What is Mathematics About?

(a) Introduction: Nature, Meaning and Definition of mathematics

Mathematics reveals hidden pattern that helps us to understand the world around us. Now much more than arithmetic, algebra and geometry mathematics today is a diverse discipline that deals with data, measurement and observation from science with inference, deduction and proof. Mathematics is an applied science. Many mathematicians focus their attention on solving problem that originates in the world of experience. Mathematics by nature is both pure, theoretical adventures of mind and a practically applied science. This dichotomy allows the theoretical mathematics to ''Do mathematics for mathematical sake'' and the applied mathematics to use mathematics as a tool to solve real problem'' Mathematics finds useful application in business, industry, music, politics, sport, medicine, agriculture, engineering and social and natural science. The result of mathematical theory and theorem are both significant and useful. Through its theorem mathematics offers science both a foundation of truth and a standard of certainty. In addition the theorem and theories offers distinctive models of thought which are versatile and powerful including modeling, abstraction, optimization, logical analysis, interference from data and use of symbols. Experience with mathematical models of thought builds mathematical power, a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to asses risk and to suggest alternatives. Mathematics empowers us to understand the world better. Many mathematicians have given their contribution in coming modern phase of mathematics. For example Euclid studied about geometry. Newton, Leibniz studied about calculus. Gauss, Joseph Fourier, Simeon Poisson, Augustine Louis Cauchy etc gave their contribution in Algebra and geometry. Some definitions given by the mathematics are presented below.

'' Mathematics is a way to settle in the mind a habit of reasoning." –Lock David Hilbert said, "Mathematics is nothing more than a game played according to certain simple rule with meaningless mark on paper." According as Russell, ''Mathematics may be defined as the subject in which we never know what we are talking about nor whether what we are saying is true." Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences." Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Wikipedia writes Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions". Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Although incorrectly considered part of mathematics by many, calculations and measurement are features of accountancy and arithmetic. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Being an open intellectual system, mathematics continued to develop, in fitful bursts, until the Renaissance, when mathematical innovations interacted with new scientific discoveries, leading to acceleration in research that continues to the present day. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Numerology is considered an application of mathematics by many but differs from mathematics in that it holds a mystical view of numbers. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered later.

There is strong relationship between science and mathematics. Science provides mathematics with interesting problem to investigate and mathematics provides science with powerful tools to use in analyzing the data. Also the mathematics is chief language of science. The symbolic language of mathematics has turned out to be externally valuable for expressing scientific idea unambiguously. We can list the nature of mathematics as follows:

Ø Mathematics is an inductive science

Ø Mathematics is a way of thinking

Ø Mathematics is an organized structure of knowledge

Ø Mathematics is an science and art both

Ø Mathematics is a language

Ø Mathematics is a study of patters.

As stated above mathematics can be categorized in to two parts.

(i) Pure Mathematics: 'Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigor, abstraction, and beauty. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering, and so on'-Wikipedia

(ii) Applied mathematics: 'Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments; the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modeling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data' -Wikipedia

(b) Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in our lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. There are mainly three types of philosophy of mathematics referred as logistic school of thought, intuitionist school of thought and formalist school of thought whose brief description are presented below:

(i) Logicism: Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic. All mathematical concepts are to be formulated in term of logical concept. All the theorem of mathematics is to be developed as theorems of logic. Gottlob Frege was the founder of logicism however Dedikind, Bertrand Russell, Whitehead have also given their contribution on school of logic. Every theorem is of the form logic. So logic is the fundamental bases of mathematics. Without logic mathematics is no more remains mathematics. Logicism holds that logic is the proper foundation of mathematics, and that all mathematical statements are necessary logical truths. For instance, the statement "If Socrates is a human, and every human is mortal, then Socrates is mortal" is a necessary logical truth. To the logicist, all mathematical statements are precisely of the same type; they are analytic truths, or tautologies. Logicism is just the claim that the theorems of mathematics are logically necessary or analytic. Logicism does not belive on formality, and mathematical discovery. If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects. But what, one might ask, are the objects of logical concepts? In this sense, logicism can be seen as shifting questions about the philosophy of mathematics to questions about logic without fully answering them.

