The Nature of Mathematics and the National Curriculum - Essay Example

The Nature of Mathematics and how it is Represented in the National Curriculum and Taught in Schools Introduction Mathematics is a tremendously scientific sphere of influence which is featured by the fact that the things which comprise of it are idealized rational theories. These things can, in no way, be recognized straightly through the logics. However, it is apparent when these things involve limits or infinite sets where, definitely, no straight experience can be existent with the inestimable itself. Mathematics, by nature is in cooperation an untainted and abstract escapade of the psyche and a practically implemented science. This difference in the opinions enables the abstract mathematics intellect to perform mathematical operations for the sake of mathematics itself, and to use mathematics as a tool to actually resolve the real problems. According to Kister, mathematics has grown into a tremendous structure constituting more than sixty classes of mathematical activities (Kister, 1992). The ideologies of mathematics possess a distinctly extensive verve. For instance, the Babylonian explanation for quadratic equations holds the same significance as it had past 4,000 years (The Georgia Framework, 1996). In the vein of other sciences, mathematics imitates the decrees of the material vicinity around us and serves as an authoritative instructional implement for comprehending nature. Nevertheless, mathematics is yet again classified by its autonomy from the material world. The intangible behavior of mathematics gave rise in relic to the essential difference in opinions of mathematics as a substance of discourse and also as an element for implementation. Mathematical notions are long-lasting and keep on expanding with time. New mathematical notions are developed on the other, bigger mathematical notions or propositions (The Georgia Framework, 1996). Equivalence can be brought in to existence to incessant improvisation where recent practices can be enhanced upon, provided with innovative efforts and time. More often than not, improvisation does not take place without attempting, and it quintessentially doesn't crop up swiftly. Too often, the problems are resolved, and new-fangled arenas of mathematics produced by gaping at getting on problems in new ways. A centralized way of examination in abstract mathematics is recognizing in each field of study a small set of foundation notions and regulations from which all the other appealing ideologies and regulations in that area can be rationally inferred. In the vein of other scientists, mathematicians are meticulously delighted when the earlier disparate parts of mathematics happened to be derived from one another, or from some more common abstract. Fraction of the sense of aesthetic which many people have imagined in mathematics lies not in the location of the paramount elaborateness or intricacy but on the divergence, in locating the economy and straightforwardness in apex of delineation and testimony, with the progress of mathematics, supplementary associations have been found amongst the parts of it which have been growing dissimilarly. These uncanny associations allow the thoughtfulness to be developed in to the several parts so that they, collaboratively, reinforce the conviction in the corrigibility and fundamental alikeness of the entire anatomy. According to Smith, the significance or importance of Mathematics is for its own sake, for the reason that it is a universal language and sagacious implement-kit for abstraction, generalization and synthesis, problem solving, whereby, it trains the psyche. Moreover, the mathematical education comprises of interpretation and utilization of different delineations of statistics and monitoring, recognition of errors, and a notion of what, how and when to compute. Last but not the least, it gains access to the labor market and common social and political enclosure (Smith, 2004). Mathematics possesses the ability to contribute significantly to a novice's future employment and educational skills and views, with critical enclosure in the society. It also has the caliber to add to the development of analytical and reckoning skills through which an individual can attain personal and social empowerment. Section 2: The nature of mathematics and the National Curriculum According to Davis and Hersh, the School mathematics is no more distinctly described nor yet value-free or culture free. Moreover, it is not similar to the academic or research mathematics, but a circumstantial election from the derivative regulation, which itself is a large quantity (Davis and Hersh, 1980). In addition, as per Ernest's views, some of the constituents of school mathematics possess no room in the regulation appropriately, however, are drawn from the chronology and eminent practices such as the study of proportions (Ernest, 1986). The parts which are opted for and the values and causes which underpin that selection significantly ought to determine the behaviour of school mathematics. Additional transformations are introduced by the choices regarding the ways in which the school mathematics should be arrayed, taught and examined. As a point of fact, we can construe to the fact that the nature and behaviour of school mathematics, extensively, is open and as a result, the good reason difficulty must provide accommodation to such diversity. For this reason, the good reason problem should deal with not only the justification for teaching and educating mathematics, but also for the assortment of what and how mathematics should be introduced in to teaching for the reason that these arguments are undividable from the problem. Furthermore, the efficacy of academic as well as school mathematics in the contemporary world is tremendously overvalued, and the serviceable argument facilitates a poor rationale for the universal teaching of the theme throughout the years of obligatory education. As a result, even though it is widely acknowledged that academic mathematics drives the social implementations of mathematics in such fields of education, commerce, administration and corporate, this can more be interpreted as an inversion of the times past. The mathematical implication of the contemporary society and modern living has been developing in an exponential way, for the reason that by now, nearly the total range of human actions and organizations are thought of and controlled numerically, together with sport, media, education, administration, science etc. The mechanical systems carry out