A major part of the study of diagrammatic reasoning involves learning to identify and distinguish between two basic types of reasoning – inductive and deductive. The study of diagrammatic reasoning usually involves the study of the relationship between visual representations and verbally expressed ideas. A diagrammatic example might be used to illustrate the inductive form of reasoning. The study of diagrammatic reasoning is primarily focused on the recognition and generation of connections from one representation to another.
Graphal reasoning in its inductive form generally refers to the generation of connections from the existence of other representations in order to derive certain mathematical objects, such as prime numbers, or geometric objects, such as rectangles and circles. For example, if an individual sees that a particular pair of numbers is prime, she is then motivated to infer that there must be another set of pairs that are also prime. In order to generate these connections, a person must connect the pairs by means of graphical representation (such as a circle). By this process, an individual can construct the necessary connections.
Graphical diagrams are commonly used to generate connections in inductive reasoning. For example, if an individual sees that the number of digits following a particular digit is greater than four, he will be motivated to infer that a set of digits that include the number and its two adjacent digits is also greater. He can do this by connecting all of his visual representations of the digits that are adjacent to the given digit using a circle. The circle thus represents the connections between those representations.
On the other hand, when a diagrammatic reasoning process involves the generation of connections from verbal descriptions, an individual uses a diagrammatic description to help him generate the connections in a specific manner. A diagrammatical example can be used to demonstrate the inductive form of reasoning. If an individual sees a diagrammatic description that depicts a line from a particular set of objects to another set of objects, he will be able to infer that the second set contains other sets of objects that are directly connected to the first set.
A second type of diagrammatic reasoning involves the generation of connections from verbal descriptions to visual representations by making the connections between the verbal descriptions in order to derive mathematical objects or ideas. For example, if an individual sees a diagrammatic description that depicts the path of a train from a certain point to another point in order to learn how it is related to a particular point on the map, he is able to derive a curve from the representation that shows the train traveling across the map. The diagrammatic description therefore represents a path, and the path is then associated with the idea of a particular point.
A third type of diagrammatic reasoning involves the generation of connections from verbal descriptions to visual representations in order to derive geometric objects or ideas. The use of diagrammatic reasoning in this case includes the generation of relationships between visual representations and verbal descriptions to generate a pattern, such as the generation of a triangle from the visual representation of the train or a circle from the verbal description of a circle.
Graphical diagrams are useful for various types of study in order to learn to recognize and generate the most relevant connections in any given situation. This type of reasoning can also be used in the generation of connections between verbal descriptions and visual representations in the process of inductive reasoning.