Different Mathematical Conference:

IMU
IMU is an international non-governmental and non-profit scientific organization, with the purpose of promoting international cooperation in mathematics. It is a member of the International Council for Science (ICSU).
The objectives of the International Mathematical Union (IMU) are:
1. To promote international cooperation in mathematics
2. To support and assist the International Congress of Mathematicians and other international scientific meetings or conferences
3. To encourage and support other international mathematical activities considered likely to contribute to the development of mathematical science in any of its aspects, pure, applied, or educational
ICMI
Overview of ICMI
Devoted to the development of mathematical education at all levels, the International Commission on Mathematical Instruction (ICMI) is a commission of the International Mathematical Union (IMU), an international non-governmental and non-profit-making scientific organisation whose purpose is to promote international cooperation in mathematics.
Founded at the International Congress of Mathematicians held in Rome in 1908 with the initial mandate of analysing the similarities and differences in the secondary school teaching of mathematics among various countries, ICMI has considerably expanded its objectives and activities in the years since.
ICMI offers today a forum to promote reflection, collaboration, exchange and dissemination of ideas on the teaching and learning of mathematics, from primary to university level. The work of ICMI stimulates the growth, synthesis, and dissemination of new knowledge (research) and of resources for instruction (curricular materials, pedagogical methods, uses of technology, etc.).
The Commission aims at facilitating the transmission of information on all aspects of the theory and practice of contemporary mathematical education from an international perspective. Furthermore ICMI also has as an objective to provide a link between educational researchers, curriculum designers, educational policy makers, teachers of mathematics, mathematicians, mathematics educators and others interested in mathematical education around the world.
ICMI takes initiative in inaugurating appropriate activities, publications and other programmes designed to further the sound development of mathematical education at all levels and to secure public appreciation of mathematics. It is also charged with the conduct of IMU's activities on mathematical or scientific education. In the pursuit of its objectives, ICMI cooperates with various thematic and regional groups formed within or outside its own structure.
Among international organizations devoted to mathematics education, ICMI is distinctive because of its close ties with the professional communities of mathematicians and mathematical educators as well as itsbreadth – thematic, cultural and regional.
http://www.mathunion.org/icmi/about-icmi/overview-of-icmi/
ICMI as an Organisation
The International Commission on Mathematical Instruction was established at the Fourth International Congress of Mathematicians held in Rome in 1908. After interruptions of activity around the two World Wars, ICMI was reconstituted in 1952. It was a time when the international mathematical community was being reorganized and ICMI then became an official commission of the International Mathematical Union (IMU).
As a scientific union, IMU is a member of the International Council for Science (ICSU). This implies that through IMU, ICMI is to abide by the ICSU statutes, one of which (Statute 5) establishes the Principle of the Universality of Science, whose essential elements are non-discrimination and equity. Through this principle, ICSU affirms the right and freedom of scientists to participate without discrimination and on an equitable basis in legitimate scientific activities, whether they be conducted in a national, transnational or international context, regardless of their citizenship, religion, political stance, ethnic origin, sex, etc. Apart from observing IMU and ICSU general rules and principles, ICMI works with a large degree of autonomy.
The General Assembly of ICMI meets during the International Congresses on Mathematical Education (ICMEs), held every four years. This Assembly is responsible in particular for the election of the Executive Committee of ICMI, which includes the presiding officers of ICMI.
The General Assembly of IMU formally adopts ICMI's Terms of Reference as well as the procedures for the election of the Executive Committee of ICMI. Moreover the vast majority of the current funding of ICMI comes from an IMU subvention that is approved by the IMU GA. ICMI files an annual report of its activities and a financial report to the IMU Executive Committee for endorsement. Furthermore, ICMI files quadrennial reports at General Assembly meetings of both IMU and ICMI.
ICME
It is a conference organized by ICMI in every four year.
ICME-1 (1969) — Lyon (France)
Proceedings of the First International Congress on Mathematical Education. D. Reidel Publishing Company, 1969, 286 p.
Note: The content of this volume also appears in Educational Studies in Mathematics 2 (1969) pp. 135-418,
accessible on the web through SpringerLink.
ICME-2 (1972) — Exeter (UK)
A.G. Howson, ed., Developments in Mathematical Education. Proceedings of the Second International Congress on Mathematical Education. Cambridge University Press, 1973, 318 p.
ICME-3 (1976) — Karlsruhe (Germany)
Hermann Athen and Heinz, Kunle, eds. Proceedings of the Third International Congress on Mathematical Education. Zentralblatt für Didaktik der Mathematik, Karlsruhe, 1977, 398 p.
ICME-4 (1980) — Berkeley (USA)
Marilyn Zweng, Thomas Green, Jeremy Kilpatrick, Henry Pollak and Marilyn Suydam, eds., Proceedings of the Fourth International Congress on Mathematical Education. Birkhäuser, 1983, 725 p.
ICME-5 (1984) — Adelaide (Australia)
Marjorie Carss, ed., Proceedings of the Fifth International Congress on Mathematical Education. Birkhäuser 1986, 401 p.
ICME-6 (1988) — Budapest (Hungary)
Ann & Keith Hirst, eds., Proceedings of the Sixth International Congress on Mathematical Education. János Bolyai Mathematical Society, Budapest, 1988, 397 p.
ICME-7 (1992) — Québec (Canada)
Claude Gaulin, Bernard R. Hodgson, David H. Wheeler and John Egsgard, eds., Proceedings of the Seventh International Congress on Mathematical Education. Les Presses de l'Université Laval, Québec, 1994, 495 p.
David Robitaille, David H. Wheeler and Carolyn Kieran, eds., Selected Lectures from the Seventh International Congress on Mathematical Education. Les Presses de l'Université Laval, Québec, 1994, 370 p.
ICME-8 (1996) — Sevilla (Spain)
Claudi Alsina, José Maria Alvarez, Mogens Niss, Antonio Perez, Luis Rico and Anna Sfard, eds., Proceedings of the 8th International Congress on Mathematical Education. S.A.E.M. Thales, 1998, 539 p.
Claudi Alsina, José Maria Alvarez, Bernard Hodgson, Colette Laborde and Antonio Perez, eds., 8th International Congress on Mathematical Education. Selected Lectures. S.A.E.M. Thales, 1998, 485 p.
ICME-9 (2000) — Tokyo/Makuhari (Japan)
Hiroshi Fujita, Yoshihiko Hashimoto, Bernard R. Hodgson, Peng Yee Lee, Stephen Lerman and Toshio Sawada, eds., Proceedings of the Ninth International Congress on Mathematical Education. Kluwer Academic Publishers, 2004, 430 p. + CD.
ICME-10 (2004) — Copenhagen (Denmark)
Mogens Niss, ed., Proceedings of the Tenth International Congress on Mathematical Education. IMFUFA, Roskilde University, Roskilde, 2008, 559 p. + CD.
ICME-11 (2008) — Monterrey (México)
Proceedings to appear.

