**Van Hiele model** . Theory of __teaching__ and __learning__ of __geometry__ , designed by the Dutch couple van Hiele. It is classified within the __didactics__ of __mathematics__ and specifically in the didactics of geometry.

Summary

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- 1 Origin of the Model
- 1 Forms of reasoning
- 2 Statement of the Van Hiele Model
- 3 Levels of reasoning by van Hiele
- 4 Model properties
- 5 Phases of learning from the Van Hiele Model
- 6 Characteristics of the learning phases

- 2 Sources

Origin of the Model

It originates in __1957__ , in the doctoral dissertations of Dina van Hiele-Geldof and Pierre van Hiele at the __University of Utrecht__ , The __Netherlands__ . The original __book__ where the theory is developed is “Structure and Insight: A theory of mathematics education.” The theory is classed within the __didactics__ of mathematics and specifically in the didactics of geometry.

Forms of reasoning

The model covers two aspects:

- Descriptive: through which different forms of geometric reasoning of individuals are identified and their progress can be assessed.
- Instructional: it marks some guidelines to be followed by the teachers to favor the advancement of the students in their level of geometric reasoning.

As its name indicates, this __learning__ theory describes the ways or forms of reasoning of __Geometry__ students .

The Van Hiele model consists of 2 parts:

- Description of the different types of geometric bodies of the students throughout their
__mathematical__training , ranging from the visual__reasoning__of preschool__children__to the formal and abstract of the students of the science faculties, to these types of reasoning. he calls them the levels of reasoning. - Description of how a teacher can organize the activity of his classes so that the students are able to access the level of reasoning superior to that which he currently has; it is about the learning phases.

Van Hiele Model Statement

- Several different levels of perfection can be found in the thinking of math students.
- A student can only truly understand those parts of mathematics that the teacher presents to them in a way that is appropriate to their level of reasoning.
- If a mathematical relationship cannot be expressed at the current level of students’ reasoning, it will be necessary to wait for the students to reach a higher level of reasoning to present it.
- You cannot teach a person to reason in a certain way. But you can help him, through an adequate teaching of mathematics, to come to reason as soon as possible.

Van Hiele’s levels of reasoning

Model van Hiele

In the existing description, you can find complete lists with characteristics of the different levels of Van Hiele in which 2 numberings of the five levels are used, starting at 0 and starting at 1.

Most important properties that allow to clearly characterize each level and differentiate it from its adjacent ones: Level 0: Visualization or Recognition Level 1: Analysis Level 2: Ordering or classification Level 3: Formal Deduction Level 4: Rigor

- Level 0: at this level objects are perceived in their entirety as a whole, not differentiating their characteristics and properties. The descriptions are visual and tend to be likened to familiar elements.

Example: Identify parallelograms in a set of figures. Identify angles and triangles in different positions in images.

- Level 1: properties of geometric objects are perceived. They can describe objects through their properties (not just visually anymore). But you cannot relate the properties to each other.

Example: a square has equal sides. A square has equal angles

- Level 2: describe objects and figures formally. They understand the meanings of the definitions. They recognize how some properties derive from others. They establish relationships between properties and their consequences. Students are able to follow demonstrations. Although they do not understand them as a whole, since, with their logical reasoning, they are only capable of following individual steps.

Example: In a parallelogram, equal opposite sides imply parallel opposite sides. Parallel opposite sides imply equal opposite sides.

- Level 3: deductions and demonstrations are made at this level. The axiomatic nature is understood and the properties are understood and are formalized in axiomatic systems. Van Hiele calls this level the essence of mathematics

Example: Synthetically or analytically demonstrates that the diagonals of a parallelogram intersect at their midpoint.

- Level 4: geometry is worked without the need for specific geometric objects. The existence of different axiom systems is known and can be analyzed and compared. A demonstration contrary to intuition and common sense will be accepted if the argument is valid.

Given that level 5 is thought to be unattainable for students and is often disregarded, in addition, work carried out indicates that non-university students, at most, reach the first three levels. It is important to note that one or a student may be, depending on the content studied, at one level or another.

Model Properties

In addition to shedding light on the __thinking__ that is specific at each level, the van Hiele identifies some generalities that characterize the model. These properties are particularly significant for educators because they provide guidance for making instructional decisions.

- As in most developmental theories, a person must go through the levels in an order. To function successfully at a particular level, a student must have acquired the strategies from the preceding level.
- Progress (or lack thereof) from one level to another depends more on the content and method of instruction received than on age: no method of instruction allows a student to skip a level; some methods favor progress, while others slow it down or block movement between levels.

