INTRODUCTION
The history of Mathematics, its emergence and evolution as the universal language of the Universe has progressed under the auspices of several unwitting custodians and inventors. From the earliest civilizations through the eras of the philosophers until the modern era, the development of mathematics has been synchronous with the development of Man. Man’s continued attempts to coax a whimsical Universe into supporting his existence, have led to the discoveries of various aspects of Mathematics. Plato’s old paradox goes thus: How will you set about looking for that thing, the nature of which
is totally unknown to you? Which among the things you did not know is the one you propose to look for? And if by chance you stumble upon it, how will you know that it is that thing since you are in ignorance of it? This line of thinking explains why why most of the discoveries in any field of human endeavour were accidental. accidental. Those who had epiphanies that that advanced mathematics mathematics did not previously set out to do so but in the course of pursuing something else, stumbled on things erstwhile unknown. According to Darwin’s theory of natural selection, a thing survives if it has the wherewithal to
compete in an an unforgiving unforgiving Universe and and wrest its right to live, from it.
Mathematical
discoveries and theories theories have only survived survived based on their utility to Man. Like any other other positive science, Man’s relationship with Mathematics is symbiotic and based on the usefulness of its products in Man’s own quest for survival.
Unlike other disciplines which have somewhat localized origins, the roots of mathematical enquiry are spread across the broad spectrum of intellectual pursuits thus revealing the dependence on and and relevance of Mathematics Mathematics to every sphere sphere of human existence. In this write-up, we will be examining an unlikely contributor to Mathematics – Ivan Petrovich Pavlov. Traditionally recognized as a physiologist and psychologist, his work proves the universality of Mathematics as his efforts have contributed in no mean way to the rarefied understanding of learning we have at our disposal today.
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PHILOSOPHICAL UNDERPINNINGS OF MATHEMATICS
The origin of mathematics is subject to argument. Whether the birth of mathematics was a random happening or induced by necessity duly contingent of other subjects, say for physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis. There are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Western philosophies of mathematics go as far back as Plato, who studied the ontological status of mathematical objects, and Aristotle, who studied logic and issues related to infinity (actual versus potential). Greek philosophy on mathematics was strongly influenced by their study of geometry. For example, at one time, the Greeks held the opinion that 1 (one) was not a number, but rather a unit of arbitrary length. A number was defined as a multitude. Therefore 3, for example, represented a certain multitude of units, and was thus not "truly" a number. At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: just as lines drawn in a geometric problem are measured in proportion to the first arbitrarily drawn line, so too are the numbers on a number line measured in proportional to the arbitrary first "number" or "one." These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the square root of two. Hippasus, a disciple of Pythagoras, showed that the diagonal of a unit square was incommensurable with its (unit-length) edge: in other words he proved there was no existing (rational) number that accurately depicts the proportion of the diagonal of the unit square to its edge. This caused a significant re-evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea. Simon Stevin was one of the first in Europe to challenge Greek ideas in the 16th century. Beginning with Leibniz, the focus Page | 2
shifted strongly to the relationship between mathematics and logic. This perspective dominated the philosophy of mathematics through the time of Frege and of Russell, but was brought into question by developments in the late 19th and early 20th century.
PSYCHOLOGISM Psychologism is a generic type of position in philosophy according to which psychology plays a central role in grounding or explaining some other, non-psychological type of fact or law. The most common types of psychologism are logical psychologism and mathematical psychologism. Logical psychologism is a position in logic (or the philosophy of logic) according to which logical laws and mathematical laws are grounded in, derived from or explained by psychological facts (or laws). Psychologism in the philosophy of mathematics is the position that mathematical concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws). John Stuart Mill seems to have been an advocate of a type of logical psychologism (although his rejection of a static ontology arguably makes his psychologism flexible enough to accommodate its detractors' criticisms), as were many nineteenth-century German logicians such as Sigwart and Erdmann as well as a number of psychologists, past and present: for example, Gustave Le Bon. Psychologism was famously criticized by Frege in his The Foundations of Arithmetic, and many of his works and essays, including his review of Husserl's Philosophy of Arithmetic. Edmund Husserl, in the first volume of his Logical Investigations, called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it. The "Prolegomena" is considered a more concise, fair, and thorough refutation of psychologism than the criticisms made by Frege, and also it is considered today by many as being a memorable refutation for its decisive blow to psychologism. Nevertheless, there remains an incontrovertible link between logic, mathematics and psychology.
