The Van Hiele Theory Of Geometric Thinking

This chapter will provide a brief explanation of the theoretical framework on Van Hiele theory of geometric thinking. Consequently review and discuss on literature involving van Hiele theory and dynamic geometry software, follow by review of literature on teaching and learning of geometry by dynamic geometry software Cabri 3D as an instructional tool. Then chapter conclude by reviewing literature on designing learning activities.

The Van Hiele Theory of Geometric Thinking

The van Hiele model of geometric thinking is one theory that offers a model for explaining and describing geometric thinking. This theory resulted from the Dutch mathematics educator doctoral work of Dina van Hiele-Geldof and Pierre van Hiele at the University of Utrecht in the Netherlands which completed in 1957. Pierre van Hiele formulated the five levels of thinking in geometry and discussed the role of insight in the learning of geometry in this doctoral thesis. Van Hiele reformulated the original five levels into three during the 1980’s. Dina van Hiele-Geldorf’s doctoral thesis, which was completed in 1957, focused on the role of instruction in the raising of a pupil’s thought levels. Her study centered on thinking of geometry and the role of instruction in assessing pupils to move though the levels.

The following summary of Van Hiele theory history is taken from Hanscomb,Kerry, (2005, p.77):

A convenient location for many primary sources on the Van Hiele model is Fuys et al. (1984).Other primary sources are van Hiele and Van Hiele-Geldof (1958) and Van Hiele (1986). Secondary sources for Van Hiele research are Mayberry (1983), who found that students may operate at different levels for different concepts; Mayson (1997),who claims that gifted students may skip van Hiele levels; and Clements and Battista (1992),who cite finding indicating that the van Hiele levels involve cognitive developmental factors as well as didactical factors.

The van Hiele theory has been applied to clarify students’ difficulties with the higher order cognitive processes, which is necessary to success in high school geometry. In this theory if students do not taught at the proper Hiele level that they are at or ready for it, will face difficulties and they cannot understand geometry. The therapy that offered for students by this theory is that they should go through the sequence of levels in a specific way. (Usiskin, 1982b). It is possible to generalize the Van Hiele model to the other topics such as physics, science and arts. Because the main idea of this theory is the consequence of levels and believing that each level is built on properties of the previous level as many researches has done based on this theory on science education.

Characteristics of Van Hiele level of geometric thought

Van Hiele theory argues there are some misconstructions in teaching of school mathematics and geometry, which was existed for long time based on the formal axiomatic geometry and was created by Euclid more than two thousand years ago. Euclid logical construction is based on his axioms, definitions, theorems, and proofs. Therefore, the school geometry that is in a similar axiomatic fashion assumes that students think in a formal deductive level. However, it is not usually the case and the students have the lack of prerequisite understanding about geometry. Van Hiele discusses this lack creates a gap between their level of geometric thinking that they are, and the level of geometric thinking that they required for and they expected to learn. He supports Piaget’s points of view “Giving no education is better than giving it at the wrong time”. Teachers should provide teaching that is appropriate to the level of children’s thinking. Van Hiele theory suggests: It depends on the students’ level of geometric thinking the teacher can decide in which level the teaching should be begun.(Van Hiele, 1999)

According to the van Hiele theory, a student moves sequentially from the initial level (Visualization) to the highest level (Rigor). Students cannot achieve one level of thinking successfully without having passed through the previous levels. Furthermore, Burger &Shaughnessy (1986) and Mayberry (1983) have found that the level of thinking at an entry level is not the same in all areas of geometry.

During last decades many researchers and investigators tried to support the Van Hiele model or disapprove of it and still some try to improve or adjust this model. Many of the researcher used Van Hiele level of geometric thought as a suitable and proper theory in their research using dynamic geometry software (Smart, 2008).

The Van Hiele levels have certain properties specially for understanding the geometry. First of all, the stages have fixed sequence property. The five levels are hieratically, it means students must go through the levels in order. He/she cannot fit in level N without having gone through the previous level (N-1). Students cannot engage in geometry thinking at higher level without passing the lower levels.