Van Hiele points out that it is possible to “teach a skilled student skills above their current level, just as children can be trained in arithmetic or fractions without telling them what fractions mean, or older students how to derive and integrate, although they do not know what derivatives and integrals are.

Geometric examples include memorizing the formula for an area or relationships such as “a square is a rectangle.” In situations like these what has happened is that the issue has been reduced to a lower level and there has been no understanding.

- Intrinsic and extrinsic. Objects inherent in one level become the objects of study for the next level. For example, at level 0 only the shape of a figure is perceived. The figure is, we agree, determined by its properties, but it is not until level-1 that it is analyzed and its components and properties are discovered.
- Each level has its own linguistic symbols and its own system of relationships that connect these symbols. A relationship that is “correct” on one level can be changed on another. For example, a shape can have more than one name (class inclusion) – a square is also a rectangle (and a parallelogram!). A student at level-1 does not conceptualize that this type of inclusions can occur. This type of notions and the accompanying
__language__, however, is essential at level-2. - If the student is at one level and the instruction at a different one, the desired learning and progress may not occur. In particular, if the teacher, materials, content, vocabulary, and everything else are at a higher level than the student, the student will not be able to follow the thought process used.

Learning phases of the Van Hiele Model

To complete the description of the model, it is necessary to know the steps that a teacher must follow to help his students rise to the next level of reasoning.

The “learning phases” are stages in the graduation and organization of the activities that a student must carry out to acquire the experiences that take him to the higher level of reasoning. Firstly, it is necessary to ensure that students comprehensively acquire the necessary basic __knowledge__ (new concepts, properties, vocabulary, etc.) with which they will have to work, and then focus their activity on learning to use and combine them. There are five learning phases proposed by Van Hiele:

__Information__: this is a phase of making contact: the teacher must inform the students about the field of study in which they are going to work, what kind of problems are going to arise, what materials they are going to use, etc. Likewise, students will learn to handle the material and acquire a series of essential basic knowledge to begin the mathematical work itself. This phase serves to direct students’ attention and allow them to know what type of work they are going to do, and for the teacher to discover what level of reasoning their students have on the new topic and what they know about it.- Guided orientation: in this phase, students begin to explore the field of study through research based on the material provided to them. The main objective of this phase is to get students to discover, understand and learn what are the concepts, properties, figures, etc. major in the area of geometry they are studying. The activities proposed to them must be suitably directed towards the concepts, properties, etc. they must study. The work they are going to do will be selected in such a way that the characteristic concepts and structures are presented to them progressively.
- Explanation: among the main purposes of this phase is to make the students exchange their experiences, comment on the regularities they have observed, explain how they have solved the activities, all within a context of dialogue in the group. It is interesting that divergent points of view emerge, since each student’s attempt to justify her opinion will have to carefully analyze her ideas (or those of her partner), order them and express them clearly.
- Free orientation: at this moment the students will have to apply the knowledge and language they have just acquired to other investigations different from the previous ones. The teacher must pose problems that, preferably, can be developed in different ways or that can lead to different solutions, in order to perfect the knowledge that students have about the field of study. In these problems clues will be placed that show the way forward, but in such a way that the student has to combine them appropriately, applying the knowledge and the way of reasoning that he has acquired in the previous phases.
- Integration: in this phase students must acquire an overview of the content and methods available to them, relating the new knowledge with other fields they have previously studied; it is about condensing into a whole domain that has explored her thinking. In this phase the teacher can promote this work by providing global understandings, but it is important that these understandings do not bring any new concept or property to the student: They should only be an accumulation, comparison and combination of things that he already knows.

Characteristics of the learning phases

- In general, the reasoning development process cannot be framed within the limits of a school year. The acquisition of higher levels, particularly 3 and 4, is usually a process of several years, so it is not surprising that at the end of the course students continue to be at the same level as at the beginning, although they will be closer to being able to achieve the higher level.
- It can also happen that throughout the course students reach a level, so the teacher must start the work that leads to the next level. In this sense, it must be borne in mind that the levels do not pose breaks in the learning process, so once the work of the last phase of a level is completed, the work of the first phase of the next level must be started.
- The learning phases must be reflected in a style of teaching geometry (and mathematics in general) and organization of teaching. Phases 2 and 4 mark the sequencing of activities for learning a topic and acquiring a level of reasoning. Phase 3 must cover all the activity in which the students take part. Phases 1 and 5 are also important and should not be ignored, although it is also not harmful to delete them if at any given time they are found to be unnecessary.

You should not try to follow the guidelines of any psycho-pedagogical-didactic-educational theory to the letter, since it is a field (mathematics education) in which the main element, the students, is enormously diverse and, for Therefore, it is necessary that teachers are free to make modifications according to the specific situation of the moment.