), Battista and Clements (1995), Ben-Chaim (1985), Olkun (2001), Özbellek (2003), Okur (2006) and Ng, (1998) carried out research studies related to children’s geometrical thinking.
An important study related to mathematical thinking has been conduct by Carpenter, Fennema and Franke initiated over 15 years ago in USA and the name of this study is Cognitively Guided Instruction (CGI) which is described as the teacher development program. Cognitively Guided Instruction sought to bring together research on the development of children’s mathematical thinking and research on teaching (Franke, & Kazemi, 2001). Carpenter, Fennema and Franke (1996) stated that Cognitively Guided Instruction (CGI) focuses on children’s understanding of specific mathematical concepts which provide a basis for teachers to develop their knowledge more broadly. The Cognitively Guided Instruction (CGI) Professional Development Program engages teachers in learning about the development of children’s mathematical thinking within particular content domains. (Carpenter, Fennema, Franke, Levi, & Empson, 1999). These content domains include investigation of children’s thinking at different problem situations that characterize addition, subtraction, multiplication and division (Fennema, Carpenter, & Franke, 1992). In order to understand how the children categorize the problems, Carpenter et al. (1992) conducted a study. According to this study, Fennema, Carpenter, and Franke (1996) portrayed how basic concepts of addition, subtraction, multiplication, and division develop in children and how they can construct concepts of place value and multidigit computational procedures based on their intuitive mathematical knowledge. At the end of this study, with the help of children’s actions and relations in the problem, for addition and subtraction, four basic classes of problems can be identified: Join Separate, Part-Part-Whole, and Compare and Carpenter et all. (1999) reported that according to these problem types, children develop different strategies to solve them. The similar study has been carried out by Olkun et al (2005) in Turkey. The purpose of these two studies is the same but the subjects and the grade level are different. Olkun et al (2005) studied with the students from kindergarten to 5th grade but the students who participated in Carpenter’s study is from kindergarten through 3rd grade (Fennema, Carpenter, & Franke, 1992). Furthermore, CGI is related to concepts addition, subtraction, multiplication and division but the content of the study done in Turkey is addition, multiplication, number and geometrical concepts (Olkun et al, 2005). Although the grade level and the subjects were different, for the same subjects, addition and multiplication, the solution strategies of the students in Olkun’s study are almost the same as the students in CGI. But the students in the study of Carpenter used wider variety of strategies than the students in Turkey even if they are smaller than the students who participated in Olkun’s study. This means that grade level or age is not important for developing problem solving strategies.
On the other hand, there are some studies related to children’s geometrical thinking which are interested in different side of geometrical thinking.
Ng (1998) had conducted a study related to students’ understanding in area and volume. There were seven participants at grade 4 and 5. For the study, she interviewed with all participants one by one and she presented her dialogues with students while they are solving the questions. She reported that students who participated in the study voluntarily have different understanding level for the concepts of area, and volume. She explained that when students pass from one level to another, 4th grade to 5th grade, their thinking becomes more integrated. With regard to its methodology and its geometry questions, it is valuable for my study.
On the contrary to Ng, Chang (1992) chose his participants at different levels of thinking in three-dimensional geometry. These levels were determined by the Spatial Geometry test. According to this study, students at lower levels of thinking use more manipulative and less definitions and theorems to solve the problems than high level of thinking. On the other hand, the levels of two-dimensional geometry identified by the van Hiele theory. The results were the same as the three-dimensional geometry. In this case, Chang (1992) stated that the students at the lower levels of thinking request more apparatus and less definitions and theorems to solve the problems. Moreover, for both cases, the students at the higher levels of thinking want manipulative at the later times in the problem-solving process than the students at the lower level of students. The result of this study indicated that using manipulative require higher level of thinking. By providing necessary manipulative, I hope the students use higher level of thinking and solve the problems with different strategy.
