The Vehicle Hiele Theory Of Geometric Thinking

This chapter will provide a brief explanation of the theoretical framework on Van Hiele theory of geometric thinking. Subsequently review and discuss on literature involving truck Hiele theory and powerful geometry software, follow by overview of literature on coaching and learning of geometry by energetic geometry software Cabri 3D as an instructional tool. Then section conclude by critiquing literature on developing learning activities.

The Truck Hiele Theory of Geometric Thinking

The truck Hiele model of geometric thinking is one theory that offers a model for detailing and describing geometric thinking. This theory resulted from the Dutch mathematics educator doctoral work of Dina van Hiele-Geldof and Pierre vehicle Hiele at the School of Utrecht in the Netherlands which completed in 1957. Pierre vehicle Hiele designed the five levels of pondering 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, that was completed in 1957, centered on the role of teaching in the bringing up of any pupil's thought levels. Her review centered on thinking of geometry and the role of instruction in assessing pupils to move though the levels.

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

A convenient location for many primary resources on the Truck Hiele model is Fuys et al. (1984). Other most important sources are vehicle Hiele and Truck Hiele-Geldof (1958) and Vehicle Hiele (1986). Extra sources for Vehicle Hiele research are Mayberry (1983), who found that students may operate at different levels for different concepts; Mayson (1997), who remarks that gifted students may skip truck 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 bigger order cognitive functions, which is essential to success in high school geometry. In such a theory if students do not trained at the proper Hiele level that they are at or ready for this, will face challenges and they cannot understand geometry. The therapy that offered for students by this theory is that they should go through the collection of levels in a particular way. (Usiskin, 1982b). You'll be able to generalize the Truck Hiele model to the other matters such as physics, technology and arts. Because the primary idea of this theory is the consequence of levels and thinking that every level is built on properties of the prior level as many researches did predicated on this theory on knowledge education.

Characteristics of Truck Hiele level of geometric thought

Van Hiele theory argues there are some misconstructions in teaching of institution mathematics and geometry, which was existed for long time based on the formal axiomatic geometry and was made by Euclid more than two thousand years back. Euclid logical building is dependant on his axioms, definitions, theorems, and proofs. Therefore, the institution geometry that is within 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. Vehicle Hiele discusses this lack creates a gap between their degree of geometric thinking that they can be, and the amount 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 presenting it at the wrong time". Educators should provide coaching that is appropriate to the level of children's thinking. Vehicle Hiele theory advises: It is determined by the students' degree of geometric thinking the instructor can decide where level the teaching should be begun. (Truck Hiele, 1999)

According to the vehicle Hiele theory, a student steps sequentially from the original level (Visualization) to the highest level (Rigor). Students cannot achieve one degree of thinking successfully with no passed through the previous levels. Furthermore, Burger &Shaughnessy (1986) and Mayberry (1983) have found that the amount of pondering at an basic level is not similar in all regions of geometry.

During last decades many researchers and investigators attempted to aid the Van Hiele model or disapprove of it but still some try to improve or change this model. Lots of the researcher used Van Hiele level of geometric thought as the right and proper theory in their research using active geometry software (Smart, 2008).

The Vehicle Hiele levels have certain properties specially for understanding the geometry. First of all, the levels have fixed series property. The five levels are hieratically, it means students must go through the levels in order. He/she cannot easily fit into level N with no gone through the previous level (N-1). Students cannot engage in geometry thinking at higher-level without passing the low levels.

Second property is adjacency of the levels. At each level of thought what's essential in the previous level become extrinsic in the prevailing level. Specific understanding and reflection on geometric ideas are had a need to move from one level to the next one, somewhat than natural maturation.

Third each level has its own icons and linguistic and relationships for connecting those icons. This property is a variation of the periods. For example when a professor use a dialect for higher-level of thinking than students level of thinking, students cannot understand the ideas and try to just memorizing the proofs and do the rote learning. In cases like this miscommunication emerge (Hong Place, 2005).

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

Lastly, the Truck Hiele theory emphasize on pedagogy and the value of teacher training to aid students' change through one level to the next one. This quality shows that appropriate activities which allow students to explore and discover geometric ideas in appropriate levels of their thinking are the best activities to progress students 'level of considering.