Second property is adjacency of the levels. At each level of thought what is essential in the previous level become extrinsic in the existing level. Individual understanding and reflection on geometric ideas are needed to move from one level to the next one, rather than biological maturation.

Third each level has its own symbols and linguistic and relationships for connecting those symbols. This property is a distinction of the stages. For example when a teacher use a language for higher level of thinking than students level of thinking, students cannot understand the concepts and try to just memorizing the proofs and do the rote learning. In this case miscommunication emerge (Hong Lay, 2005).

The next characteristic, clarifies two persons in different levels cannot understand each other. As each level of thinking has its own language and symbols so students in different levels cannot understand each other.

Lastly, the Van Hiele theory emphasize on pedagogy and the importance of teacher instruction to assist students’ transition through one level to the next one. This characteristic indicates that appropriate activities which allow students to explore and discover geometric concepts in appropriate levels of their thinking are the best activities to advance students ‘level of thinking.

Phases of learning geometry

Van Hiele theory defines five levels of learning geometry which students must pass in order to obtain an understanding of geometric concept. To progress from one level to next level should be involve these five levels as Usiskin argued:

“The learning process leading to complete understanding at the next higher level has five phases, approximately but not strictly sequential, entitled:

Inquiry

Directed orientation

Explanation

Free orientation

Integration” (p.6)(Usiskin, 1982a).

These five level are very valuable in designing activities and design instructional phases.

Phase one: Inquiry

First phase of learning geometry starts with inquiry or information satge. In this stage students learn about the nature of the geometric objects.in order to design appropriate activities, Teacher identify students’ prior knowledge about new concept which need to be learnt. Then teacher design proper activities to encourage and encounter students with the new concept which is being taught.

Phase two: Directed orientation

During this phase while students doing their short activities with set of outcomes like: measuring, folding and unfolding, or geometry games, teacher provides appropriate activities base on students’ levels level of thinking to encourage them be more familiar with the concept being taught.

Phase three: Explanation

As the name of this phase demonstrates, in this stage students try to describe their learning of new concept in their own words. Students in this phase start to express their conclusions and finding with their other classmates and teacher in their own words. They communicate mathematically. The role of teacher in this stage is supplying relevant mathematical terminology and language in a proper manner, by using geometrical and mathematical language accurately and correctly.

Phase four: Free orientation

In this phase geometrical tasks that appeal to numerous ways is presented to the students. This is the students who decide how to go about accomplishing these tasks. As the way of solid geometry, they have learned to investigate more complex open-ended activities.

Phase Five: Integration

In this stage students summarize completed tasks and overview whatever they have learned to develop a new network of concepts. By completing this stage it is expected that students attained a new level of geometric thought.

One of important properties of these phases of learning in Van Hiele theory is not linear in nature. Sometimes students need a cycle form of these phases by repeating more than one time to overcome certain geometrical concepts. The role of teaches here is providing suitable activities based on these five phases to develop each level of van Hiele geometric thinking.

The Van Hiele level of geometric thinking

According to Van Hiele theory, the development of student’s geometric thinking considered regarding the increasingly sophisticated level of thinking. These levels are hierarchies and able to predict future students’ enactment in geometry(Usiskin, 1982a). This model consists of five levels in understanding, which numbered from 0 to 4. However, in this research we defined these levels from 1 to 5 to be able categorize students, who are not fitted in the model as level 0.

Level 1, Visualization

Level 2, Analysis

Level 3, Informal deduction

Level 4, deduction

Level 5, rigor

Level 1: Visualization

The base stage of Van Hiele geometric thinking which is encountered with goals of mathematical domain is Level 1. The objectives of the first level are functions like the underpinning elements of everything that are going to be studied.

Understanding at this stage includes visualizing base objects. At this level visualization defines as comprehension or seeing initial objects in students’ minds. For instance, a number line in this stage could be defined as real numbers in the domain of real numbers. Vectors and matrices can be seen as basic objects in the domain of leaner algebra. So perceiving vector as a directed segment or matrices as a rectangular table of numbers lies in level 1.

