The general aim is to validate the cognitive competence evaluation model of the children in their first instructional levels designed by Gardner. We also pretend to identify students with possible intellectual exceptionality following the criteria established by Gardner and colleagues (1998a; Krechesky and Gardner, 1990) from the theory of Multiple Intelligences and by Castello and Batlle (1998). In general terms our study goes in two stages: The first, consist on the evolution of the cognitive competence of the students in their first instructional levels using the evaluation model designed by Gardner and colleagues (1998a, b & c). In this part of the research we will proceed to the evaluation of the psychometric activities proposed by Gardner for evaluating such competence and we will determine the structural validity of the Gardner's theory model. We as well will study the participants' cognitive characters, the existence of differences between groups of level and gender through differential and descriptive statistics.The second stage will be for the identification of exceptional students (precocious, gifted and talented) using the identification protocol proposed by Castelló (1998) and the multiple intelligences model proposed by Gardner, that has been widely analysed in the previous chapter; in this stage and once the evolution model goodness has been examined, we can determine the number of exceptional students and the kind of exceptionality they have, having support in the theory models stabilised by such authors and commented in the first chapter of this study.The research is done with 294 students (from 5 until 8 years old) belonging to three different schools of the provinces of Murcia and Alicante (Spain). The results show that the identification protocol takes us closer to the valuation of different typologies on which high ability is framed. We have to be cautious in the definition of extremely gifted and/or talent because as Castelló and Batlle (1998) point out we cannot confirm that a student is extremely gifted just because he/she has scored certain marks. Although we can conclude with the opposite that an insufficient delivery in a determined measuring situation shows that such person is not extremely gifted or talented. In general, we can get closer to the hypothesis that nothing opposes to that the students identified are extremely gifted or talented, but never demonstrate with full security such fact, as the underlining cognitive processes measuring is not made.As it is obvious due to the measure used in Castelló's model is from the structure and concept different to the evaluation scales proposed by Gardner there is neither agreement nor disagreement as the Kappa concordance index shows in the identification of high ability. Although we can conclude that can be used as complementary, as the Gardner's model allows obtaining relevant information on intellectual dimensions, something that the Castelló and Batlle model doesn't have, such dimensions refer to the naturalist, corporal and musical competences.