Most probably you are familiar with Piaget's final stage of cognitive development i.e. the Formal Operational Stage. According to Piaget, the stages of cognitive development come to a close with the attainment of this stage which consists in being capable of the 16 binary operations of propositional logic such as that of Russel and Whitehead. Atomic propositons are combined into molecular propostions in terms of conjuntion disjuction and negation and when all combinatory possiblities are considered exhaustively by the child and hence is able to identify logical contradictions in his arguments , he is said to be formal operational. The onset age for the Genevan children he studied was around 12 to 13. When several years ago I investigated the same among secondary school children in Malaysia I found that a large percentage  children even  around the ages of 16 or so were not  Formal Operational. Furtheremore I also noticed that Abstract Thinking as such is DIFFERENT from formal operational thinking and is more INCLUSIVE.  I found that it is children who are abstract thinkers who are good learners and a large percentage of them end up in the institutions of higher learning.

NBT helped me considerably in the systematic investigation of such abstract thinking. Here it is necessary to distinguish TWO stages.
 

1. Early adolescence say children around the ages 13 to 15.
 

2. Late adolescence say children around the ages 15 to 18 or so.

The presence for the capacity for abstract thinking among the early adolescence is shown by Baums of the following description:
pillar-like trunk opening up at both ends and absolutely clean  i.e. without any strokes or shading inside and outside. There is globular envelope covering completely the open ended top of the trunk. Inside the crown are ball-like fruits just globular with stick-like stems but just hanging in the air.

For the late adolescence it is somewhat different and more differentiated. We have the same clean trunk opening up at the bottom but at the top now we have radiating branches each ending with its own cloud-like envelopes!

These Baums were found to occur  mostly in A and B; very rarely in C and D.

I have named them " KalaivaaNith tharu" .i.e. Baums indicating the presence of Goddess of learning , KalaivaaNi or Saraswathi.  The whiteness and clarity is common to both; KalaivaaNi is always seen to wear WHITE  garments, sit on WHITE  lotus  and the vehicle is annappaRavai, also Pure White in colour.

I noticed  the following intellectual capacities among such children no matter boys or girls.

1. Normally such chidren are described by teachers as 'BRIGHT"( notice the mataphor here) and QUICK to understand whatever presented in the class.

2. There is penetrative GAZE and along with it a PERCEPTUAL ACUMEN which I think is the central cognitive factor in abstract thinking. They listen attentively , digest quite fast whatever presented and extract the ESSENCE from it all and lodge them in their memory all in good speed so that they can FOLLOW the teacher and provide the correct answers when demanded of them by the teacher.

3. They may not spend much time pouring over books but spend considerable amount of time on THINKING about the lessons which they may do even while playing.

4. Not only they learn in a systematic and organised way but also are self-disciplined and well behaved. They have more control over their emotions than  an average child  of their age group.

5. On the whole they are inclined in the direction of science and mathematics and when streaming  into the arts and sciences begins , as it does in Malaysia around the age 16 or so, they tend to be put in the science stream though a small number also end up in the arts.

It appears to be that the essence of abstract thinking consists in being able to establish a psychic fissure between self and the sensory stimulations that one gets bombarded with always. It is this  fissure that enables them to SEE objectively whatever that is presented in the class and EXTRACT the essence and throw away the nonessential. In this way they manage to put more information into their long term memory compared to children who are incapable of this and hence unable to RETAIN much of what is presented during instruction.

It appears to be me, on the basis of such studies, in order to make a student a succesful learner we have to train his perceptual acumen, the way he SEES things. The development of intelligence belongs to the phenomenology of perception, of making the child SEE with a penetrating GAZE. This has interesting implications for educational technology that I shall not venture into at the moment.