Issue of Mathematics education: Gender issue, Ethnomathematics and so on..........

Will be published soon.........

R.M Gange and his Learning Theory:

Robert Mills Gagne (August 21, 1916– April 28, 2002) was an American educational psychologist whose theory of instruction has provided a great number of valuable ideas to instructional designers, trainers, and teachers. It is really useful to everyone for their study. Gagne (1985) described the nature of an instructional theory as an attempt to relate the external Events of Instruction to the outcomes of learning by showing how these events lead to appropriate support or enhancement of internal learning processes. Gagne was, of course, not the first theorist to suggest that all learning is not alike, that learning might be analyzed into different types of learning. Various psychologist including carr, Melton, Tolman, had given some contribution in categorizing the types of learning however Gagne very strongly categorized the types of learning in hierarchical order. His types of learning were not his own creation. He began by reviewing learning theory and research, such as Watson, Thorndike, Tolman, Pavlov,kindler, Köhler, etc in determining the types of learning. Pavlov described about singal learning, thorndike described about stimulus response learning, skinner about chaining, underwood about verbal association, postman about multiple discrimination, kindler about concept learning, he himself defined about principle learning, katona about problem solving. So we can say that he was the learning psychologist who collected the important idea for learning. So his theory was no more his own creation however a important step in development of learning psychology. He described all types but the first, signal learning (classical conditioning), as having prerequisite relationships with one another. Gagne carefully referenced researchers that had examined these eight types of learning: They are as follows:

