Ethnomathematical Issue in Mathematics Education:

Shiva Thapa,

M.Ed 2^nd Year, University Campus, Kritipur, Roll .no. 984

· Background/Introduction:

The term ethnomathematics was first coined by Brazilian mathematician Prof. Dr. Ubiratan D’Ambrosio( Sao Paulo, Brazil) however a German historian and philosopher Oswald Spengler (1980-1936) whose interest was also in mathematics and science had tried to write about mathematics in different cultural eras. Later on Prof. Ambrosio led the concept of ethnomathematics in logical end and established as an emerging issue of modern mathematics. Ubiratan D'Ambrosio in 1977 during a presentation for the American Association for the Advancement of Science presented the concept of ethnomathematics. How ever it was established as a valid research program in 1984 in the session of ICME. He presented various document between 1985 and 2001 about ethnomathematics.

· Meaning of Ethno-mathematics:

Prof. D'Ambrosio deconstructs the word ethnomathematics as: ethno + mathema + tics (techné) where ethno is natural, social, cultural and imaginary environment, mathema is explaining learning knowledge and tics is models, style, art and techniques. So ethnomathematics is combine form of mathematics, cultural anthropology and mathematical modeling. We know anthropology simply refers the holistic study of human being and cultural anthropology refers cultural variation among all human being of the world.

We know that mathematics is universe. It has its own world and own language. The way of learning mathematics is different in each individual due to their culture, society and environment. Different people of different cultural group have different way of doing simple mathematical operation, counting, estimating, calculation, measurement etc. We know that with out mathematical concept everything remains incomplete. Every social task is going on in the base of mathematics. A carpenter uses mathematics in his own style, a farmer uses mathematics in his farming, and a mechanic uses the mathematics in making and repairing the machine. In the practice of Nepal generally the teacher and student uses the numeration system in base 10 where as some specific cultural group of Nepal as well as some ethnic group of Mexico and United State of America uses the numeration system of base 20. There are various way of measurement used universally. We use Sexagesimal system (English System) measured in degree where as other uses French system called centesimal system measured in grade, and some uses radian or circular system. The rural area's people of Nepal are traditionally using their own tools of measurement called mana, pathi, dhak, taraju, hat, bitta, etc. So we can say that ethnomathematics is the mathematics practiced by different cultural group such as urban and rural communities, group of worker, professional classes, children in a given age group, indigenous societies, and so many group of the world. We have accepted Standard Unit in measurement but also we are practicing it in different manner. Such type of mathematical practice is called ethno-mathematics.

"The mathematics which is practiced among identifiable cultural groups such as national-tribe societies, labor groups, children of certain age brackets and professional classes" -D‘Ambrosio, 1985).

Simply we can say the ethnomathematics is:

Ø The way of mathematical representation in cultural aspects.

Ø Mathematical practice in indigenous groups.

Ø Mathematics for non mathematician.

Any concept of ethno-mathematics must eventually meet philosophical debates about the nature of mathematics. For more than two thousand years mathematics has been regarded as the epitome of rational truth, the study of the essential features of quantity, relationships and space. However the concept of ethno-mathematics has broken that conviction. The term ethnomathematics is used to express the relationship between culture and mathematics. The term requires a dynamic interpretation because it describes concepts that are themselves neither rigid nor singular—namely, ethno and mathematics (D’Ambrosio1987). The term ethno describes “all of the ingredients that make up the cultural identity of a group: language, codes, values, jargon, beliefs, food and address, habits, and physical traits.” Mathematics expresses a “broad view of mathematics which includes ciphering, arithmetic, classifying, ordering, inferring, and modeling”.

Prof. D'Ambrosio writes in his article entitled, The Program Ethnomathematics: A Theoretical Basis of the Dynamics of Intra – Cultural Encounters that the word Ethnomathematics may be misleading. It is often confused with ethnic-mathematics. I see ethno is a much broader concept, focusing on cultural and environmental identities.

· Development of Ethnomathematics as the issue in mathematics Education

v Overview of Ethnomathematical Issue from different international Convention:

ICMI( International Commission of Mathematical Instruction) is an international organization founded in 1908 to foster efforts to improve worldwide the quality of mathematics teaching and learning. Since 1969 it has been organizing the ICME( International Congress of Mathematics Education). It has completed its 11^th Congress and 12^th will be held on 2012 in Seoul, South Korea and the past conference of ICME are: 1969- France, 1972-UK, 1976-Germany, 1980-USA, 1984-Australiya, 1988-Hungary, 1992-Canada, 1996 Spain, 2000-Tokio, 2004-Copenhegun Denmark, 2008-maxico. From third ICME the issue of Ethnomathematics has been discussing how ever from 5^th Congress of ICME has given the shape of Ethomathematics formally.

