Mathematical Thinking

What Is Mathematical Thinking?

Mathematics is a discipline with its own definition of valid knowledge and its own conventional signs, such as numbers, signs for operations (e.g., +, −, ÷, √), and signs for relations (e.g., =, ≠, >, <). People who are not mathematicians use ways of thinking considered valid in mathematics: One does not need to be a mathematician to know that if Andrew is taller than Jon, and Jon is taller than Peter, then Andrew is taller than Peter. The ability to combine two relations (such as taller than) to arrive at a conclusion is deductive reasoning, which is a typical form of logical reasoning used in mathematics. Logical reasoning can be found among children and adults who have not learned mathematics in school. Mathematical thinking is the use of logical reasoning in the domain of mathematical objects (e.g., quantities, sets, numbers, measures, and space) using mathematical signs. Because of the strong association between logic and mathematics, the expression logico-mathematical thinking is very common, but not all logical thinking is connected to mathematical objects (e.g., logic is also basic to scientific reasoning).

Mathematical signs are not exclusive to mathematicians: Many children and adults who never attended school learn to count and know how to use numbers to arrive at conclusions on the basis of numbers. For example, if they are told that someone had four things and lost one, they will know that the person now has three things. The word things is used here intentionally: It reveals the abstract nature of thinking about numbers. It indicates that, no matter what one is talking about, four minus one is three. This is so by definition: in a number system, the relations between numbers are defined by the system.

Mathematical thinking works with relations that are already known, and it seeks to establish relations through systematic investigation and to express relations using mathematical signs. For example, one might have cloth in three different colors and have four different icons that can be used to make flags; if each color is systematically combined with each icon, how many different flags could be created? Mathematical thinking in this context is characterized by the systematic approach used in creating the combinations and by the attempt to express the relation among the quantities numerically (the number of flags is equal to the number of background colors times the number of icons).

When people pursue assumptions to logical conclusions, think with abstractions, approach information systematically, and use numerical signs in rigorous ways, they are thinking mathematically. Many people engage in these sorts of activity in different contexts, not just when learning and using mathematics formally. Thus, to understand the development of mathematical thinking, one needs to consider how it develops informally, outside school, and how informal knowledge is changed by mathematics learned in school.

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