**Concrete Operational Stage**

Characteristic |
Description |
Example |

Conservation | Concrete operational children recognize that certain physical characteristics of objects remain the same even when their outward appearance changes. | After spilling 10 pennies stacked on her desk, Lizzie bent down to search for them. "I know there has to be ten," she said to herself, "because that's how many I put in that little pile on my desk yesterday." |

Decentration | Concrete operational children coordinate several important features of a task rather than centering on only the perceptually dominant one. | After getting two glasses of lemonade from the kitchen, one for her brother and one for herself, Lizzie remarked, "Don't worry, I gave you just as much. My glass is tall but thin. "Yours is short but wide." |

Reversibility | Concrete operational children can think through the steps in a problem and then go backward, returning to the starting point. | Lizzie understands that addition and subtraction are reversible operations. In other words, when you add 7 plus 8 to get 15, then this tells you that 15 minus 8 must be 7. |

Hierarchical classification |
Concrete operational children can flexibly group and regroup objects into hierarchies of classes and subclasses. | Lizzie discussed how to display her rock collection with her friend Marina. Marina suggested, "You could divide them up by color. Or, you could use shape and color." |

Seriation | Concrete operational children are guided by an overall plan when arranging items in a series. | Lizzie decided to arrange her rocks by size. She quickly lined up all 20 rocks in a row, selecting the smallest and then the next smallest from the pile, until the arrangement was complete. |

Transitive inference | Concrete operational children can seriate mentally. After comparing A with B and B with C, they can infer the relationship between A and C. | "I saw Tina's new lunch box, and it's bigger than mine," Marina said while eating her sandwich with Lizzie one day. "Well, it must be bigger than mine too, because look - my box isn't even as big as yours," said Lizzie. |

Spatial operations | Concrete operational children conserve distance; understand the relations among distance, time, and speed; and create organized cognitive maps of familiar environments. | Lizzie realizes that a truck blocking the sidewalk does not change the distance to the end of her street. She also knows that if she runs faster than Marina for the same amount of time, she will travel farther. In addition, she can draw a map that depicts the route from her house to Marina's house with major landmarks along the way. |

Horizontal decalage | Logical concepts are mastered gradually over the course of middle childhood. | Conservation of number and liquid are mastered before conservation of area and weight. |