Piaget came to understand that the ability to conserve depended upon two more fundamental cognitive or thinking skills: Decentration and Reversibility.
Decentration involves the ability to pay attention to multiple attributes of an object or situation rather than being locked into attending to only a single attribute. When children are asked to compare the volume of juice in two glasses, it is their ability to decentrate that enables them to flexibly consider both the height and the width of the glasses in arriving at their decision. Younger children tend to get fixed on only one dimension or attribute of a situation, such as the height of a container, and to make their judgment of how much stuff can be fit into that container based on that single dimension. Other dimensions simply are not attended to. Through the development of decentration skills, older children start to be able to pay attention to more than one thing at at time.
You might think of decentration as being the basis for multitasking. It involves the development of attentional flexibility so that attention can be shifted from one aspect of a situation to another and then back again. It also involves the development of short term memory abilities to the point where memory of one aspect of a situation is able to be preserved while another aspect of that situation is also being attended to. This overlaying of memories enables children to develop an integrated perspective on what they are looking at.
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In everyday life, decentration helps children to advance their math and reading skills by making it possible for them to go beyond simple memorization of symbols and to begin to understand how symbols can be arranged to convey meaning. Decentrated children can look at words or math formulas and simultaneously consider the individual symbols (letters, numbers) these structures are composed of, as well as the overall meaning of the words or formulas. Decentration is fundamental to more advanced cognitive skills such as reading. Children who have not yet achieved decentration would not see and appreciate words when they look at written sentences; instead, they would more likely pay attention to the individual letters.
Reversibility takes conservation one step further. Children capable of conservation appreciate that an object's quality is not altered simply by transforming how that object appears. Children capable of reversibility appreciate that if an object's quality is altered through some true subtraction or addition, the object's original quality can be restored by reversing the alteration. This capability is enabled in large part through the maturation of children's memory so as to enable their retention of awareness of a series of events and their ability to run backwards through those remembered events so as to see how something transformed could be restored to its initial state. For example, children demonstrate conservation when they appreciate that if Mom takes four apples from a bowl in the middle of the table and puts one at each family member's plate, there are still four apples present. They demonstrate reversibility when they appreciate that when Mom puts one of the apples in her pocket that an actual transformation of the quantity of apples on the table has occurred, which can be reversed by Mom taking the apple back out of her pocket and returning it to the table. Alternatively, children demonstrate an appreciation of reversibility when they recommend that a scale that has been unbalanced by the placement of a weight can be restored to balance by the removal of that weight.