Besides of these studies, Ben-Chaim et all. (1985) carried out the study to investigate errors in the three-dimensional geometry. They reported four types of errors on the problem related to determining the volume of the three-dimensional objects which are composed of the cubes. Particularly, they categorize these errors two major types which students made. These major types of errors defined as “dealing with two dimensional rather than three and not counting hidden cubes” (Ben-Chaim, 1985). The similar study was conducted by Olkun (2001). The aim of this study is to explain students’ difficulties which they faced with calculating the volume of the solids. He concluded that while students were finding the volume of the rectangular solids with the help of the unit cubes, most of the students were forced open to find the number of the unit cubes in the rectangular solids. Also, the students found the big prism complicated and they were forced open to give life to the organization of the prism which was formed by the unit cubes based on the column, line and layers in their mind, i.e. they got stuck on to imagine the prism readily. (Olkun, 2001). The categorization of students’ difficulties will be base for me to analyze difficulties related to geometry problems of the students who are participant of my study.
Besides of these studies, Battista and Clements (1996) conducted a study to understand students’ solution strategies and errors in the three-dimensional problems. The study of Battista and Clements (1996) was different from the study of Ben-Chaim (1985) and Olkun (2001) in some respect such as Battista and Clements categorized problem solving strategies but Ben-Chaim and Olkun defined students’ difficulties while reaching correct answer. Categorization of the students’ problem solving strategies in the study of Battista and Clements (1996) is like the following:
“Category A: The students conceptualized the set of cubes as a 3-D rectangular array organized into layers.
Category B: The students conceptualized the set of cubes as space filling, attempting to count all cubes in the interior and exterior.
Category C: The students conceptualized the set of cubes in terms of its faces; he or she counted all or a subset of the visible faces of cubes.
Category D: The students explicitly used the formula L x W x H, but with no indication that he or she understood the formula in terms of layers.
Category E: Other. This category includes strategies such as multiplying the number of squares on one face times the number on other face.” (Battista& Clements ,1996).
At another study of Battista and Clements (1998), their categorization was nearly the same but their names were different than the study which has done in 1996. In this study, they categorized the strategies as seeing buildings as unstructured sets of cubes, seeing buildings as unstructured sets of cubes, seeing buildings as space filling, seeing buildings in terms of layer and use of formula. Battista and Clements (1996, 1998) concluded that spatial structuring is basic concept to understand students’ strategies for calculating the volume of the objects which are formed by the cubes. Students should establish the units, establish relationships between units and comprehend the relationship as a subset of the objects. Actually, these studies are important for my study since they gave some ideas about different solutions for solving these problems. Also, different categorization of students’ geometry problems strategies will help me about how I can categorize students’ strategies. Also,
In addition to these studies, Seçil (2000), Olkun (2001), Olkun, Toluk (2004), Özbellek (2003) and Okur (2006) have been conducted studies in Turkey. Seçil (2000) has investigated students’ problem solving strategies in geometry and Okur (2006) have studied the reason of failure in geometry and ways of solution. In the study of Özbellek, the misconceptions and missing understandings of the students related to the subject angles at grade 6 and 7. Also, studies has been done to investigate the difficulties of students related to calculating the volume of solids which are formed by the unit cubes (Olkun, 2001) and the effects of using materials on students’ geometric thinking (Olkun & Toluk, 2004).
As a result, in order to understand children’ thinking, several studies has been conducted. Some of them were related to children’ mathematical thinking and some of them were interested in children’s geometrical thinking. These studies dealt with children’s thinking in different aspects and so their findings are not related to each other. But the common idea is that spatial ability and geometrical thinking are correlated positively. Since spatial reasoning is intellectual operation to construct an organization or form for objects and it has important role to for constructing students’ geometric knowledge (Battista, 1998).
Section 3: Spatial Ability
The USA National Council of Teachers of Mathematics (2000)explained that the spatial ability is useful tool to interpret, understand and appreciate our geometric world and it is logically related to mathematics (Fennema&Tartre, 1985). On the other hand, McGee (1979) describes spatial ability as “the ability to mentally manipulate, rotate, twist or invert a pictorially presented stimulus object”. Since spatial ability is important for children’s geometric thinking, the development of it has been investigated by several studies. First and foremost study has been carried by Pia