MATHEMATICAL REASONING

MATHEMATICAL REASONING

Reasoning is fundamental to knowing and doing mathematics. Reasoning enables children to make use of all their other mathematical skills and so reasoning could be thought of as the 'glue' which helps mathematics makes sense. There are various terms used to refer to "reasoning": critical thinking, higher-order thinking, logical reasoning, or simply reasoning. Different subject areas tend to use different terms.

Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. Mathematical Reasoners are able to reflect on solutions to problems and determine whether or not they make sense. They appreciate the pervasive use and power of reasoning as a part of mathematics.

Inductive reasoning involves looking for patterns and making generalizations. For example, students use this type of reasoning when they look at many different parallelograms, and try to list the characteristics they have in common. The reasoning process is enhanced by also considering figures that are not parallelograms and discussing how they are different.  Students may use inductive reasoning to discover patterns in multiplying by ten or a hundred or in working with exponents. Learning mathematics should involve a constant search for patterns, with students making educated guesses, testing them, and then making generalizations.

Deductive reasoning involves making a logical argument, drawing conclusions, and applying generalizations to specific situations. For example, once students have developed an understanding of "parallelogram," they apply that generalization to new figures to decide whether or not each is a parallelogram. This kind of reasoning also may involve eliminating unreasonable possibilities and justifying answers. Although students as young as first graders can recognize valid conclusions, the ability to use deductive reasoning improves as students grow older. More complex reasoning skills, such as recognizing invalid arguments, are appropriate at the secondary level.

STRUCTURE OF MATHEMATICAL REASONING

Mathematical reasoning involves more than just deduction. Mathematical theories are systematized by axioms and definitions in a way exemplified by Euclid in his famous compilation of geometric knowledge in the Elements. Euclid's model of a how to structure a mathematical theory still dominates today. Euclid divided his theory into four parts, each of which he gave explicitly:

- Definitions

- Common Notions (Logic)

- Postulates (Axioms)

- Theorems

The Definitions are supposed to clarify the concepts used in terms of primitives that are completely clear and familiar. The Common Notions are to provide rules of logic, that is, rules for making inferences which preserve truth. The Postulates, or Axioms, are the substance of the theory. They provide the sum total of all that one need assume in order to derive the rest of the theory, which is separated into Theorems.

DEFINITIONS

        Definition is a precise and unambiguous description of the meaning of a mathematical term . It characterizes the meaning of a word by giving all the properties and only those properties that must be true. In the book ‘Elements’ Euclid gave definitions for the phenomena observed with regards to solid objects and their parts. For example he defined ‘point’ as follows. Point is that which has no part. Line was defined as breadthless length. These definitions acted as the starting point for the logically bound structure.

AXIOM

The word "axiom" comes from the Greek word ‘axioma’, a verbal noun from the verb ‘axioein’ meaning "to deem worthy", but also "to require", which in turn comes from ‘axios’ meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof.

Axiom is a premise so evident as to be accepted as true without controversy. In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.

THEOREM

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.  

Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.