Phases of learning geometry

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

"The training process leading to complete understanding at another more impressive range has five stages, approximately but not firmly sequential, entitled:

Inquiry

Directed orientation

Explanation

Free orientation

Integration" (p. 6)(Usiskin, 1982a).

These five level are incredibly valuable in developing activities and design instructional phases.

Phase one: Inquiry

First phase of learning geometry starts with inquiry or information satge. Within this stage students find out about the nature of the geometric objects. in order to design appropriate activities, Tutor identify students' preceding understanding of new principle which need to be learnt. Then tutor design proper activities to encourage and face students with the new theory which has been taught.

Phase two: Directed orientation

During this period while students doing their brief activities with group of final results like: measuring, folding and unfolding, or geometry game titles, educator provides appropriate activities foundation on students' levels degree of thinking to encourage them be more familiar with the idea being taught.

Phase three: Explanation

As the name of the phase demonstrates, in this level students make an effort to express their learning of new notion in their own words. Students in this phase start to point out their conclusions and finding using their other classmates and instructor in their own words. They connect mathematically. The role of instructor in this level is supplying relevant mathematical terminology and language in an effective manner, by using geometrical and mathematical dialect accurately and effectively.

Phase four: Free orientation

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

Phase Five: Integration

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

One of important properties of these phases of learning in Vehicle Hiele theory is not linear in dynamics. Sometimes students desire a cycle form of these phases by repeating more than one time to defeat certain geometrical ideas. The role of shows here's providing appropriate activities based on these five stages to build up each degree of vehicle Hiele geometric thinking.

The Vehicle Hiele level of geometric thinking

According to Vehicle Hiele theory, the introduction of student's geometric thinking considered about the increasingly sophisticated level of thinking. These levels are hierarchies and able to forecast future students' enactment in geometry(Usiskin, 1982a). This model involves five levels in understanding, which numbered from 0 to 4. However, in this research we identified these levels from 1 to 5 to have the ability 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 level of Vehicle Hiele geometric thinking which is encountered with goals of mathematical website is Level 1. The goals of the first level are functions like the underpinning elements of everything that will be studied.

Understanding at this stage includes visualizing bottom objects. As of this level visualization defines as comprehension or seeing first objects in students' imagination. For instance, lots series in this level could be defined as real figures in the website of real quantities. Vectors and matrices is seen as basic items in the domains of leaner algebra. So perceiving vector as a directed section or matrices as a rectangular stand of numbers is based on level 1.

Elementary professors know that it requires a couple of years of institution for pupils to master visualization level. For example, it takes long time for students to see real amounts in a number line format. Likewise, perception of the bought list or selection of numbers, or an ordered pair of items is not something that occurs to the untaught head and eyes. Hence, serious teaching effort and advantages needed to students achieve Level 1and it is not assumed the visualization of initial items to be evident or trivial for students.

Geometry in Iran begins in elementary college and remains until level 8 with bringing out geometry figures like circles, squares, triangles, straight lines, etc. At the level 1 student learn to acknowledge geometric characteristics in things that can be physically seen. At this time scholar are assumed to be able to categorize geometric styles by visual acknowledgement, and know their names, for example, in solid geometry in level 1, if shown an image of a polyhedron just like a cube, students can say that it is a cube because it looks like one for her or him. At this time, it isn't required to think of the cube, or any other geometric subject, in conditions of its properties, like declaring a cube has 6 faces and 12 edges.

With visual acknowledgement a student can make a copy, by pulling, plotting or using some sort of powerful geometry software, of any shape or construction of shapes if indeed they could be shown or informed what it is they were supposed to be copying. In this stage, the teaching should be predicated on the name the college student has memorized for the thing and not the object's properties. For example, maybe it's "draw a cube" not bring a "polygon with 12 equal edges that are perpendicular to the base and 6 equal faces.

Level 2: Research Stage

At analysis stage, students begin to analysis things which were only visually identified at pervious level, discovering their parts and relations among these parts. They concentrate on the properties of the objects. For instance, concentrate on Real Amounts in this level can be closure under businesses. This property can be leading to distinguishing subsets of Real Statistics inside the set in place that are Integers and Rational Numbers.