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THE PSYCHOLOGY OF LEARNING
Psychology is a child of philosophy as are many other disciplines. The Ancient Greeks divided the branches of knowledge or philosophy into the trivium and the quadrivium. The quadrivium was considered superior to the trivium and it included Arithmetic, Geometry, Astronomy and Music. Having established the philosophical foundations of Mathematics, its link to Psychology can be found in the area of learning. It has been found that the only areas where age and psychological development have no bearing on ability and giftedness are in the areas of music and mathematics. In other words, genius children can outperform adults only in these areas which almost entirely compose the Greek quadrivium. This knowledge was discovered by psychologists inspired by the precedents of philosophy. The area of learning and memory is important to Mathematics in that it being a science must be replicable and in order to be replicable it must be learned and remembered. Educational psychology seeks to understand, predict and improve the determinants of learning and remembering in students of all disciplines. Due to the cultural assessment of Mathematics as a difficult subject, the findings of learning theorists gain added relevance to its assimilation and retention.
WHAT IS LEARNING? Learning is acquiring new or modifying existing knowledge, behaviors, skills, values, or preferences and may involve synthesizing different types of information. The ability to learn is possessed by humans, animals and some machines. Progress over time tends to follow learning curves. Human learning may occur as part of education, personal development, school or training. It may be goal-oriented and may be aided by motivation. The study of how learning occurs is part of neuropsychology, educational psychology, learning theory, and pedagogy. Learning may occur as a result of habituation or classical conditioning, seen in many animal species, or as a result of more complex activities such as play, seen only in relatively intelligent animals. Page | 4
Learning may occur consciously or without conscious awareness. There is evidence for human behavioral learning prenatally, in which habituation has been observed as early as 32 weeks into gestation, indicating that the central nervous system is sufficiently developed and primed for learning and memory to occur very early on in development.
LEARNING IN EDUCATION PSYCHOLOGY Educational psychology is the study of how humans learn in educational settings, the effectiveness of educational interventions, the psychology of teaching, and the social psychology of schools as organizations. Educational psychology is concerned with how students learn and develop, often focusing on subgroups such as gifted children and those subject to specific disabilities. Researchers and theorists are likely to be identified in the US and Canada as educational psychologists, whereas practitioners in schools or school-related settings are identified as school psychologists. Educational psychology can in part be understood through its relationship with other disciplines. It is informed primarily by psychology, bearing a relationship to that discipline analogous to the relationship between medicine and biology. Educational psychology in turn informs a wide range of specialties within educational studies, including instructional design, educational technology, curriculum development, organizational learning, special education and classroom management. Educational psychology both draws from and contributes to cognitive science and the learning sciences. To understand the characteristics of learners in childhood, adolescence, adulthood, and old age, educational psychology develops and applies theories of human development. Often represented as stages through which people pass as they mature, developmental theories describe changes in mental abilities (cognition), social roles, moral reasoning, and beliefs about the nature of knowledge.
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For example, educational psychologists have researched the instructional applicability of Jean Piaget's theory of development, according to which children mature through four stages of cognitive capability. Piaget hypothesized that children are not capable of abstract logical thought until they are older than about 11 years, and therefore younger children need to be taught using concrete objects and examples. Researchers have found that transitions, such as from concrete to abstract logical thought, do not occur at the same time in all domains. A child may be able to think abstractly about mathematics, but remain limited to concrete thought when reasoning about human relationships. Perhaps Piaget's most enduring contribution is his insight that people actively construct their understanding through a self-regulatory process. Two fundamental assumptions that underlie formal education systems are that students (a) retain knowledge and skills they acquire in school, and (b) can apply them in situations outside the classroom. But are these assumptions accurate? Research has found that, even when students report not using the knowledge acquired in school, a considerable portion is retained for many years and long-term retention is strongly dependent on the initial level of mastery. One study found that university students who took a child development course and attained high grades showed, when tested ten years later, average retention scores of about 30%, whereas those who obtained moderate or lower grades showed average retention scores of about 20%. There is much less consensus on the crucial question of how much knowledge acquired in school transfers to tasks encountered outside formal educational settings, and how such transfer occurs. Some psychologists claim that research evidence for this type of far transfer is scarce, while others claim there is abundant evidence of far transfer in specific domains. Several perspectives have been established within which the theories of learning used in educational psychology are formed and contested. One of such perspectives is behaviourism to which Ivan Pavlov belonged.
