The Van Hiele theory, also known as the Van Hiele model or the Van Hiele levels of geometric thinking, is a theory of mathematical development that was proposed by two Dutch educators, Dina van Hiele-Geldof and Pierre van Hiele. The theory is based on the idea that students progress through a series of levels as they develop their understanding of geometry, and that these levels are hierarchical, with each level building upon the understanding and skills acquired at the previous level.
According to the Van Hiele theory, there are five levels of geometric thinking, each corresponding to a different stage of development. These levels are:
Visualization: At this level, students are able to recognize and draw basic geometric shapes, such as circles, squares, and triangles. They are able to identify these shapes in their environment and distinguish between them based on their characteristics, such as size and number of sides.
Analysis: At this level, students are able to classify shapes based on their properties, such as size, symmetry, and angles. They are also able to recognize and describe the relationships between shapes, such as congruence and similarity.
Deductive: At this level, students are able to use logical reasoning to prove geometric statements and theorems. They are able to apply their understanding of geometric concepts to solve problems and make predictions.
Informal Deductive: At this level, students are able to apply their understanding of geometry in more abstract and formal settings, such as in coordinate geometry and transformations.
Formal Deductive: At this highest level, students are able to apply their understanding of geometry in more complex and abstract settings, such as in non-Euclidean geometries.
The Van Hiele theory suggests that students typically progress through these levels in a linear fashion, with each level building upon the understanding and skills acquired at the previous level. However, the rate at which students progress through the levels can vary greatly, and some students may never reach the highest levels of geometric thinking.
One of the key implications of the Van Hiele theory is that teaching geometry should not be limited to simply memorizing formulas and theorems, but should instead focus on helping students develop their understanding of geometric concepts and their ability to apply them in real-world situations. This can be achieved through activities such as hands-on exploration, problem-solving, and the use of visual aids and manipulatives.
In conclusion, the Van Hiele theory is a valuable framework for understanding the development of mathematical thinking, particularly in the area of geometry. By recognizing the different levels of geometric thinking and the skills and understanding that students acquire at each level, educators can tailor their instruction to meet the needs and abilities of their students, and help them progress through the levels of geometric thinking in a meaningful and effective way.
Teaching Geometry for a Deeper Understanding
Studies have found that many children reason at multiple levels, or intermediate levels, which appears to be in contradiction to the theory. And this is all just piled on to the usual social distractions that we all experience without the technology. Even students of two different levels working together may have a hard time communicating about the geometric properties at play. Teaching activity examples with more clarification of Van Hiele levels Moving from level 0 to level 1: Sorting shapes into groups by properties. For example, in Again, be sure to include a variety of input information.
It implies that children should be starting to do activities like the shape sorting task described above that are intended to encourage level 1 thinking, though without the goal of using standard vocabulary. A huge challenge can be tackling two-column proofs, which are seemingly detached from practical life applications, for the first time. Children can discuss the properties of the basic figures and recognize them by these properties, but generally do not allow categories to overlap because they understand each property in isolation from the others. An individual needs to achieve the formal operational stage in order to understand, formally reason and build proofs. Properties are in fact related at the Analysis level, but students are not yet explicitly aware of the relationships. The description of the groups in a shape sorting activity can reflect the level of geometric thinking that was used for the shape sorting: Level 0 descriptions often have orientation or visual similarity cited as sorting rules Level 1 descriptions often have a lot of properties listed more than are needed Level 2 descriptions often have more efficient lists of properties.
Level 5: Rigour The fifth and final level is described by students being able to go between geometries Euclidean vs non-Euclidean and think outside of one axiomatic system Pusey, 2003. The student in this case associates a rectangle with other objects that are shaped like a rectangle, such as a door, window, and so on, from his previous encounters. Because of the inability to understand definitions which are based on the properties of the figures , the student at this level is not able to, for example, recognise a square as a special case of a rectangle. The following is a description of the Van Hiele levels: Level 1: Recognition Students at this level, which is also called visualisation, have the ability to learn the names of figures and view the figures as a whole according to their appearance only. That sounds a lot like wishful thinking, however.
The Van Hiele Student Development Levels for Geometry Learners
Detailed accounts and summaries of this early, but still highly relevant, work can be found in the following, e. Fixed sequence: the levels are hierarchical. Those who fall behind will only be able to memorize and scrape by. Problems may be more complex and require more free exploration to find solutions. When you're looking for a set of shapes, think about what might be tricky about an attribute For example right angles are harder to see if they aren't horizontal and vertical--if your sorting attributes include right angles, you want some tilted ones in your sorting set. Listen to the children so you know what they know and what they don't know yet.
At Level 2 a square is a special type of rectangle. These tasks will not have set procedures for solving them. Children at the visualization level can insepect specific examples of shapes and figure out what properties they have. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero. The five van Hiele levels are sometimes misunderstood to be descriptions of how students understand shape classification, but the levels actually describe the way that students reason about shapes and other geometric ideas. A teacher might say, "This is a rhombus. The shapes to be sorted should include shapes that might be problematic squares, rectangles, etc.
Challenges: The student may have difficulty progressing from one property to another. European researchers have found similar results for European students. Set diagrams are helpful for showing relationships between different kinds of shapes. She reported that by using this method she was able to raise students' levels from Level 0 to 1 in 20 lessons and from Level 1 to 2 in 50 lessons. Third grade Preassess: See comments on first grade. The discoveries are made as explicit as possible. A teacher might say, "This is a rhombus.
Properties of The Levels The van Hiele levels have five properties: 1. Ah, but there are no right angles, and opposite pairs of angles are congruent, so it must be a parallelogram. Deepening level 1 and moving to level 2 understanding: Guess my rule is an activity where children try to guess the rule that produced a particular group or sorting of shapes. They need a lot of practice with recognizing figures and reinforcement of the properties before they can begin to apply the more abstract concepts. Students cannot "skip" a level.
Students can reason with simple arguments about geometric figures. Give good examples of shapes to sort by given attributes. These tasks will not have set procedures for solving them. Thought Process: This set of angle measures is from an isosceles triangle because two of the measures are the same, and the sum is 180 degrees. Children at the analysis stage can not only identify whether a property is true of a particular shape children at level 0 can also do this , but can also identify whether a property is true of all of the shapes in a known group for instance--do all squares have line symmetry? The student does not understand the teacher, and the teacher does not understand how the student is reasoning, frequently concluding that the student's answers are simply "wrong".
Visualization is holistic: children identify shapes by recognizing that they look like examples they have seen. Later, children learn properties such as "having straight sides" or " having square corners", and use them to describe shapes. They will also be able to participate and understand informal deductive discussions about the shapes and their different characteristics. Properties are in fact related at the Analysis level, but students are not yet explicitly aware of the relationships. If children have trouble with tricky triangles or tilted squares or maybe even if they don't show misconceptions , make sure you include some tricky examples when you talk about shapes. A teacher might ask, "What happens when you cut out and fold the rhombus along a diagonal? Level 1: Analysis Description Students start to learn and identify parts of figures as well as see figures in a class of shapes.