In sound geometry, the evaluation level is where students begin seeing the properties associated with the several designs or configurations. A cube will now turn into a shape with 6 equal faces which opposite faces are parallel and 12 sides and adjacent angles right perspectives and having opposite faces identical, as well as having the diagonals intersect in their middle. However, at this time, it isn't assumed that students will be seeking logical relationships between properties such as understanding that it is enough for a Parallelepiped as a solid with parallel opposing faces and the rest of the properties follow. Nor is it assumed that students will look at a cuboid as a special type of Parallelepiped. Therefore, students will identify patterns and solids based on the wholeness of the properties. Quite simply, relationships between patterns and configurations stay only on the list of properties they may have.

At this level if a student were asked to describe a form or sound, the information would be based on the object's properties. At the same time, if students were asked to replicate a shape or solid predicated on the list of properties, they might manage to do it. Students would also be able to verify information and solids hieratically by analyzing their properties. Within this stage university student can recognize the interrelation between figures and their properties. For instance, knowing the house that the Parallelepiped the scholar would be able to deduce that cuboid is special kind of Parallelepiped.

Level 3: Informal Deduction Stage

Informal deduction is recognized as the third degree of geometric thinking. A few of analysts name this level as abstract/Connection level too(Battista, 1999; Cabral, 2004). Within this stage students can reason logically. This level is achieved whenever a university student can operate with the relation of statistics and solids and can apply congruence of geometric statistics to demonstrate certain properties of a total geometric configuration of which congruent figures are a component. They become aware of sufficient and necessary condition for a concept. Students fit as of this level after obtaining pervious levels (visualization and evaluation).

At this level more attention directed at relations among properties. Quite simply, in this stage target is "properties of units of properties". In such a level relating to romantic relationship between properties of items 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 original foundation elements. They intend to categorize properties that are equivalent using situation. The numerical human relationships between properties are the main concentrate in this stage. Understanding and finding these interactions is a kind of casual deduction.

For example, in this level students would start to improve the proven fact that some operations in real volumes employs from other pieces like natural numbers. Then they would start making a strategy understanding the Real Quantities axiom as a systematic commutative field. However they cannot make proofs for such casual observation. Just within 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 almost all of the students hop to the 3rd level, informal deduction, is challenging. Now they can group the properties and identify the least amount of the needed properties. For instance a cube, which might experienced at level 2 the properties of six equal square encounters, twelve equal edges with similar diagonal, parallel corners, perpendicular Adjacent sides, now would explain with the smaller amount of the properties such as form composed of six equal squares. Since it sometimes appears, students in this level start formulation meanings for classes of items and figures. For example, the right triangle can be defined as a particular kind of triangle that has two perpendicular sides or a right angle. As with this stage parallelogram and rectangle are not independent patterns, cube and cuboid also will be a special model of Parallelepiped.

In this level students could give informal arguments to demonstrate geometric results. They start deductively considering geometry which is one of important areas 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 equipped students in the informal deduction level would able to justify quarrels that they offered before with casual logic interactions. Therefore, at this level they can give informal logical connections and utilize them about earlier discovered properties. Overall, students now learn to recognize the significance of the deduction and logic in the Geometry.

Level 4: Deduction

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

The goal of the prior level was exploring the relationships among properties of the bases element by the students. At level 4 those relationships are being used to deduce theorems about basic elements predicated on laws and regulations of deductive reasoning. The main reason for level 4 is the business of the claims about relations from level 2 and 3 into deductive proofs.

Discussing to the real number example, as of this level, it is expected of the students to demonstrate, for real statistics if. Students are ready to accept a system of axioms, theorem, and explanations. They are able to create the proofs form the axioms and just using the models or diagrams to aid their quarrels. Thus, students have the ability to formally verify what that they had proved previously in level 3 using diagrams and casual arguments. They also start to distinguish the need for undefined terms in Geometry, which is very difficult concept to comprehend in purely logical system.

Another point in this level is the fact, students begin to become aware, understand and identify the differences between contrapositive, converse, and a theorem. They can also verify or disprove some of those relationships. In such a level students notice relationships and relationships between theorems and group them correspondingly. These level is the level at which high school students are educated in Iran. Mesal 3d

Level 5: Rigor

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

"For example, as of this level the questions of "will be the proofs consistent with each other", "how strong of the relationship is defined in the substantiation and "how do they equate to other proofs" would be asked. The level of Rigor entails a profound questioning of all of the assumptions which have come before.