Elementary teachers know that it takes a few years of school for pupils to master visualization level. For example, it takes long time for students to see real numbers in a number line format. Similarly, perception of an ordered list or array of numbers, or an ordered pair of points is not something that occurs to an untaught mind and eye. Hence, serious teaching effort and introduction needed to students achieve Level 1and it is not assumed the visualization of initial objects to be obvious or trivial for students.

Geometry in Iran starts in elementary school and continues until level 8 with introducing geometry shapes like circles, squares, triangles, straight lines, etc. At the level 1 student learn to recognize geometric characteristics in objects that can be physically seen. At this stage student are assumed to be able to categorize geometric shapes by visual recognition, and know their names, for example, in solid geometry in level 1, if shown a picture of a polyhedron like a cube, students would be able to say that it is a cube because it looks like one for him or her. At this stage, it is not required to think of a cube, or any other geometric object, in terms of its properties, like saying a cube has 6 faces and 12 edges.

With visual recognition a student would be able to make a copy, by drawing, plotting or using some sort of dynamic geometry software, of a shape or configuration of shapes if they could be shown or told what it is they were supposed to be copying. In this stage, the instruction should be based on the name the student has memorized for the object and not the object’s properties. For instance, it could be “draw a cube” not draw a “polygon with 12 equal edges that are perpendicular to the base and 6 equal faces.

Level 2: Analysis Stage

At analysis stage, students begin to analysis objects that were only visually perceived at pervious level, identifying their parts and relations among these parts. They focus on the properties of these objects. For example, focus on Real Numbers in this stage can be closure under operations. This property can be leading to distinguishing subsets of Real Numbers inside the set which are Integers and Rational Numbers.

In solid geometry, the analysis stage is where students begin seeing the properties associated with the different shapes or configurations. A cube will now become a shape with 6 equal faces which opposite faces are parallel and 12 edges and adjacent angles right angles and having opposite faces equal, as well as having the diagonals intersect in their middle. However, at this stage, it is not assumed that students will be seeking logical relationships between properties such as knowing that it is enough for a Parallelepiped as a solid with parallel opposite faces and all the other properties follow. Neither is it assumed that students will think about a cuboid as a special type of Parallelepiped. Therefore, students will identify shapes and solids based on the wholeness of their properties. In other words, relationships between shapes and configurations remain merely on the list of properties they have.

At this stage if a student were asked to describe a shape or solid, the description would be based on the object’s properties. At the same time, if a student were asked to reproduce a shape or solid based on the list of properties, they would be capable of do so. Students would also be able to verify figures and solids hieratically by analyzing their properties. In this stage student can recognize the interrelation between figures and their properties. For example, knowing the property that the Parallelepiped the student would be able to deduce that cuboid is special kind of Parallelepiped.

Level 3: Informal Deduction Stage

Informal deduction is known as the third level of geometric thinking. Some of researchers name this level as abstract/Relation level too(Battista, 1999; Cabral, 2004). In this stage students can reason logically. This stage is achieved when a student can operate with the relation of figures and solids and is able to apply congruence of geometric figures to prove certain properties of a total geometric configuration of which congruent figures are a part. They become aware about sufficient and necessary condition for a concept. A student fit at this level after achieving pervious levels (visualization and analysis).

At this level more attention given to relations among properties. In other words, in this stage focus is “properties of sets of properties”. In this level according to relationship between properties of objects students attempt to group these properties into subgroups. Students try to find out what are the minimum of properties that needed to describe of the initial base elements. They intend to categorize properties which are equivalent in certain situation. The mathematical relationships between properties are the main focus in this stage. Understanding and finding these relationships is a kind of informal deduction.

For example, in this stage students would start to improve the idea that some operations in real numbers follows from other sets like natural numbers. Then they would start making an approach understanding the Real Numbers axiom as a systematic commutative field. But they cannot make proofs for such informal observation. Just in the next stage student would be able to produce proofs and deductions. That is where using the tools like Cabri 3D as a dynamic geometry software play very important roles.