1) Signal Learning 2) Stimulus-Response Learning 3) Chaining 4) Verbal Association 5) Multiple Discrimination 6) Concept Learning 7) Principle Learning 8) Problem Solving

As previously explained Gagne's theory of instruction is commonly broken into three areas. The first of these areas that I will discuss is the taxonomy of learning outcomes. Gagne's taxonomy of learning outcomes is somewhat similar to Bloom's taxonomies of cognitive, affective, and psychomotor outcomes (some of these taxonomies were proposed by Bloom, but actually completed by others). Both Bloom and Gagne believed that it was important to break down humans' learned capabilities into categories or domains. Gagne's taxonomy consists of five categories of learning outcomes - verbal information, intellectual skills, cognitive strategies, attitudes, and motor skills. Gagne, Briggs, and Wager (1992) explain that each of the categories leads to a different class of human performance.

To tie Gagne's theory of instruction together, he formulated nine events of instruction. Which are 1) Gain attention 2) inform learning objective 3) stimulate recall of prior learning) 4) present the content 5) provide learning guidance 6) Elicit performance 7) provide feedback 8) Asses performance 9)Enhance performance and transfer to the job. When followed, these events are intended to promote the transfer of knowledge or information from perception through the stages of memory. Gagne bases his events of instruction on the cognitive information processing learning theory.

The way Gagne's theory is put into practice is as follows. First of all, the instructor determines the objectives of the instruction. These objectives must then be categorized into one of the five domains of learning outcomes. Each of the objectives must be stated in performance terms using one of the standard verbs (i.e. states, discriminates, classifies, etc.) associated with the particular learning outcome. The instructor then uses the conditions of learning for the particular learning outcome to determine the conditions necessary for learning. And finally, the events of instruction necessary to promote the internal process of learning are chosen and put into the lesson plan. The events in essence become the framework for the lesson plan or steps of instruction. Similarly the Gagne has identified the different phases of learning. They are 1) Apprehending phase 2) The acquisition phase 3) The storage phase 4) Retrieval phase. Robert Gagne in his book ‘On the conditions of Learning’, has given a taxonomy of learning types (Gagne, 1970 Chap.4). That he has arranged hierarchically. Robert Gagne suggests a hierarchical list of eight category of learning. The list is hierarchical in sense that it proceeds from very simple conditioning types of learning up to complex learning such as that involves in problem solving learning. Additionally the list is also hierarchical in that sense that lower level of learning is prerequisites of higher level of learning.

1) Signal learning: This is a type of associative learning that has been initially studied by Pavlov who has called it conditioned reflex. A subject that responds in a certain way (R) to a stimulus S1 is given two stimuli (S1 and S2) simultaneously. After sufficient number of repetitions he learns to give the response (R) to S2 even in the absence of S1. Much of the learning that we do without giving conscious thought is of this type. Much of the initial learning of early childhood is signal learning. As the dinner bell was the signal for the Pavlov's dog. So he responds to the signal as he would to the event. i.e. instead of salivating when dinner arrives the dog did so at the signal. It is characteristics of these types of learning that stimulus and response must be closely associated in time.

2) Stimulus-response learning: This is another type of associative learning that has been called trial and error learning by Thorndike. Skinner has used the term operant learning for it. It involves some goal or objective that the subject attempts to achieve. The process is essentially a successive approximation process. The initial efforts are almost random. The subject modifies his approach in every attempt. Each successful attempt is remembered while failed attempts are forgotten. The success rate improves with more attempts. A good example is a child learning to walk. Initially he falls down often. But with more attempts he is able to master the skill. It is different from signal learning but signal learning is the base of this learning so we say there is hierarchical dependency in lower types of learning of higher types of learning. The condition of stimulus response is that the learning is typically gradual. i) The repetition of association of stimulus and response is usually necessary. ii) The response become more precise as the repetition takes place.(This is what skinner called shaping) iii) There is reward or reinforcement for exhibiting the required response and there is no reward when the behavior is incorrect. It is second types of learning according as Gagne.