This kind of questioning also entails an evaluation to other mathematical systems of similar attributes. For example, in Level 5if we considered Real Statistics we would begin to compare them as a field to other fields in general. It really is fair to say that level is usually only carried out by professional mathematicians. "(Smart, 2008)

At Level 5 of vehicle Hiele theory students can work in non-Euclidean of geometric system. So this level does not fulfilled by the high school students which is usually allocated to college or university or university or college students in higher education. At non-Euclidean geometry constructing visual models for recognition is challenging and useful, therefore the target is more on abstract ideas. So, the majority of geometry which is performed in this level is dependant on abstract and proof-oriented. Students in this level are capable to compare axioms systems such as Euclidean and Non-Euclidean.

Most of the students who've fitted in this level become professionals in geometricians and geometry so they could carefully develop the theorems in several axiomatic geometric systems. Therefore as smart (2008) emphasis, this level usually is the task of professional mathematicians and their students in advanced schooling that conduct research in the areas of the geometry.

The Vehicle Hiele began his research after he found that most of the students have a problem with learning geometry. He discovered that these students struggled with geometry, although they easily grasped other mathematics topics. The results of these study showed, most of the Students are taught at level 3and 4. Then truck 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 issue more focus is necessary at second level, analysis level plus more focus on third stage, casual deduction. Then it can be expected that they are in a position to success at the deduction level. (Battista, 1999)

Van hiele mentioned that students should pass through lower levels of geometric thinking properly and get good at them before attaining higher levels. Vehicle Hiele theory suggests achieving higher-level of thought requires a precise designed instructions. Since students are not able to bypass levels and achieve understanding, completely interacting with formal facts can cause students to relay on memorization without understanding. Furthermore, geometric thinking is inherent in the types of skills you want to nurture in students.

Research relating to the van Hiele Model of Geometric Thinking and Conversation with powerful geometry software

Van Hiele identified in his article (1999) that the training geometry can be started in a playful environment to explore geometrical ideas with certain figures, and properties, parallelism, and symmetry. He encouraged some mosaic puzzles in this goal. In the type of his work, geometry centered software provide the more powerful environment that can be used to enhance the level of geometric thinking. There are many studies carried out on effects of using some energetic geometry software such as (geometers' Sketchpad) GSP on degrees of vehicle Hiele.

Different researches have been involving the Truck Hiele geometric thinking since last decades. Some experts used truck Hiele Model as the theoretical framework while some used it as an analytic tool. Moreover many researches execute analysis on geometric softwares like: Geometry Scratchpad used truck Hiele theory to learn their effects on geometric reason, geometric thinking and other aspects.

In order to learn whether powerful geometry software can enhance the level of geometric thinking or not several studies has been conducted. In general, the truck Hiele Model has been found in their research as an analytic tool and theoretical framework. For example, July (2001) noted and described 10th-grade students' geometric thinking and spatial abilities as they used Geometer's Sketchpad (GSP) to explore, create, and evaluate three-dimensional geometric things. 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 truck Hiele theory. He found there is research that students' geometric thinking was improved by the finish of the analysis. The teaching episodes using GSP inspired level 2 thinking of the truck Hiele theory of geometric thinking by aiding students to look beyond the visible image and attend to the properties of the image. Via GSP students could resize, tilt, and change solids so when students investigated cross parts of Platonic Solids, they found that they could not rely on their perception alone. Furthermore teaching episodes using GSP prompted level 3 of the truck Hiele thinking by aiding students find out about connections within and between composition of Platonic solids(July, 2001).

Noraini Idris (2007) also discovered the positive effects of using GSP on degree of Truck 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 compare Moyer, T(2003) in his PhD thesis used a non-equivalent control group design to investigate the consequences of GSP on truck Hiele levels. His research carried out in 2 control groupings and 2 experimental teams in one senior high school in Pennsylvania. He had used Truck Hiele' tests written by Usiskin. However, Evaluation of pre-test and post-test did not show a big change on increasing Van Hiele degree of geometric thinking(July, 2001; Moyer, 2003).