For most of the students jump to the third level, informal deduction, is not easy. Now they can group the properties and identify the minimum amount of the needed properties. For example a cube, which might have had at level 2 the properties of six equal square faces, twelve equal edges with equal diagonal, parallel edges, perpendicular Adjacent edges, now would describe with the smaller amount of the properties such as shape composed of six equal squares. As it is seen, students in this level start formulation definitions for classes of objects and figures. For instance, a right triangle can be defined as a special kind of triangle that has two perpendicular sides or a right angle. As in this stage parallelogram and rectangle are not independent shapes, cube and cuboid also would be a special model of Parallelepiped.

In this level students could give informal arguments to prove geometric results. They start deductively thinking about geometry and it is one of important aspects of the present stage. Some simple rules may be using here, because students follow just simple logics. For example, if A=B and B=C then A=c.

Most of fitted students in the informal deduction level would able to justify arguments that they presented before with informal logic relationships. Therefore, at this level they can give informal logical relationships and use them about earlier identified properties. All in all, students now start to recognize the significance of the deduction and logic in the Geometry.

Level 4: Deduction

Deduction is the fourth level of Van Hiele theory of geometric thinking. In this level students start to construct rather than just memorize the proofs. They are able to find differences between the same proofs.

The goal of the previous level was discovering the relations among properties of the bases element by the students. At level 4 those relations are used to deduce theorems about base elements based on laws of deductive logic. The main purpose of level 4 is the organization of the statements about relations from level 2 and 3 into deductive proofs.

Discussing to the real number example, at this level, it is expected of the students to prove, for real numbers if. Students are ready to accept a system of axioms, theorem, and definitions. They can create the proofs form the axioms and just using the models or diagrams to support their arguments. Thus, students are able to formally prove what they had proved previously in level 3 using diagrams and informal arguments. They also start to distinguish the need for undefined terms in Geometry, which is very hard concept to understand in purely logical system.

Another point in this stage is that, students begin to become aware, understand and identify the differences between contrapositive, converse, and a theorem. They can also prove or disprove any of those relationships. In this level students become aware of relationships and connections between theorems and group them correspondingly. These level is the stage at which high school students are taught in Iran. Mesal 3d

Level 5: Rigor

In level fifth which named rigor, traditionally students hyper analysed the deductive proofs from level 4. They are looking to find the relationships between proves. This level looks to identified organizations of pervious level.

“For example, at this level the questions of “are the proofs consistent with each other”, “how strong of a relationship is described in the proof and “how do they compare with other proofs” would be asked. The level of Rigor involves a deep questioning of all of the assumptions that have come before.

This type of questioning also involves a comparison to other mathematical systems of similar qualities. For example, in Level 5if we considered Real Numbers we would begin to compare them as a field to other fields in general. It is fair to say that this level is usually only undertaken by professional mathematicians.”(Smart, 2008)

At Level 5 of van Hiele theory students can work in non-Euclidean of geometric system. So this level does not met by the high school students and it is usually assigned to college or university students in higher education. At non-Euclidean geometry constructing visual models for recognition is not easy and useful, so the focus is more on abstract concepts. So, most of geometry which is done in this level is based on abstract and proof-oriented. Students in this stage are capable to compare axioms systems such as Euclidean and Non-Euclidean.

Most of the students who have fitted in this level become professionals in geometricians and geometry so they are able to carefully develop the theorems in different axiomatic geometric systems. Therefore as smart (2008) emphasis, this level usually is the work of professional mathematicians and their students in higher education that conduct research in other areas of the geometry.

