Reasoning is based on previous established facts. To establish a new fact or truth one has to put it on test of reasoning. If the new fact coincides with the previously established facts, it is called logical or rational. Logical reasoning is beyond subjectiveness. In the process of logical reasoning, we approach everything with a question mark in our mind. For each question we make a hypothesis and this hypothesis is tested empirically or theoretically with the help of previously proved or established truths or facts. In mathematical working we also move upwards by the process of reasoning.
From our observation of physical and social environment we form certain intuitive ideas or notions called postulates and axioms. These postulates and axioms are self-evident truths and need no further proof or explanation. Thus, postulates and axioms are assumed to be true without reasoning. But this does not mean that here we ignore the process of reasoning. Actually self-evident truths are beyond reasoning. That is why we can not assume any evidence to be true. Only those evidences can be assumed as true that could not be proved untrue or irrational by existing logical knowledge. Thus, postulates and axioms are bases of mathematics as-well-as of our process of logical reasoning. In mathematics we make several propositions and while proving a proposition we base our arguments on previously proved proposition. Thus, each proposition is supported by another proposition that has already been proved or established. Consequently if we go back one-by-one, we reach to a propositions that is based on postulates and axioms. Thus, in mathematics we always use the process of logical reasoning. Therefore, mathematics may be called as the science of logical reasoning.
In mathematics two types of reasoning is used. These prominent types of reasoning are:
- Inductive Reasoning and
- Deductive Reasoning
Inductive Reasoning: Generally human knowledge arises from observations and experiences. In the beginning mathematics also arises out of practical applications and it is mostly inductive and intuitive. When the statements or propositions are based on general observations and experiences, the reasoning is called inductive. From our observation we can get that some particular properties hold good in the sufficient number of cases and by this we may conclude that these properties will also hold in all other similar cases. This type of logical reasoning is called inductive reasoning. Here inductive reasoning. Here inductive means that a particular theme, theory, rule, formulae is induced from general experience or observation. Thus, in inductive reasoning we proceed from several particular examples or experiences to a general agreement. In mathematics this type of reasoning is very much used.
Deductive Reasoning: In deductive reasoning we proceed from general to a specific. This type of reasoning is based on self-evident truth, established facts, postulates and axioms etc. Here a particular statement or proposition is proved with the help of already established general rules. Therefore in deductive reasoning we proceed from a premise. We make several statements or propositions in our mind. This reasoning consists in comparing the statements and drawing a conclusion therefrom. Thus, here we deduce the solution or proof of a particular problem or statement on the basis of a general premise. In mathematics, the inductive reasoning is useful for beginners but, afterward mostly deductive reasoning is more fruitful. As-far-as the place of inductive and deductive is concerned,’ the following saying is good enough to clear it.
“Mathematics in the making is not a deductive science, it is an inductive, experimental science and guessing is the tool of mathematics. Mathematician like all other scientists, formulate their theories form bunches, analogies and simple examples. They are pretty confident that what they are trying to prove is correct, and in writing these, they use only the bulldozer of logical deduction”.
Whitehead has also emphasised the importance of deductive reasoning in mathematics by saying, “Mathematics in its widest sense is the development of all types of deductive reasoning.”
D’ Alembert says, “Geometry is a practical logic, because in it, rules of reasoning are applied in the most simple and sensible manner.
Pascal says, “Logic has borrowed the rules of geometry. The method of avoiding error is sought by everyone. The logicians profess to lead the way, the geometers alone reach it, and aside from their science there is no true demonstration.”
Geometry is a true demonstration of logic Mathematics is the only branch of knowledge, in which logical reasoning or logical laws are applied and the results can be verified by the method of logical reasoning.
W.C.D. Whetham- “Mathematics is but the higher development of Symbolic Logic.”
C.J. Keyser- “Symbolic Logic is Mathematics; Mathematics is Symbolic Logic.”
“The symbols and methods used in the investigations of the foundation of mathematics can be transferred to the study of logic. They help in the development and formulation of logical laws. In mathematics the symbol has got a meaning, e.g., a < b means ‘a’ is less than ‘b’. In logic, the meaning of this symbol has been extended. Let ‘a’ denote the class denoted by the cows and ‘b’ stand for the class denoted by the animals then a < b is easily interpreted to mean “a is included in b”, that is, all cows are animals.
For another example, take the symbol ‘x’. Let A denote the class. “Teachers’ and B the class, ‘Ladies.’ AXB may be interpreted to mean the class of persons who are both Teachers and Ladies.
‘Thus the meanings of mathematical symbols have been extended to represent the relationship of propositions in logic. The aims of the mathematician and those of the logician are practically the same.
What is Science of Teaching or Pedagogy?
Now question arises, what is the science of teaching? Does our teaching need some sort of science for it’s effective functioning and better output? In this connection, it is true that in our daily life we usually feel the need of the services of science for making the things simpler and more workable for our living, prosperity and existence. As a result we can save our time and energy in accomplishing out task and try to get maximum output with the help of minimum input. Therefore anything referred to as science of teaching is also expected to help a teacher in his task teaching in the same way as science helps in doing a task related to our day to day life and world of work. Hence by utilizing the services of a functionary named as science of teaching one must be able to reach in a quite effective way-be putting the-least input in terms of the men-material resources. As a conclusion the term science of teaching stands for the ways and means provided to or utilised by a teacher for managing his task of teaching as smoothly and effectively as possible by involving his least efforts for drawing the maximum possible better teaching outcomes.
Before we try to understand the concept of meaningful learning, it will be better here to know once again what learning stands for. The definition given by R.S. Woodworth, appears befitting, which runs as “An activity may be called learning in so far it develops the individual in any way good or bad and makes his environment and experiences different from what it would otherwise have been.”
Learning can produce both good and bad developments in the learner. But the learner, his guardians, his teachers and the society in general want the process of learning to lead to good results and healthy outcomes. That learning is to be avoided, curbed or replaced which is likely to prove harmful in any way. Not only informal learning, but even formal learning can be injurious to the learner and his well wishers in some way. Child comes to school for better, experiences, where the situations should not be worse for him. Every activity of the process of learning should be governed by certain aims and objectives. There cannot be anything of wasteful, useless, aimless or meaningless nature.
Meaningful learning is therefore that learning which is oriented towards good experiences and outcomes. In it there is no place for meaningless and harmful experiences. It must ensure positive results. It is constructive, productive, purposeful and progressive in nature.
Meaningful learning in mathematics can consist of the mathematical experiences of the following type:
- Which are helpful in mental, emotional and social development.
- Which have utilitarian, practical and behavioural values.
- Which are useful in learning higher and advanced aspects of the subject.
- Which are helpful in the proper learning of other subjects and activities of the curriculum.
- Which stimulate and maintain interest in the subject.
- Which lead to the development of proper attitude towards the subject.
Meaningful Learning in Mathematics
In the above sense the learning in the subject mathematics may be termed meaningful if it serves in the following ways:
- Helps in the developments of mental and intellectual powers.
- Helps in the learning of other subjects.
- Helps in building base for the future learning in mathematics.
- Helps in solving problems related to life activities.
- Helps in earning livelihood or prepares for useful and productive vocation.
- Helps in developing positive attitude towards the subject mathematics.
- Helps in inculcating interest or motivates the students for self-learning and independent research in mathematics.