4.1 The stages of learning as Dienes
The learning process is a process based on abstraction, generalization and communication. This process of abstraction is to accurately analyze and Dienes identifies six different stages in it:
Stage 1: introduces the individual in the middle => Game Free
2nd stage: review, manipulate, get rules => Structured Games
3rd stage: becoming aware of the common structure to games made
4th stage: representation of the common structure graphically or schematically => Stage representative
5th stage study of the properties of abstract structure, which implies the need to invent a language => Stage symbolic
6th stage: Construction of axioms and theorems => formal Stage
Her pedagogical approach is: to achieve the handling of a formal system to always start from reality.
Van Hiele Model in Geometry
The van Hiele, from the consideration of mathematics as an activity and the learning process as a process of reinvention characterizing formulated his theory a hierarchy of levels which facilitates an orderly transition possible teaching of geometry.
The model compares the inductive learning process and suggests 5 levels of knowledge in geometry outlined below:
Level 0: Visualization
* A geometric figure is seen as devoid of any components or attributes.
* A student at this level can learn geometric vocabulary, can identify specific shapes of a set of them and, given a set, you can play.
Level 1: Analysis
* The student will analyze in an informal manner the properties of figures collected through observation and experimentation processes.
The student is able to:
- See relationships between properties and between figures
- Develop or understand definitions
Level 2: Informal Deduction (ordination)
* The student:
. logically orders the properties of the concepts
. begins to build abstract definitions
. can go on and give informal arguments
. not understand the meaning of the deduction the role of the axioms
Level 3: Formal Deduction
At this level the student is able to build, not just memorize, demonstrations.
Level 4: Rigor The student can:
. compare different systems based on axiomatic
. explore different geometries in the absence of concrete models
This level is practically unreachable by high school students. The van Hiele claim that only respect for the hierarchy of levels allows a correct learning.
Regarding methodological aspects should be noted that:
. students progress through the levels in the order listed.
. if a level has not been sufficiently established prior to Jan. l next level instruction, students will work only at the highest level, so algorithmic.
The contents to be working in this area are essentially the following:
1.-The logical operations
. Zoning / Planning
And the first 4 lead us to the mathematical formalization.
The operation RANK is essential for the toddler begins to
properly structuring your mindset and to organize their thinking
Qualifying is organizing a given information criterion. And
classification in child's classroom understand the organization of collections of physical objects in terms of its attributes to analyze then YES that new objects belong and not belonging to that collection.
Child-level attributes are the physical characteristics. It is therefore necessary
many activities of description of objects by their physical characteristics to select after a fixed attribute and other objects for containing it.
Activity to begin sorting with 3 years based on the story of Tommy the Turkey
Defined collections begin forming a common attribute, and then passed to describe the elements that make up the collection. After the exercise do not, that is, given a collection to find the attribute that is defined and characterized. The following exercises in degree of difficulty are the collections of more than one attribute.
3.-The Management of collections management by quality criteria.
4.-The logical quantifiers work in the classroom through activities designed correctly the
meaning and use of quantifiers OR, AND, NOT.