Fyhn (2008) categorized students' responses in line with the van Hile levels in a narrative form of an climbing trip(Fyhn, 2008). The theoretical construction used Smart(2008) for his research" Introducing Perspectives in Level Four" was a combination of a coaching theory called Natural Mathematics Education (RME) and a learning theory called the vehicle Hiele Model of Geometric Thinking. His research conclusions suggest the usefulness of using lessons plans based on both theoretical frameworks in assisting students develop an analytical conceptualization of mathematics. In this research the model was neither demonstrated nor disproved but just accepted as an analytic framework.

Gills, J (2005) looked into students' ability to create geometric conjectures in both statistic and powerful geometry environments in his doctoral thesis. All participates were subjected to both environment and take parted, up to eight laboratory activities. He also used truck Hiele theory as the primary theoretical framework with an increase of emphasis on geometric reasoning. (Gillis, 2005)

Research that used the vehicle Hiele Model as an accepted framework covers variety of different subject areas. For example, Gills, J (2005) find out the numerical conjectures made by high school geometry students when given indistinguishable geometric characters in two different, vibrant and statistic of geometric surroundings. Burger and Shaughnessy (1986) analyzed students from grade one to first 12 months of university to find out in what level the students are working regarding triangles and quadrilaterals.

Cabri 3D

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

Cabri 3D as energetic and interactive geometry provides a significant improvement over those attracted on the whiteboard by allowing the user to animate geometric information, relationships between tips on the geometric subject may easily be exhibited, which is often useful in the learning process. There are also graphing and display functions, which allow exploration of the relationships between geometry and algebra. This program can be run under Home windows or the Apple pc OS(CABRILOG SAS, 2009).

From Euclidean geometry, Compass, straightedge and ruler, for quite some time, have been found in as the unique method of teaching and learning geometry, and tools used to aid people in expressing their knowledge. Together with the creation of computer systems, new world opened up to instructing and learning geometry. The swiftness and storage area of modern Personal computers, together with lessening prices, have permitted the development of `virtual certainty' computer games taking a 3D graphics chips included on modern' design credit 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 review, instructing and learning geometry. While all the active geometry software attempt to model use of straightedge, compass and ruler in Euclidean geometry, other futures like measuring potential and dragging opportunities and changing the view of items in 3 Dimensional (Gonzalez & Herbst, 2009).

Cabri 3D launched in September 2004 by Cabrilog, this software has the capacity to revolutionize teaching and learning of 3D geometry, by any means levels, in the same way that vibrant geometry software has for 2D (CABRILOG SAS, 2009). Cabri 3D can show the same aptitude for making new discoveries as a study tool. There are some important practical top features of Cabri 3D. First, This program is competent to store the data as wording in Cabrilog's development of the 'Extensible Markup Vocabulary' (XML). XML is the easiest version of the SGML standard for creating and making HTML documents (ideal for use on Websites). XML created by the internet Consortium as a far more flexible alternative to HTML. Next, as Oldknow mentioned, Files developed in Cabri 3D can be placed as active items in web-pages, pass on sheets, phrase documents and etc. It is a fascinating future because this objects which put in the data can be manipulated by users who do not own a backup of Cabri 3D in their pcs. (Oldknow, 2006)

One of the top charactirisitc of Cabri package deal is draging. Arzarello, Olivero, Paola, &Robutti (2002) discovered that dragging in Cabri allows students to validate their conjectures. They said that work in Cabri will do for the students to be convinced of the validity of the conjectures. When the teacher does not motivate students to learn why a conjecture holds true, then your justifications distributed by students may continue to be at a perceptive-empirical level. Students would declare that the proposition is true because the house witnessed on the Cabri figure continues the same when dragging the drawing, given the hypotheses do not change. When such a perception is distributed in the class, then Cabri might become an obstacle in the move from empirical to theoretical thinking, as it allows validating a proposition with no need to employ a theory. These researcher asserted, if teacher makes explicit the role of facts in justification, then students will be motivated to show why a certain proposition holds true (within a theory), once they know within the Cabri environment, that it is true. To paraphrase Polya (1954), first we have to be convinced that a proposition holds true, then we can verify it. (Arzarello, Olivero, Paola, & Robutti, 2002).