The Van Hiele started his research after he found that most of the students have difficulty with learning geometry. He observed that these students struggled with geometry, although they easily understood other mathematics topics. The results of their study showed, most of the High school students are taught at level 3and 4. Then van Hiele deduced most of the students had difficulty in learning geometry at level 3 and 4, because they could not understand geometry at level 2 to be able to move onto grasping level. Therefore, for melting this problem more focus is needed at second stage, analysis level and more emphasis on third stage, informal deduction. Then it can be expected that they are able to success at the deduction level.(Battista, 1999)

Van hiele noted that students should pass through lower levels of geometric thinking smoothly and master them before attaining higher levels. Van Hiele theory recommends achieving higher level of thought needs a precise designed instructions. Since students are not able to bypass levels and achieve understanding, permanently dealing with formal proof can cause students to relay on memorization without understanding. In addition, geometric thinking is inherent in the types of skills we want to nurture in students.

Research involving the van Hiele Model of Geometric Thinking and Interaction with dynamic geometry software

Van Hiele described in his article (1999) that the learning geometry can be started in a playful environment to explore geometrical concepts with certain shapes, and properties, parallelism, and symmetry. He advised some mosaic puzzles in this purpose. In the line of his work, geometry based software provide the more powerful environment which can be used to enhance the level of geometric thinking. There are several studies carried out on effects of using some dynamic geometry software such as (geometers’ Sketchpad) GSP on levels of van Hiele .

Different researches had been involving the Van Hiele geometric thinking since last decades. Some researchers used van Hiele Model as the theoretical framework while others used it as an analytic tool. Moreover many researches conduct study on geometric softwares like: Geometry Scratchpad used van Hiele theory to find out their effects on geometric reason, geometric thinking and other aspects.

In order to find out whether dynamic geometry software is able to enhance the level of geometric thinking or not several researches has been conducted. In general, the van Hiele Model has been used in their research as an analytic tool and theoretical framework. For example, July (2001) documented and described 10th-grade students’ geometric thinking and spatial abilities as they used Geometer’s Sketchpad (GSP) to explore, construct, and analyze three-dimensional geometric objects. Then he found out the role that can dynamic geometry software, such as GSP, play in the development of students’ geometric thinking as defined by the van Hiele theory. He found there was evidence that students’ geometric thinking was improved by the end of the study. The teaching episodes using GSP encouraged level 2 thinking of the van Hiele theory of geometric thinking by helping students to look beyond the visual image and attend to the properties of the image. Via GSP students could resize, tilt, and manipulate solids and when students investigated cross sections of Platonic Solids, they learned that they could not rely on their perception alone. In addition teaching episodes using GSP encouraged level 3 of the van Hiele thinking by aiding students learn about relationships within and between structure of Platonic solids(July, 2001).

Noraini Idris (2007) also found out the positive effects of using GSP on level of Van Hiele among Form Two students in secondary school. In addition she reported the positive reaction of students toward using this software in learning geometry.

In contrast Moyer,T(2003) in his PhD thesis used a non-equivalent control group design to investigate the effects of GSP on van Hiele levels. His research carried out in 2 control groups and 2 experimental groups in one high school in Pennsylvania. He had used Van Hiele’ tests written by Usiskin. However, Comparison of pre-test and post-test did not show a significant difference on increasing Van Hiele level of geometric thinking(July, 2001; Moyer, 2003).

Fyhn (2008) categorized students’ responses according to the van Hile levels in a narrative form of a climbing trip(Fyhn, 2008). The theoretical framework used Smart(2008) for his research” Introducing Angles in Grade Four” was a combination of a teaching theory called Realistic Mathematics Education (RME) and a learning theory called the van Hiele Model of Geometric Thinking. His research findings suggest the usefulness of using lesson plans based on the two theoretical frameworks in helping students develop an analytical conceptualization of mathematics. In this study the model was neither proved nor disproved but just accepted as an analytic framework.

Gills,J (2005) investigated students’ ability to form geometric conjectures in both statistic and dynamic geometry environments in his doctoral thesis. All participates were exposed to both environment and take parted, up to eight lab activities. He also used van Hiele theory as the main theoretical framework with more emphasis on geometric reasoning.(Gillis, 2005)

Research that used the van Hiele Model as an accepted framework covers variety of different topics. For example, Gills,J (2005) find out the mathematical conjectures formed by high school geometry students when given identical geometric figures in two different, dynamic and statistic of geometric environments. Burger and Shaughnessy (1986) tested students from grade one to first year of university to determine in what level the students are functioning regarding triangles and quadrilaterals.

