Monday 23 July 2012

unit - 1 Inter-disciplinary Approach


UNIT- I
NATURE, CHARACTERISTICS AND OF MATHEMATICS
Meaning of mathematics
Mathematics is commonly defined as the study of patterns of structure, chance, and space; more informally, one might say it is the study of figures and numbers. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in philosophy of mathematics.
Development of mathematics
The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d 3d millennia b.c., it was used for surveying and menstruation; estimates of the value of pi are found in both locations.
Characteristics of mathematics
Children may exhibit feelings in insecurity, as well as fears of failure, punishment, ridicule, or stigmatizing labels. Children with math anxiety may also have a negative attitude or negative emotional reaction to math. Teachers need to provide students with experiences that they will be successful in, in order to promote a more positive attitude. They learning bridge or strategies are good ways to help prevent the early development of math anxiety.
Logical sequence in mathematics
The problem is predicting the next term of a partially specified sequence. The user shall input the rest few terms of a mathematical sequence. The expert system will rest try you understand the pattern and using the found pattern it would predict the next term.
Integer sequences
Integer sequences are the most commonly seen sequences. For integer sequences, it’s the addition, subtraction and multiplication operators that play the major role in Xing up in the function f. so, in order to discover the function f, we need to perform various operations on the integers that are the first few given terms of the sequence.
For example, consider the sequence 3;7;11;15;…… the way this sequence is understood is by taking the deference between adjacent terms of the sequences.


Structure of mathematics
The focus of my presentation will be on such structural aspects of mathematics that are known or likely to cause problems or challenges for the process of learning mathematics, and hence for its teaching. I shall interpret the term ―the structure of mathematics in a somewhat broad sense, by taking into account also the nature of mathematics and its characteristics as a discipline and not solely its architectural features as reflected in one or more possible construction(s) of edifice.
Abstraction
Mathematical thinking often begins with the process of abstraction-that is, noticing a similarity between two or more objects or events. Aspects that they have in common, whether concrete or hypothetical can be represented by symbols such as numbers, letters, other marks, diagrams, geometrical constructions, or even words.
Whole numbers are abstractions that represent the size of sets of things and events or the order of things within a set. The circle as a concept is an abstraction derived from human faces, flowers, wheels, or spreading ripples; the letter a may be an abstraction for the surface area of objects of any shape, for the acceleration of all moving objects, or for all objects process of addition, whether one is adding apples or oranges, hours, or miles per hour.

Symbolism of mathematics
The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle-they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.
The symbolism of mathematics is in truth the outcome of the general ideas which dominate the science.
Mathematics as science of measurement
Mathematics and science education, including the metric system of measurement, will be strengthened throughout the system, especially in the early grades.
Strategies
1. Implement the Missouri academic performance standards and frameworks for math and science.
2. Develop and implement statewide assessments aligned to the state’s content, performance, and skills standards.
3. Expand active learning opportunities through the use of technology.

4. Evaluate and disseminate effective math and science programmers.

Objectives
The number of teachers with a substantive background in mathematics and science, including the metric system of measurement, will increase by 50 percent.
Strategies
1. Ensure authentic assessment training for all teachers.
2. Increase the availability of math and science professional development activities aligned with the state’s knowledge, performance, and skill standards.
3. Institute competency-based teacher certification

Mathematics and its relationship
Mathematics with arts
The arts and mathematics involve students understanding of relationships between time and space, rhythm and line through the experience of these abstract concepts in various arts forms and mathematical ideas. Mathematically related aesthetic considerations, such as the golden ratio, are used across visual, performing and multi-modal arts forms.
Mathematics with civics and citizenship
The concepts developed in the study of mathematics are applicable to a range of civic and citizenship understandings. Mathematical structure and working play essential roles in key aspects of our society as well as key civics concepts. Particular aspects of civics and citizenship require mathematical understanding, including concepts of majority rule, absolute majority, one vote one value representation and proportional voting systems.
Mathematics with communication
Mathematics structure and working mathematically play essential roles in understanding natural and human worlds. Developments of the languages of mathematics are crucial to its practical application. Students learn to use the language and concepts of mathematics both within the discipline itself and also its applications to modeling and problem solving across the other domains. In this process they employee a range of communication tools for illustrating relationships and displaying results such as Venn diagrams and tree diagrams.
Mathematics with English
Mathematics, including the use of conjectures and proof, has clear links to the development of structures and coherent argument in speaking writing. Mathematical structure is strongly related to semantics syntax and language and to the use of propositions and quantifiers embedded in principled argument in natural languages.
Mathematics with health and physical education
In health and physical education, mathematics provides tools and procedures which can be used to model situations and solve problems in areas such as:
1. Scoring different sporting events involving time distance, weight and number as variables.
2. Calculating percentage improvement in results from data collected through fitness testing or performance in physical activities.

