Logical and Lateral Thinking

In the study of mathematics, students frequently create and extend patterns and draw conclusions from those patterns. Patterns are not limited to numbers or shapes. It is the process of noticing relationships among elements of patterns that help students think critically. When students draw general conclusions from a limited set of observations, they are developing inductive reasoning skills (Jacobs, 1994).

As students extend their observations to apply to formal proof, they begin to discover that inductive reasoning has it limitations. At some point, their conjectures might start to "crash and burn" and this is the point at which frustration can occur. It's natural and expected for this frustration to occur.  The process of inductive reasoning is more important than always arriving at the correct solution immediately. By engaging in this process, students can see that there is often more than one method for achieving a solution. As a teacher you need to help your students understand that inductive reasoning is of great importance in suggesting conclusions in mathematics. However, to be certain that the conclusions are reliable, the method of deductive reasoning should be used (Jacobs, 1994).

Jacobs defines deductive reasoning as, "A method of drawing conclusions from facts we accept as true by using logic"(Jacobs, 1994, p.32). Most people think of the formal proofs that occur in high school Geometry and use theorems, postulates, axioms, and so on, as the ultimate application of deductive reasoning. There are, however, a number of different ways in which to plant the seeds for deductive thinking long before high school.

Examples of Deductive Reasoning: All polar bears are white. This bear is a polar bear therefore it is white. or All equilateral triangles have sides of equal length. This triangle is a equilateral triangle therefore it's sides are all the same length.

Deductive reasoning is based on information or facts that are known and it allows us to determin what is logically true. In contrast, inductive reasoning allows us to discover "what may be true" (Jacobs, 1994, p.32). In inductive reasoning, observations on a specific item or object are applied as generalizations to all similar objects.

Inductive Reasoning Examples: This iced tea is bitter --> All iced tea must be bitter or This triangle has equal sides --> All triangles have equal sides.

Inductive reasoning does not apply formal logic. When you arrive at a conclusion using inductive reasoning it cannot be proved with formal logic in the same way as a conclusion derived by deductive reasoning.

Students can learn these thinking skills from logic problems, puzzles, brain teasers, and lateral thinking problems. In this module's learning activities, you will review a few websites and resources that have many examples of these problems.