A set of concepts, methods and procedures to clarify ways of teaching and learning understanding and to define criteria for credibility in a perspective adopted research is called theoretical framework. A theoretical framework is a dynamic structure comprising: theoretical analysis, design and practice, gathering and analyzing data. The theoretical analysis presents conceptions about what it means understanding a concept and as such understanding is achieved by student. Design and practice presents how choose a task and how to organizing the practice teaching based on theoretical analysis, but it accepts not previous procedures or ideas. Gathering and analyzing data related the data emerged from practice to that present in the theoretical analysis using qualitative, quantitative or both techniques research. The refinement of the data observed require, in a dynamics way, return and review the other components of the framework.

Conceptions of Theoretical Analysis in T.L.P.S.
Problem: task that involves procedures of puzzles and variety of approaches; runs up not only with elements already known.
Understanding of Mathematics: understanding is to be able to relate a conception presents in a mathematical problem with several other implied therein.
Attitude mathematics: an individual demonstrates and is able to bring into practice skills related to knowledge and learning of mathematics if he responds to perceived situations in a task acting such as: curious to know what he does not know, doing attempts, using known and alternatives techniques, being persistent, identifying patterns, thinking about what and why he does to get a task answer, doing comparison of, organizing ideas and formalizing concepts.
Goals of learning: The TLPS’s goals are to achieve knowledge, to develop positive mathematical attitudes, to engage the learning process

Conceptions of Design and Practice:
A Problem: it is the starting point for new concepts; it is planned so that the resolution will lead to: new ideas from empirical knowledge; recalled peripherals subjects; justify why studying the concept; create first conceptions.
Instruction: A problem situation is placed. Groups of students discuss, write down and present their solutions for the entire class without interference of the teacher. The teacher guides the analysis and synthesis of the resolution’s common points and differences. The latest stage of the study is formalization.

Conceptions of Gathering and Analyzing Data:
Spontaneous Vision: a open or not open problem triggers a process of cognitive constructions, it raises spontaneous notions, it promotes trial and error activities to build
understanding ideas.
Learning Evidence: do the student’s actions are related to mathematical attitudes? Which goals were achieved?
Teaching Evidence: which aspects of validation and acceptance of the instruction
directions emerge from teacher’s practices?
Analysis and Synthesis: they support the review of the practices, the tasks selection and the necessary basic knowledge.