Cabri 3D

Most of the dynamic geometric software until 2005 has been constructed in 2 dimensions. Just a few dynamic geometry software, has constructed on Three-dimensional dynamic geometric software such as, Autograph and Cabri 3. Focus of present study is on Cabri 3D, which is a new version of Cabri II (2 dimensional software). Cabri 3D is a commercial interactive geometry software manufactured by the French company Cabrilog for teaching and learning geometry and trigonometry. It was designed with the ease-of-use in mind.

Cabri 3D as dynamic and interactive geometry provides a significant improvement over those drawn on a whiteboard by allowing the user to animate geometric figures, relationships between points on a geometric object may easily be demonstrated, which can be useful in the learning process. There are also graphing and display functions, which allow exploration of the connections between geometry and algebra. The program can be run under Windows or the Mac OS(CABRILOG SAS, 2009).

From Euclidean geometry, Compass, straightedge and ruler, for many years, have been used in as the unique method of teaching and learning geometry, and tools used to aid people in expressing their knowledge. With the creation of computers, new world opened up to teaching and learning geometry. The speed and memory of modern PCs, together with decreasing prices, have made possible the development of `virtual reality’ computer games making use of the 3D graphics chips included on modern’ graphics cards. some educational spin-off from this has been the development of 3D interactive geometry software such as Cabri 3D, Autograph ,etc…

But tools can contain particular conceptions so; the aim of designing a dynamic geometry software package is to provide new instructional tools to study, teaching and learning geometry. While all the dynamic geometry software attempt to model use of straightedge, compass and ruler in Euclidean geometry, other futures like measuring capability and dragging possibilities and changing the view of objects in 3 Dimensional (Gonzaílez & Herbst, 2009).

Cabri 3D launched in September 2004 by Cabrilog, this software has the capacity to revolutionize teaching and learning of 3D geometry, at all levels, in the same way that dynamic geometry software has for 2D (CABRILOG SAS, 2009). Cabri 3D can share the same aptitude for making new discoveries as a research tool. There are some important practical features of Cabri 3D. First, This program is capable to store the files as text in Cabrilog’s development of the ‘Extensible Markup Language’ (XML). XML is the simplest version of the SGML standard for creating and designing HTML documents (suitable for use on Internet sites).XML designed by the World Wide Web Consortium as a more flexible replacement for HTML. Next, as Oldknow discussed, Files developed in Cabri 3D can be inserted as active objects in web-pages, spread sheets, word documents and etc. It is an interesting future because this objects which inserted in the files can be manipulated by users who do not own a copy of Cabri 3D in their computers.(Oldknow, 2006)

One of the important charactirisitc of Cabri package is draging.Arzarello, Olivero, Paola, &Robutti (2002) found that dragging in Cabri allows students to validate their conjectures. They claimed that work in Cabri is enough for the students to be convinced of the validity of their conjectures. If the teacher does not motivate students to find out why a conjecture is true, then the justifications given by students may remain at a perceptive-empirical level. Students would claim that the proposition is true because the property observed on the Cabri figure stays the same when dragging the drawing, given the hypotheses do not change. When such a belief is shared in the classroom, then Cabri might become an obstacle in the transition from empirical to theoretical thinking, as it allows validating a proposition without the need to use a theory. These researcher asserted, if teacher makes explicit the role of proof in justification, then students will be motivated to prove why a certain proposition is true (within a theory), after they know within the Cabri environment, that it is true. To paraphrase Polya (1954), first we need to be convinced that a proposition is true, then we can prove it.(Arzarello, Olivero, Paola, & Robutti, 2002).

In some researches the centrality has given to dragging in 2D dynamic geometry software and its implications for developing different types of reasoning (Arzarello et al. 2002).in addition because dragging is something which might make motion in 3D (on the 2D screen), it is more difficult to interpret and understand by the user. The various aspects of dragging in 3D DGE are issues that could usefully be the focus for research.(Hoyl