(ii) Intuitionism: Intuitionism is the immediate apprehension with out intervention of any logical process of knowledge of mental perception. Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived through mental construction that proves it to be true and the communication between mathematicians only serves as a means to create the same mental process in different minds. According as intuitionist philosopher mathematics is the production of human mind. From this school of thought we can say that mathematic is to be built solely by finite constrictive method on the intuitively given sequence of natural number. Thus Brouwer's intuitionism stands apart from other philosophies of mathematics; it is based on the awareness of time and the conviction that mathematics is a creation of the free mind, and it therefore is neither Platonism nor formalism. It is a form of constructivism, but only so in the wider sense, since many constructivists do not accept all the principles that Brouwer believed to be true. So intuitionism is no more like other philosophy of mathematics. It believes on the inner capacity of learner. How the individual perceive the knowledge is important than what they learn. Learning is the product of individuals mind.

(iii) Formalism: The formalist thesis is that mathematics is concerned with formal symbolic systems. If fact, mathematics is regarded as a collection of such abstract development, in which the terms are mere symbols and the statement are formulas involving these symbols are the ultimate base of mathematics. Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. A major early proponent of formalism was David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics. Recently, some formalist mathematicians have proposed that all of our formal mathematical knowledge should be systematically encoded in computer-readable formats, so as to facilitate automated proof checking of mathematical proofs and the use of interactive theorem proving in the development of mathematical theories and computer software. Because of their close connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition (see below). It is usually said that formalist philosophy is the realist philosophy in present context. The intuitionism and logicism may not be sufficient of the learning of mathematics. To study about the geometry in class 10 the individual must study some basic knowledge about geometry in previous class. Every mathematical concept is in sequential form.

Short not on: (I) Reception vs. Discovery learning (II) Meaningful vs. Rote learning:

David P. Ausubel formulated a new theory called meaningful learning theory. He was a very well known mathematician in 1950's decade who propounded meaningful learning theory. In his time the lecture method was being so criticized however he emphasized on the same nature of learning. In his time lots of mathematician was arguing about problem solving and discovery learning in against of lecture method of learning. They was saying that lecture method of learning is cause of rote learning however Dr. Ausuble with some experiment said that rote learning is the product of problem solving method of teaching. He further says that some problems are given to the student; if they don't have any idea to solve it then they try to remember the way of solving the problem guided by their teacher. There are some ideas to make the learning meaningful but exposition/reception method is the first and fundamental base of that learning. We can study his theory into two parts. One is Reception vs. Discovery learning and other is meaningful vs. rote learning. Some short notes about these are presented below:

Reception vs. Discovery Learning:

It is a one way of studying the theory of David P. Ausuble. He is not totally in against of discovery learning however he focuses that reception learning is much more effective than discovery learning. In which situation the reception and discovery learning are effective? Read both of the scenarios found below and then answer the questions. Answers may be found following the questions.

Scenario A: Ram, a three and a half year old boy, is fascinated by the glowing red color of the burners on his parents stove. He has been repeatedly told not to touch the burners on the stove but doesn't seem to care. After having many of his attempts to touch the stove burners thwarted by his parents, Ram finally succeeds in touching a glowing red burner. His efforts earn him a severely burned finger and some valuable knowledge. "Never touch the burners on a stove, especially when their red."

Scenario B: An eighth grade science class is beginning a new section on astronomy and the solar system. They were instructed to read the chapter specifically dealing with this information outside of class while the teacher begins his/her verbal lectures over the material.

Questions

1 What type of learning is taking place in each scenario?

2 From scenario (A) we can see that Ram's parents did an inadequate job of teaching him the dangers of touching a hot burner. What was wrong with their approach? What could they have done better if anything at all?

3 Which scenario requires a later stage of cognitive development? Why?

4 Do the scenarios take place on an individual basis or do they require outside assistance?

5 If you were teaching a class about the solar system would you use "Discovery" or "Reception" learning? Why?

6 Which type of learning do you think Ausubel was more focused?

Response

1 For the most part, large bodies of subject matter are acquired through Reception learning, whereas everyday problems of living are solved through Discovery learning.

2 Reception learning, although phenomenological simpler than discovery learning, paradoxically emerges later developmentally and particularly in its more advanced and pure verbal forms, implying a high level of cognitive maturity.

Example: Using the above scenario (A), the child does not have to have any knowledge of concepts such as heat or the mechanical warnings of the stove to learn the lesson. On the contrary, in scenario (B), students must have the proper verbal communication abilities to comprehend the information taught by the teacher. (As children, we tend to learn by way of Discovery while as adults we learn through Reception.)

