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.

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.