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IVAN PETROVICH PAVLOV (1849-1936)
Ivan Pavlov was born in Ryazan in the Central Federal District of Russia, where his father, Peter Dmitrievich Pavlov, was a village priest. He began his higher education as a student at the Ryazan Ecclesiastical Seminary, but then dropped out and enrolled at the University of Saint Petersburg to study the natural sciences and became a physiologist. In 1875 Pavlov completed his course with an outstanding record and received the degree of Candidate of Natural Sciences. However, impelled by his overwhelming interest in physiology, he decided to continue his studies and proceeded to the Academy of Medical Surgery. He received his doctorate in 1878 and completed the third course in 1879, again being awarded a gold medal. After a competitive examination, Pavlov won a fellowship at the Academy, and this together with his position as Director of the Physiological Laboratory at the clinic of the famous Russian clinician, S. P. Botkin, enabled him to continue his research work. In 1883 he presented his doctor's thesis on the subject of the centrifugal nerves of the heart. In this work he developed his idea of "nervism", using as example the intensifying nerve of the heart which he had discovered, and furthermore laid down the basic principles on the trophic function of the nervous system. In this as well as in other works, resulting mainly from his research in the laboratory at the Botkin clinic, Pavlov showed that there existed a basic pattern in the reflex regulation of the activity of the circulatory organs. Pavlov was invited to the Institute of Experimental Medicine in 1890 to organize and direct the Department of Physiology. Over a 45 year period, under his direction it became one of the most important centers of physiological research. In the 1890s, Pavlov was investigating the gastric function of dogs, and later children, by externalizing a salivary gland so he could collect, measure, and analyze the saliva and what response it had to food under different conditions. He noticed that the dogs tended to salivate before food was actually delivered to their mouths, and set out to investigate this "psychic secretion", as he called it.
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Pavlov was highly regarded by the Soviet government, and he was able to continue his research until he reached a considerable age.
PAVLOV’S WORK
Classical conditioning (also Pavlovian or respondent conditioning, Pavlovian reinforcement) is a form of conditioning and learning that was first demonstrated by Ivan Pavlov (1927). The typical procedure for inducing classical conditioning involves presentations of a neutral stimulus along with a stimulus of some significance, the "unconditional stimulus." The neutral stimulus could be any event that does not result in an overt behavioral response from the organism under investigation. Conversely, presentation of the significant stimulus necessarily evokes a natural response. Pavlov called these the unconditional stimulus (US) and unconditional response (UR), respectively. If the neutral stimulus is presented along with the unconditional stimulus, it would become a conditional stimulus (CS). Pavlov used the term conditional because he wanted to emphasize that learning required a dependent or conditional relationship between CS and US. If the CS and US always occur together and never alone, this perfect dependent relationship or pairing, causes the two stimuli to become associated and the organism produces a behavioral response to the CS. Pavlov called this the conditional response (CR). This theory was expanded upon by J.B. Watson in 1913 in his book Psychology as a Behaviourist views it . Pavlov’s work formed the basis for the school of behaviourism.
Operant conditioning, put forth by B.F. Skinner, was to take this principle a step further. It is a form of psychological learning during which an individual modifies the occurrence and form of its own behavior due to the association of the behavior with a stimulus. Operant conditioning is distinguished from classical conditioning (also called respondent conditioning) in that operant conditioning deals with the modification of "voluntary behavior" or operant behavior. Operant behavior "operates" on the environment and is maintained by its consequences, while classical conditioning deals with the conditioning of reflexive (reflex) behaviors which are elicited by Page | 8
antecedent conditions. Behaviors conditioned via a classical conditioning procedure are not maintained by consequences.
THE BEHAVIOURIST PERSPECTIVE TO EDUCATION Applied behavior analysis, a set of techniques based on the behavioral principles of operant conditioning, is effective in a range of educational settings. For example, teachers can alter student behavior by systematically rewarding students who follow classroom rules with praise, stars, or tokens exchangeable for sundry items. Despite the demonstrated efficacy of awards in changing behavior, their use in education has been criticized by proponents of selfdetermination theory, who claim that praise and other rewards undermine intrinsic motivation. There is evidence that tangible rewards decrease intrinsic motivation in specific situations, such as when the student already has a high level of intrinsic motivation to perform the goal behavior. But the results showing detrimental effects are counterbalanced by evidence that, in other situations, such as when rewards are given for attaining a gradually increasing standard of performance, rewards enhance intrinsic motivation. Many effective therapies have been based on the principles of applied behavior analysis, including pivotal response therapy which is used to treat autism spectrum disorders.