3) Chaining: Chaining is the process of establishing a sequential connection of a set of stimulus-response pairs for the purpose of attaining a particular goal. For example, the opening of a lock involves a number of simpler steps connected in a sequence (locate the key-hole -insert the key - turn the key clockwise - watch for lever unlocking - take off the lock). Successful chaining requires prior learning of each component response. Algorithms are generally such chaining sequences. The condition for the chaining is i) the individual's link must be established first. ii) Again time is the factor, the event in the chain must occur close together in time.

4) Verbal Association: Human beings have the ability to encode and express knowledge through sound patterns. Verbal association here refers to the most elementary kind of verbal behavior - learning of verbal associations (object « name) and verbal sequences (chains of verbal associations). Gagne says verbal association will be classified as only the sub verity of chaining but this chain should be verbal. Verbal association has some unique characteristics. Condition for verbal association: i) each link must be established previously the link in the individuals mind to make the association. ii) There should be response differentiation.

5) Multiple Discrimination Learning: Discrimination is the ability to distinguish between two or more stimulus objects or events. There are two different kinds of capabilities involved. The first is where the learner is able to make different responses to different members of a collection of stimulus events and objects. The second type involves the capability of the learner to respond in a single way to a collection of stimuli belonging to a single set. (This involves recognition of the defining rule for the set and responding accordingly. Condition for multiple discriminations are i) Stimulus/Response must already be established. ii) Interference from conflicting stimulus must be reduced in to minimum form. That is distinction must be emphasized.

6) Concept learning: Concept learning involves discrimination and classification of objects. We will distinguish between two types of concept learning: concrete and abstract. Concrete concepts are those that are formed through direct observation. For example, consider the edge of a table, the edge of a razor blade and the edge of a cliff. It is possible to formulate a rule that defines an edge. But the concept of edge is formed more easily through direct observation of several examples. A learner can respond to a set of stimulus objects in two ways – one by distinguishing among them and the other by putting them into a class and responding to any instance of that class in the same way. Both these types are examples of concept learning. The significance of concept learning is that it frees the learner from the control by specific stimuli. Condition for concept learning is i) The variety of stimulus must be presented so that conceptual property common to all can be discriminated. ii) The stimulus response must be established.

7) Principle (or rule) learning: In a formal sense rule is a chain of two or more concept. If the two angles in a triangle are equal then the side opposite to the angels is also equal. This may be distinguish from a simple verbal fact to be memorized its that if the rule is correctly learned than the learner will be able to apply it in all relevant situation. Some concepts are not concrete. They are based on rules that involve other concepts. So they have to be learnt through definition. Definitions are statements that express rules for classifying, i.e. rules that are applicable to any instance of a particular class. Definitions are used for objects as well as for relations. A salient feature of principle learning is that the learner cannot acquire the concept through memorizing its statements verbatim unless he knows the referential meanings of the component concepts. For example, 2H2 + O2 = 2H2O is meaningless unless you understand what the symbols H2, O2, and H2O represent and are familiar with the mole concept. The concept formation process is cumulative. It weaves the different objects into a semantic web. A simple chaining should be taken place as the condition of rule learning.

8) Problem solving: Problem solving, here, refers to something more than classroom mathematical drills. Also referred to as heuristics purpose. The process of problem solving is one in which the learner discovers a combination of previously learnt rules that can be applied to achieve a solution for a novel situation. The following sequence of events is typically involved in problem solving. (1) Presentation of the problem, (2) definition of the problem, (3) formulation of hypothesis, (4) verification of hypothesis. The learning outcome of problem solving is a higher order rule that becomes a part of the student’s repertory. According to Gagne, cognition and concept formation is a multi-layered phenomenon, each layer consisting of a particular learning type. Signal learning, Stimulus-response learning, Chaining, Verbal Association and Multiple Discrimination Learning are all pre requisites for the formation of concepts and the ability to solve problems. The process of concept formation involves all these eight processes. A very important point here is that if the learning has not been sufficiently accomplished at any level, then there are perceptible deterioration at all higher levels. The condition for problem solving is i) The learner must be able to identify the essential features of response that will be the solution. ii) Relevant rules should be used and recalled. iii) The recalled rules are combined so that a new rule emerges. In this way R.M Gagne has identified his each learning with hierarchical dependency on lower level of learning of higher level of learning.

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.