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
Deļ¬nition 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.