What has recent psychological research taught us about learning and how can we best apply these findings to improve teaching and enhance student learning?
Research in neuroscience, cognitive psychology, and neurobiology and behavior identified several key factors validated by empirical research.
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Engagement
What are the factors that enhance or inhibit involvement in learning? Engagement tends to decline if an activity is motivated by the promise of a reward (as opposed to an intrinsic motivation, such as a desire to increase one’s competence). Motivation is also reduced if individuals engage in more than one activity at a time, or if they attribute their failure to a lack of ability (rather than a lack of effort).
Emotional Factors Affecting Cognition
Learners have distinct styles that influence learning. Especially important is whether a student has a prevention and promotion focus. A student with a prevention focus is especially sensitive to negative outcomes, seeks to avoid errors, and is driven by security concerns, while a student with a promotion focus is more sensitive to positive outcomes. Learning is enhanced when there is regulatory fit, when fit when the manner of in which a student engages in an activity sustains their goal orientation or interests regarding that activity.
Grounded Cognition
Learning, memory, and reasoning are enhanced when students have the opportunity to perceive and interact with real-world examples. Thus, simulations and problem solving activities can play a valuable role in promoting understanding and recall.
Mental Modeling
A mental model is a representation or a conceptualization of a larger reality which allows an individual to readily acquire, code, store, recall, and decode information. By allowing an Page | 10
individual to structure knowledge, mental models play a crucial role in cognition, recall, learning, and decision making.
The Zone of Proximal Development
The early 20th century developmental psychologist Lev Vygotsky wrote about “the zone of proximal development,” a phrase that refers to the level of understanding that a student can reach with a teacher’s help. Thus, an instructor seeks to stretch and broaden a student’s understanding (i.e. scaffold) by identifying those areas that are within a student’s grasp: not too easy, but also not too difficult.
Repeated Testing
Testing can be a valuable learning tool. It can focus on evaluation, or it can be used in other ways: to motivate study, consolidate learning, combat overconfidence, and assist students in monitoring their own understanding. Testing enhances long-term memory and helps students retrieve and apply knowledge.
Spacing Recent research has demonstrated that a student’s ability to remember, retrieve, and utilize
information is greater when an instructor’s presentations of difficult material are spread out over time rather than concentrated intensively.
Generation Effect
Studies have shown that when students generate their own answers to a problem, their mastery of a topic is greater than when an instructor shows them how to solve a problem. Page | 11
Metacognition Metacognition refers to one’s self -awareness of one’s own thought processes. It also involves
the ability to monitor comprehension and accurately evaluat e one’s learning. Metacognition helps students avoid distractions, sustain effort, and modify their learning strategies based on their awareness of the strategies’ effectiveness. Strategies for encouraging metacognition include having students: • Ask reflective questions; • Recount their thought processes as they attempt to solve a problem; and • Make graphic representations of their thoughts and knowledge (e.g. concept maps, flow
charts, semantic webs).
The above are higher order legacies of the work of Ivan Pavlov. Their significance is most noticeable in the teaching of Mathematics.
CONCLUSION: PUTTING IT ALL TOGETHER Ivan Pavlov’s dalliance with Mathematics was brief during his time at the University of Saint
Petersburg.
However, his contributions to psychology and invariably learning transcend
intellectual boundaries. The serendipitous manner in which he made his first discovery in conditioning is consistent with the manner in which other contributors to science chanced upon epiphanies; e.g. Newton’s apple and the discovery of gravity. Despite the artificial separations between fields of study, all knowledge is truly one unified whole and none can be exhaustively considered in isolation from others. According to Pythagoras, “Mathematics is the language of the Universe”. Ivan Pavlov and what he represents, is a signpost on Man’s journey to self -
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BIBLIOGRAPHY
www.wikipedia.com www.encyclopediabritannica.com www.microsoftencarta.com Richard R. Skemp, The Psychology of learning Mathematics, Routledge, (2011) Waerden, B. L. Van Der, Science Awakening, New York, Oxford University Press, (1961) Holt, John, How Children Learn UK: Penguin Books. (1983)
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