In some researches the centrality has directed at dragging in 2D active 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 display), 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 concentration for research. (Hoyles & Baptiste Lagrange, 2010)

Jones, Keith and Mackrell, Kate and Stevenson, Ian in section four of Mathematics Education and Technology-Rethinking the Surfaces book (2010) reviewed that we now have some place decisions that the custom made of powerful geometry software should make. One of these group of decisions is about how precisely the 3D objects would look on-screen. For an thing and its floors some characteristics should be observed to have a 3D point of view and appearance. The artist should decide the ways that the aesthetic appearance of your 3D on-screen subject will depend not only after its geometry but also after the viewpoint by using lighting, shading, and, where appropriate, feel. These characteristics in programing and making software call making. In terms of perspective, the default for the Cabri 3D is one-point perspective. The default browsing distance is 50 cm, representing the display screen at arm's length from the viewer's attention, chosen as it was thought to be natural (Hoyles & Baptiste Lagrange, 2010). See Figure------

Figure :

A Snapshot of Cabri 3D

Figure :

Snapshot of Autograph 3

If we compare Cabri 3D with the Autograph, naturally the viewing distance chosen for Autograph 3 is more subjectively and shorter than Cabri. Furthermore, both Cabri 3D and Autograph 3 use shading, In terms of rendering. It means in both these DGS, brightness of your surface would depend on the way where it is facing relative to the inferred observer. But fogging is a distinctive potential for Cabri 3D which is not are present in Autograph. Fogging is a computer graphics conditions for the result by which items far away appear to be fainter than objects readily available. These characteristics is shown in figures ----

Using the mouse is further set of decisions relate to dragging objects, which should be turning over by the software designers. On a flat screen Dragging only can give movement in 2 sizes. For Cabri 3D strong software a conclusion was made that normal pressing and dragging can move the idea and thing parallel with the bottom aircraft. While dragging and pressing Switch key in the same time can move the object and point perpendicular with the base plane.

Designing Learning Activities to Engage

Teachers can build the jobs by dynamic geometry software that are not possible with paper-and-pencil technology. Furthermore they can design more interactive and interesting activities with 3D geometry software (both Euclidean and co-ordinate). the developer of such activities must be aware of the intricacy of the images on-screen, and the necessity for learners to orient themselves to a flat-screen representation of 3D. There may also be issues for users moving from 2D DGE to 3D software. For example, in two dimensional dynamic geometry software, the perpendicular tool just produce a line or portion, while in Cabri 3D this tool produce a perpendicular plane to a line and perpendicular to a lines is a airplane. (Hoyles & Baptiste Lagrange, 2010).

Such new issues about using tools in 3dimesonal geometry software are just start to analyze. Because these issues surfaced while teachers and researchers began to use these software as a mediate the learners' knowledge of geometry. for example, Accascina and Rogora (2006) found that Cabri 3D as a very effective for quickly bringing out students 3D geometry and providing them with enough intuitive support for understanding non trivial. Then they repeated the test of plane section with class Eight students, who had not any pervious understanding of three-dimensional geometry. They detected students enjoyed the activities and were able to grasp the key intuitive ideas. They advised, instructors should be mindful about confusions and myths which may come up when interpreting a Cabri 3D diagram. Often scholar will face confusions in their thoughts (Accascina & Rogora, 2006).

Another example of review on using Cabri 3D process in learning geometry has done by Mackrell (2008). in this research she has discovered that students at Quality 7 and 8 students get highly motivated to use Cabri 3D to build their own constructions. That structure was comprising real-world objects. Overall, mathematics education has benefited from some useful associations between technology designers and users, perhaps forget about so than in the region of geometry education. (Hoyles & Lagrange, 2009)

Summary:

In this chapter Van Hiele theory as the primary theoretical frame use its characteristics and phases of learning mentioned, then the related books on truck Hiele theory including using dynamic geometry software outlined. Subsequently relevant litrature on Cabri 3D which have been completed have disseised, pursuing by issues in building learning activates.

All in every, The Truck Hiele Model provides us with a distinctive learning theory that may be related to Geometry and other areas of mathematics as well. The expectation is that instructors who understand different degrees of the van Hiele Model can recognize the level their students are currently performing at and adjust their teaching appropriately. At exactly the same time, the van Hieles always professed that success is very much indeed based on adequate teaching.

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