3 While Reception learning can only take place with outside assistance (teacher), Discovery learning is on a more individual basis.

4 Reception learning is usually a much more effective way of teaching in a classroom setting than Discovery learning.

Example: In scenario (B), it would take an incredible amount of time for the students to learn all of the information concerning Astronomy and our solar system if learned through Discovery.

5 Ausubel believed meaningful reception learning to be the best form of learning in a classroom. In fact, he did very little research concerning

F.H Bell writes in his book that, The main idea of Ausubel is reception instead of discovery. The distinction between reception and discovery is not difficult to understand. In reception learning the principle content of what is to be learned is presented to the learner in more or less final form. The learning doesn't contain any discovery in his part. He is required only to internalized the material or incorporate it in to his cognitive structure so that it is available for reproduction or other use at some future date. The essential future of discovery learning on the other hand is that the principle content of what is to be learned isn't given but must be discover by the learner before he can internalize it. Main principle of that learning is to discover something. But this may not be possible in all situations.

Meaningful vs. Rote learning:

Learning must be meaningful instead of rote memorization. Ausubel focused that learning will be meaningful through verbal exposition rather than problem solving. For example: 2x+4=8 is a given equation. The students of class five are asked to solve it. Then student will do 2x=8-4 or 2x=4 or x=2. Answer is correct but they mayn't know how the value of x becomes 2 and they try to remember that if we carry 4 into right hand side from left hand side of the equation then it will be negative. But reality is not like that. In that process they may not know that equation is equality. Also they may not know if we add equal quantity to both side of the equation then the resulting equation will also be true. So at the very first class of equation, if the teacher explains it clearly through verbal exposition then only student will understand clearly. So as a conclusion in Ausubel's view we can say that problem solving learning is caused of rote learning instead of verbal exposition.

Here also F.H Bell writes that the distinction between rote and meaningful learning is frequently confused with reception and discovery learning. Actually each distinction constitutes an entirely independent dimension of learning. Hence both reception and discovery learning can each be rote or meaningful depending condition under which learning occurs. Ausubel has observed that discovery learning and problem solving teaching techniques are result in rote learning. Just as poor expository teaching can cause student to memorize materials which has no meaning to them. When learning to solve statement problem to algebra, many student memorize problem types and set of rules for solving each types. Good expository teaching is only the best teaching for meaningful learning. The primary idea of Ausubel's theory is that learning of new knowledge is dependent on what is already known. In other words construction of knowledge begins with our observation and recognition of events. Ausubel's meaningful learning is concerned with how student learn large amount of meaningful materials from verbal/textual presentation is a school setting through which the meaningful learning occurs.

A critical comment on any one mathematics textbook of grade 8

Introduction:

The textbook is valuable when it is used properly so the textbook is designed in such a way that it can be used properly. Appraisal of textbook can be taken as the answer of what should be included in the textbook? What types of textbook should be there? Is it appropriate in academic, physical and psychological point of view?

For the appraisal of mathematics textbook, I have taken the textbook of grade 8 published by curriculum development centre (CDC), Ministry of Education (MOE) and printed by Janak Shiksha Samagri Kendra limited. The presented compulsory mathematics textbook of graded 8 had been written by Dr. Santoshman Makey, Harinarayan Upadhaya and Sunma Tuladhar in 2053 B.S and later on it has been modified by Dr. Siddhi Prasad Koirala, Bhojraj Sharma, Salikram Bhusal, Barun Prasad Baida, Indira Aryal and Nirmala Gautam with the current modified curriculum of Nepal. The first edition of that book was in 2053 B. S and the last edition of the book is of 2061 B.S. The appraisal of textbook from Academic, Physical and psychological point of views are presented below respectively.

Academic Aspects: In this aspect mostly the structure of content, organization of content, presentation, rigor, vocabulary, correctness, theorems and proofs, generalizations, illustrations, example, teaching methods and teaching materials, exercise and review, reference and index etc. are evaluated. The presented textbook of grade 8 has some strength however it may not be free from some shortcomings also. Saying about the strength of the book it is written by well qualified and experience teacher whose aims of writing the textbook is very genuine. The strength and weakness of the textbook are presented below as follows:

Strengths of the textbook:

Ø The language used in the textbook is very simple and clear.

