Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt. Generally, there are questions asked related to basic concepts, Facts and Formulae of the Mathematics.
To let you make the most of Mathematics section, we are providing important facts related to the Mathematics Pedagogy. At least 10-15 questions are asked from Mathematics Pedagogy in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.
The Van Hiele Model of Geometric Thinking
Van Hiele theory
The theory has three aspects: the existence of levels, the properties of the levels, and the progress from one level to the next level.
Van Hiele levels
According to the theory, there are five levels of thinking or understanding in geometry:
- Level 0 Visualization
- Level 1 Analysis
- Level 2 Abstraction
- Level 3 Deduction
- Level 4 Rigor
Level 0 Visualization (Basic visualization or Recognition)
At this level pupils use visual perception and nonverbal thinking. They recognize geometric figures by their shape as “a whole” and compare the figures with their prototypes or everyday things (“it looks like door”), categorize them (“it is / it is not a…”). They use simple language. They do not identify the properties of geometric figures.
Level 1 Analysis (Description)
At this level pupils (students) start analyzing and naming properties of geometric figures. They do not see relationships between properties, they think all properties are important (= there is no difference between necessary and sufficient properties). They do not see a need for proof of facts discovered empirically. They can measure, fold and cut paper, use geometric software etc.
Level 2 Abstraction (Informal deduction or Ordering or Relational)
At this level pupils or students perceive relationships between properties and figures. They create meaningful definitions. They are able to give simple arguments to justify their reasoning. They can draw logical maps and diagrams. They use sketches, grid paper, geometric SW.
Pierre van Hiele wrote: “My experience as a teacher of geometry convinces me that all too often, students have not yet achieved this level of informal deduction. Consequently, they are not successful in their study of the kind of geometry that Euclid created, which involves formal deduction.”
Level 3 Deduction (Formal deduction)
At this level students can give deductive geometric proofs. They are able to differentiate between necessary and sufficient conditions. They identify which properties are implied by others. They understand the role of definitions, theorems, axioms and proofs.
Level 4 Rigor
At this level students understand the way how mathematical systems are established. They are able to use all types of proofs. They comprehend Euclidean and non-Euclidean geometry. They are able to describe the effect of adding or removing an axiom on a given geometric system.
Properties of levels
The levels have five important characteristics:
Fixed sequence (order): A student cannot be at level N without having gone through level (N−1). Therefore, the student must go through the levels in order.
Adjacency: At each level, what was intrinsic in the preceding level becomes extrinsic in the current level.
Distinction: Each level has its own linguistic symbols and its own network of relationships connecting those symbols. The meaning of a linguistic symbol is more than its explicit definition; it includes the experiences which the speaker associates with the given symbol. What may be “correct” at one level is not necessarily correct at another level.
Separation: Two persons at different levels cannot understand each other. The teacher speaks a different “language” to the student at a lower level. The van Hieles thought this property was one of the main reasons for failure in geometry.
Attainment: The learning process leading to complete understanding at the next level has five phases – information, guided orientation, explanation, free orientation, integration, which are approximately not strictly sequential.
Five phases of the learning process
Van Hieles believed that cognitive progress in geometry can be accelerated by instruction. The progress from one level to the next one is more dependent upon instruction than on age or maturity. They gave clear explanations of how the teacher should proceed to guide students from one level to the next. However, this process takes tens of hours.
- Information or Inquiry: Students get the material and start discovering its structure. The teacher holds a conversation with the pupils, in well-known language symbols, in which the context he wants to use becomes clear. (A teacher might say: “This is a rhombus. Construct some more rhombi on your paper.”)
- Guided or directed orientation: Students deal with tasks which help them to explore implicit relationships. The teacher suggests activities that enable students to recognize the properties of the new concept. The relations belonging to the context are discovered and discussed. (A teacher might ask: “What happens when you cut out and fold the rhombus along a diagonal? Along the other diagonal?”)
- Explanation or Explication: Students formulate what they have discovered, and new terminology is introduced. They share their opinions on the relationships they have discovered in the activity. The teacher makes sure that the correct technical language is developed and used. The van Hieles thought it is more useful to learn terminology after students have had an opportunity to become familiar with the concept. (A teacher might say: “Here are the properties we have noticed and some associated terminology for the things you have discovered. Let us discuss what these mean: The diagonals lie on the lines of symmetry. There are two lines of symmetry. The opposite angles are congruent. The diagonals bisect the vertex angles.”)
- Free orientation: Students solve more complex tasks independently. It brings them to master the network of relationships in the material. They know the properties being studied, but they need to develop understanding of relationships in various situations. This type of activity is much more open-ended. (A teacher might say: “How could you construct a rhombus given only two of its sides?” and other problems for which students have not learned a fixed procedure.)
- Integration: Students summarize what they have learned and keep it in mind. The teacher should give to the students an overview of everything they have learned. It is important that the teacher does not present any new material during this phase, but only a summary of what has already been learned. (A teacher might say: “Here is a summary of what we have learned. Write this in your notebook and do these exercises for homework.”)
Pierre van Hiele wrote: “A definition of a concept is only possible if one knows, to some extent, the thing that is to be defined.”