How developed the concept of Ethomathematics?

Mathematics in culture and society, Why teach mathematics?? -1976 on third ICME, (Ubi D'Ambrosio)

Dared use of the word Ethomathematics. If there are ethnobiology, ethnomusicology, ethnopsychiatry, then why not Ethomathematics ? -1977 AAAS

Concept of Ethomathematics, Its etymological meaning-1978 ICM

Preparation of Backgroud for ethnomathematics/Holistic concept of curriculum, cycle of knowledge from reality to action -1780 ICME

Recognition of Ethomathematics as a valid research program-1984 ICME

ISGEm( International Study Group on Ethnomathematics) was launched at the 1985 meeting of the National Council of Teachers of Mathematics (NCTM) in San Antonio, Texas and ISGEm has been organizing the ICEm(Interantional Conference of Ethnomathemtics) in every four years.

Ø The First International Congress of Ethnomathematics (ICEm-1) took place in Granada, Spain, from 2 to 5 September 1998

Ø The Second international Conference of Ethnomathematics (ICEm-2) took place in Ouro Preto, Brazil.

Ø Third International Conference on Ethnomathematics (ICEm-3) Auckland, New Zealand 12-16 February, 2006.

Ø Forth international Conference of Ethnomathematics (ICEm-4) is going to held on July 25-30, 2010, Maryland USA. (http://pages.towson.edu/shirley/ICEM-4.htm)

v How did Prof Ambrosio initiate the concept of ethnomathematics? In his own word:

''I first started to look at mathematics of different cultural environments, back in the mid-seventies, when preparing the essay for ICME 3, on “Why teaches Mathematics?” My main focus was the relations between Mathematics and Society. My view of Mathematics was as the science that emerged from the Mediterranean basin and organized in antiquity, mainly by the Greeks, and of Society as communities, cultures and civilizations organized according to the model of urban, economic and social relations that emerged in post-feudal Europe, since the Late Middle Ages and Renaissance. Both, Mathematics and Society shared this historical background and became prevalent all over the World. In the Age of Enlightenment the philosophical foundations of both Mathematics and Society were well established. This foundation was supposed to guide my paper for ICME 3. But, after much travel in Brazil, in the Americas and Africa, I was, for some time, curious about traditional ways of dealing with numbers and forms, as well as with the presence of traditions in societal arrangements and in religious practices. These recognitions were responsible for the impressive development of anthropology in the 19th and 20th centuries, particularly of ethnography. Recognition of the possibility of different mathematics in different cultures and the relations of these different approaches to space and time within models of society and education were timidly suggested in the classics of anthropology. Then I came across the precious and pioneering book of Claudia Zaslavsky, Africa Counts. I became very interested in finding something similar in other regions of the World. Many years of working as a consultant with UNESCO and the Organization of American States favored my ideas. Thus I was courageous enough to deal with the theme of “Why Teach Mathematics?” taking into account these very broad factors. The paper was received with mixed feelings. How could someone question the Mathematics that was in curricula all over the World? The answer to “Why Teach Mathematics?” should stress how important was to teach that mathematics. It was correct to discuss new approaches of teaching and learning that mathematics. Cultural roots and social tensions had not much to do with that mathematics. Against the expected, I decided to open up reflections on mathematics education as related to cultural and social issues. It was a very controversial session during ICME 3. An important issue, in defense of that Mathematics, was, and continues to be, its cognitive value. They claim it is essential in developing cognitive abilities. This privileged position may be a myth, as suggested by current research in Artificial Intelligence, as well as recent advances in studies of the mind and the brain. Of course, traditional mathematics may be a very important intellectual exercise, the same as poetry, music and Ethnomathematics. After ICME 3, I was strongly motivated to find theoretical support for the views I had exposed in my paper. This led me to look for new approaches to the History and Philosophy of Mathematics. To emphasize Mathematics as related to culture and society, during the 1977 annual meeting of the AAAS in a session on Native American Science, I dared to use the word Ethnomathematics. In the meeting, many speakers were talking about Ethnobiology, Ethnopharmacology, Ethnopsychiatry and many others “ethno-disciplines.” Why not Ethnomathematics? I pronounced the word, to the surprise of many, just as a word to design a very broad idea. Obviously, the academic environment attending the meeting identified Ethnomathematics as ethnic-mathematics. Regrettably, many people still react this way. Clearly, this is not what I had in mind when I used the word. What I had in mind, which emerged in preparing the essay for ICME 3, was a broader view of the History and Philosophy of Mathematics, emphasizing its relation to culture and society. Then, during the ICM 78, in Helsinki, came the idea of the etymological exercise that led to ethno-mathema-tics, where the words were used with special meaning. Thus, ethno means cultural environment, mathema means teaching, understanding, explaining, and tics is used as a reminder of techné, the root for arts and techniques. Thus came the word Ethnomathematics, which I claim to be a conceptual word. In the formation of the word resides the concept of Ethnomathematics. Of course, the abusive and imprecise appropriation of etymology was criticized by some. But how can we propose the new, if we are afraid of criticism, even if sometimes acid? At ICME 5, in Adelaide, Australia, in 1984, these ideas were spelled out. Thus Ethnomathematics was recognized as a valid research program, with obvious pedagogical implications. All I reported above happened within the last 30 years. Now, Ethnomathematics is recognized and practiced all over the word. Much research, practical and theoretical, of a methodological or an ethnographic character, is reported. The number of journals, books, associations, congresses is growing. The International Study Group on Ethnomathematics/ISGEm was created in 1985, and later national study groups, such as the pioneering North American Study Group on Ethnomathematics, were created. Many classroom proposals are available, and a number of teacher training programs are offered all over the world. Criticism is still present, which is an indication of the growing presence of Ethnomathematics. Criticism helps ethnomathematicians to reflect upon their work. A question is always present. What about the future of Ethnomathematics? Is it a fading proposal? Is Ethnomathematics to be replaced by the variants Critical Mathematics Education, or Mathematics and Society, History and Pedagogy of Mathematics, or some branches of Psychology of Mathematics Education? Groups are organized with these labels. Ethnomathematics, the way I conceive it, is intrinsic to all of these, as well as to Pure and Applied Mathematics. I insist in claiming that the essence of the Program of Ethnomathematics is to understand how knowledge is generated, how it is organized and how it is diffused in different cultural environments. Once we recognize influence of culture in knowing and doing, we are within the scope of the Program of Ethnomathematics. This is clear when we understanding the etymological construction of the word. It is ethno+mathema+tics. It is much more than ethno+mathematics. This very subtle difference is deep in its meaning.''

Ethnomathematics and mathematics education

The word ethnomathematics is itself mathematics education. It is an absolutely essential key for Mathematics Education. Of course, "the way of doing" mathematics, which means the way of teaching and learning, it cannot be reduced unique and universal at least in the very early elementary levels of learning mathematics. In this stage there is no difference between "using mathematics" and "doing mathematics", in fact what we do in the early elementary levels of mathematical education is to explain and to understand in a mathematical language those simple operations which we use to manage the every-day-live: counting, estimating, calculating etc. Needless to say how native algorithms to perform these operations are culturally-dependent and, therefore, are different. That is why the (Ethno)-Mathematics becomes absolutely essential for mathematics education.

Prof. D'Ambrosio further writes in his article as mentioned above, The Program Ethnomathematics: A Theoretical Basis of the Dynamics of Intra – Cultural Encounters, that once he had presented Ethnomathematics as a Research Program in the History and Philosophy of Mathematics with Pedagogical Implications. From it we can conclude that his intention of describing about enthnomathematics was in application of mathematics in teaching field.

The introduction of Ethnomathematical approach by Ubiratan D’ Ambrosio affected and continues to affect in various ways mathematics education. The researchers of Ethnomathematics field, among others, are interested in: the nature of mathematics teaching/ learning, legalization of mathematical cognition, discovering and use of informal mathematics—that are met in everyday contexts—in teaching mathematics. The current trends of Ethnomathematics have categorized, by Renuka Vital and Ole Skovsmose (1997), in four main strands: They are 1) Pure anthropological approach 2) The historical anthropology approach 3) The socio-psychological approach 4) Ethnomathematics and mathematics education.

In this way ethnomathematics has been developed as the emerging issue in the universe of mathematics.

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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.