Mathematics with humanities-economics
The economics and mathematics are related through the use of mathematics to model a broad range of economic, political and social phenomena. Examples include the use of statistical modeling and analysis in a census, sampling populations to predict election outcomes, and modeling and forecasting economic indicators such as the consumer price index and business confidence.
Mathematics with geography
The application of mathematical skills plays a key role in financial literacy, in particular the use of ratio, proportion and percentage in related calculations such as percentage increase or decrease in price of a commodity or personal income. Mathematics provides the basis of measurement, scale and spatial representation used in maps and plans. Geography also uses the concepts of direction, length, angle and bearing, gradient and contour and area.
Mathematics with history
The study of history includes the analysis and interpretation of a range of historical information including population charts and diagrams and other statistical information. The concepts and skills developed in mathematics support student understanding and interpretation of a range of history sources and their presentation as evidence in demonstration historical understanding.
Mathematics with science
The knowledge and skills students engage within the various dimensions of mathematics support students in their studies of all aspects of science. In science students use measurement and number concepts particularly in data collection estimation of error analysis and modes of reporting. The mathematics domain supports students in developing number handling skills.
To collect the records interpret and display data appropriately, looking for patterns, drawing conclusions and making generalizations. Predictions for further investigations, extrapolations and interpolations may be made from their own experimental results or from reliable second and data.
Materials sciences is concerned with the synthesis and manufacture of new materials, the modification of materials, the understanding and prediction of material properties, and the evolution and control of these properties over a time period. Until recently, materials science was primarily an empirical study in metallurgy, ceramics, and plastics. Today it is a vast growing body of knowledge based on physical sciences, engineering, and mathematics. For example, mathematical models are emerging quite reliable in the synthesis and manufacture of polymers. Some of these models are based on statistics or statistical mechanics and others are based on a diffusion equation in finite or infinite dimensional spaces. Simpler but more phenomenological models of polymers are based on Continuum Mechanics with added terms to account for ‘memory.’ Stability and singularity of solutions are important issues for materials scientists. The mathematics is still lacking even for these simpler models. Another example is the study of composites.
Motor companies, for example, are working with composites of aluminium and silicon-carbon grains, which provide lightweight alternative to steel. Fluid with magnetic particles or electrically charged particles will enhance the effects of brake fluid and shock absorbers in the car. Over the last decade, mathematicians have developed new tools in functional analysis, PDE, and numerical analysis, by which they have been able to estimate or compute the effective properties of composites. But the list of new composites is ever increasing and new materials are constantly being developed. These will continue to need mathematical input. Another example is the study of the formation of cracks in materials. When a uniform elastic body is subjected to high pressure, cracks will form. Where and how the cracks initiate, how they evolve, and when they branch out into several cracks are questions that are still being researched.
 Mathematics in Biology
Mathematical models are also emerging in the biological and medical sciences. For example in physiology, consider the kidney. One million tiny tubes around the kidney, called nephrons, have the task of absorbing salt from the blood into the kidney. They do it through contact with blood vessels by a transport process in which osmotic pressure and filtration play a role. Biologists have identified the body tissues and substances, which are involved in this process, but the precise rules of the process are only barely understood. A simple mathematical model of the renal process, shed some light on the formation of urine and on decisions made by the kidney on whether, for example, to excrete a large volume of diluted urine or a small volume of concentrated urine. A more complete model may include PDE, stochastic equations, fluid dynamics, elasticity theory, filtering theory, and control theory, and perhaps other tools. Other topics in physiology where recent mathematical studies have already made some progress include heart dynamics, calcium dynamics, the auditory process, cell adhesion and motility (vital for physiological processes such as inflammation and wound healing) and biofluids. Other areas where mathematics is poised to make important progress include the growth process in general and embryology in particular, cell signalling, immunology, emerging and reemerging infectious diseases, and ecological issues such as global phenomena in vegetation,
modelling animal grouping and the human brain.

Mathematics in Digital Technology
The mathematics of multimedia encompasses a wide range of research areas, which include  computer vision, image processing, speech recognition and language understanding, computer aided design, and new modes of networking. The mathematical tools in multimedia may include stochastic processes, Markov fields, statistical patterns, decision theory, PDE, numerical analysis, graph theory, graphic algorithms, image analysis and wavelets, and many others. Computer aided design is becoming a powerful tool in many industries. This technology is a potential area for research mathematicians. The future of the World Wide Web (www) will depend on the development of many new mathematical ideas and algorithms, and mathematicians will have to develop ever more secure cryptographic schemes and thus new developments from number theory, discrete mathematics, algebraic geometry, and dynamical systems, as well as other fields.

Mathematics in the Army
Recent trends in mathematics research in the USA Army have been influenced by lessons learnt during combat in Bosnia. The USA army could not bring heavy tanks in time and helicopters were not used to avoid casualty. Also there is need for lighter systems with same or improved requirement as before. Breakthroughs are urgently needed and mathematics research is being funded with a hope to get the urgently needed systems. These future automated systems are complex and nonlinear, they will likely be multiple units, small in size, light in weight, very efficient in energy utilisation and extremely fast in speed and will likely be self organised and self coordinated to perform special tasks.  Research areas are many and exciting. They include: (i) Mathematics for materials (Materials by design - Optimisation on microstructures; Energy Source - compact power, Energy efficiency; Nonlinear Dynamics and Optimal Control). (ii) Security issues (needs in critical infrastructure protection, mathematics for Information and Communication, Mathematics for sensors, i.e. information/ data mining and fusion, information on the move i.e. mobile communication as well as network security and protection). (iii) Demands in software reliability where mathematics is needed for computer language, architecture, etc. (iv) Requirements for automated decision making (probability, stochastic analysis, mathematics of sensing, pattern analysis, and spectral analysis) and (v) Future systems (lighter vehicles, smaller satellites, ICBM Interceptors, Hit before being Hit, secured wireless communication systems, super efficient energy/ power sources,  modelling and simulations, robotics and automation.

During the last 50 years, developments in mathematics, in computing and communication technologies have made it possible for most of the breath taking discoveries in basic sciences, for the tremendous innovations and inventions in engineering sciences and technology and for the great achievements and breakthroughs in economics and life sciences. These have led to the emergency of many new areas of mathematics and enabled areas that were dormant to explode. Now every branch of mathematics has a potential for applicability in other fields of mathematics and other disciplines. All these, have posed a big challenge on the mathematics curricula at all levels of the education systems, teacher preparation and pedagogy. The 21st Century mathematics thinking is to further strengthen efforts to bridge the division lines within mathematics, to open up more for other disciplines and to foster the line of inter-discipline research.
 

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