Ø It is written in desired curriculum.

Ø Here are not so many examples so it has helped to the student to get adequate opportunity through initiative and independent effort.

Ø Here are the sequence and consistency in organization of the subject matter.

Ø The way of writing the textbook is interesting and comprehensive.

Ø Here is not any irrelevant subject matter.

Ø The textbook satisfies the demand of examinations also.

Ø The proof presented on the textbook is appropriate for the maturity of the student.

Ø The vocabulary used in the textbook is appropriate and suitable.

Ø The homework and assignment can easily be given from the textbook.

Ø The illustrations presented on the textbook are simple.

Ø The exercise and their answer are given on the textbook is simple.

Ø The exercise and their answers are given on the textbook appropriately.

Ø It enables to give the new knowledge to the student.

Ø The subject matter is presented in simple to complex form.

Ø The level wise organization of the subject matter is genuine.

Ø It is sufficient to give the fundamental bases for the next class.

Weaknesses of the textbook;

Ø The steps of solving the problem are not given sufficiently so that the normal student can't understand clearly the problems in some pages of the textbook. We know the textbook should be written in such a way that the normal student can understand the matter from self study also. Here are given some steps to solve the problem which can't be understood by students even the teacher may get the problem in understanding the matter. For example in the lesson 'Square root and cube root' the method of finding the square root from division method is very complex. The direction and steps given in the textbook are quite difficult and complex to understand. So there is less probability of understanding the problem and steps given in the textbook for the student of grade 8. Even the vocabulary given in the textbook is not suitable. We know the symbols and terms used in the textbook must be those which are popular so that they may have no confusion. All new term should be clearly and accurately defined but authors has not concerned in this unit.

Ø The subject error as well as printing mistake is not avoided carefully.

Ø There are not included few difficult exercise to challenge the most intelligent student.

Ø There are not sufficient provision for practice and review; it may not meet the demand of students' ability and interests.

Ø There are not sufficient suggestions to improve the study habit.

Ø It does not facilitate to use the inductive, analytic, heuristic, laboratory method.

Ø Here are presented the suitable proof of the theorem but it has not given any alternative proof of the student.

Physical Aspects:

It is an aspect of textbook which is commonly important as the academic aspects. The presented textbook of the grade 8 is suitable in looking, easy to carry and of around 200 pages whose price can be said moderately priced. Except this strength, here are given some more strength and weakness as follows:

Strengths of the textbook:

Ø The shape and size of the textbook is suitable for desired class.

Ø It is very easy to carry.

Ø It is easy to fold while studying.

Ø The printing of the book is suitable.

Ø The line spacing of the book is suitable.

Ø The letters used in the textbook is suitable.

Ø The figures sketch in the textbook is appropriate.

The weakness of the textbook:

Ø The cover of the book is not attractive.

Ø The block of the book is not printed properly.

Ø The quality of the book in not good.

Ø Only one color is used in the book.

Ø Very few figures are drawn in the textbook.

Ø The book is prepared keeping the intention to be cheap.

Psychological Aspect:

It is also an important aspect of textbook. The textbook should be appropriate form academic and physical aspect as well as psychological aspect. The textbook should be able to arouse the interest to the students. If the book is suitable from the physical and academic point of view the book will be suitable from psychological point of view otherwise it may not be suitable and appropriate. The presented text book of grade 8 is suitable from psychological point of view how ever it can't free from some weakness. If we can remove the weaknesses of physical and academic aspects definitively the weaknesses of psychological aspect will remove automatically. Comparatively the textbook of grade 8 is suitable with contemporary other books because it is written by well qualified, established and experienced author so as a conclusion we can say the book is appropriate psychological point of view.

Suggestions:

We have already said that the textbook is valuable when it is used properly. The teacher should not feel that his work is confined to transferring the contents of the textbook into the head of students. The textbook shouldn't be used as only source of instructional materials. It should be used as an aid in teaching not a substitute for teaching. Its place in teaching can only be real if the teacher supplements it by his oral exposition. So to be the good textbook the above knowledge should be followed carefully and intelligently meanwhile writing the textbook. Those matter and steps should not be put in the textbook which may arouse the confusion to the students. If we remove the weaknesses of the textbook mentioned in the physical, academic and psychological aspects then the text book will be appropriate and effective.