intricate tasks of knowledge imprison, policy implementation and resource allotment. We can construe to the fact, that the complex mathematics can be brought into use for controlling many features of our lives, finances, banking, with the least of human inspection and interference, just the once the systems are in proper position. The aims of mathematical teaching cannot be cannot be significantly taken concern of in solitude from their socialistic framework. The targets are forms of intent where the purposes belong to communes or individuals. As a point of fact, educational targets are a form of delineating values, interests and ideologies of particular individuals or communes. More to it, the interests and notions of some of such groups tend to clash or disagree. In another place, with meticulous reference to William's analysis, in the history of educational and social conviction, there can be distinguished five groups which illustrate that each of the groups has unique goals for mathematics education and disparate outlooks of the behaviour of mathematics (Williams, 1961). The first interest group comprises of the Industrial Trainers who look forward to attain knowledge of the basic mathematical skills and social training in compliance. Second group consists of Technological Pragmatists who look forward to gain knowledge about learning the foundation skills and to learn the techniques to problem resolving with the collaboration of mathematics and information technology. The third group constitutes of Old Humanist Mathematics veterans who seek to comprehend the advanced mathematics with some admiration of pure mathematics. Furthermore, Progressive Educators make up the fourth group in order to attain confidence, creativity and self formation with the help of mathematics. Lastly, the fifth group consists of Public Educators who take interest in the empowerment of educators as serious and educate the literate citizenry of the society. These social communes are employed in the National Curriculum for mathematics in which the first three groups have managed to bag a room for their targets in the national curriculum. However, the fourth group which is of the Progressive Educators put to rights themselves with the addition of a personalized information-implementation width, that is to say the procedures of utilizing and implementing mathematics which adds on to one of the National Curriculum acquirement of aims. Nevertheless, regardless of delineating progressive self-actualization targets by means of mathematics, this element exemplifies serviceable targets. Notwithstanding this concession over the behaviour of the procedure constituent enclosed in the curriculum, the room for the constituent has been abridged over consecutive assessments and is in process of total eradication. According to Ernest, the consequences of the chronological challenges and procedures are that the National Curriculum may be interpreted to render service to three chief causes out of which, firstly, much of the National Curriculum in mathematics is loyal to interacting numeracy and fundamental mathematical skills and know-how across an assortment of mathematical schemes that comprise of numbers, algebraic titles, form and scope, and regulating statistics (Ernest, 2000). Secondly, for the sake of advanced or high achieving pupils, the comprehension and utilization of these fields of mathematics at higher gradations are incorporated as an aim. As a point in fact, there is an augmentation in to a set of academic representational practices of mathematics. Lastly, there is a realistic process filament that runs through the National Curriculum mathematics which is looked forward to establish the serviceable dexterities of utilizing and implementing mathematical calculations to the real world difficulties and problems. Each of the consequences is serviceable to an enormous restraint for the reason that they establish usual or expert mathematical capabilities which, on one hand, engross the apprentice with helpful implements and on the other, are pertained to the practical problems. The lean of this consequence appears to be a subject of astonishment to probably none for the reason that the entire throve of the National Curriculum is cognizant to be diverted towards technical and technological capabilities and competence. This throve is sustained by the New Labor education guiding principle. According to the planning stages of the Qualifications and Curriculum Authority, for the duration of the key stage, the individuals must be offered enormous opportunities which are essential for their learning in order to enhance their commitment with the concepts, processes and constituents of the subject. Hence, the curriculum ought to provide opportunities to the students for developing confidence in an incrementing series of methods as well as techniques. They should be provided with the benefits of working upon the sequences of tasks which engross the utilization of the similar mathematical concepts in gradually more challenging or untried contexts, or growingly demanding mathematics in similar frameworks. The curriculum ought to facilitate the students with working on open and clogged chores in an assortment of real and theoretical contexts which enable them to pick up the mathematics to use. Moreover, the students be supposed to work on problems which crop up in additional subjects and in contexts which are beyond the school. Students would then be able to implement their knowledge on tasks which bring together the diversified features of concepts, processes and mathematical elements. Moreover, they would be able to work collaboratively as well as in a self-governed way in a series of circumstances for the reason that they would become familiar with a range of resources which they can pick up suitable (National Curriculum, 2009). The representation and analysis of statistics becomes much easier with the help of modus operandi and relationships and for comprehending to the number anatomy and currency exchange in contemporary alien dialects. Moreover, the measurement and making accurate edifications in design as well as technology, thereby, managing finances in economic comfort and financial abilities. Also, the mathematical skills add on to the financial capabilities and to other supplement features of gearing up for the adult life. Students work collaboratively which includes interactions with regards to mathematics, computing their own and others' works and responding in a way more constructive way, thereby, presenting the ideologies to a wider commune. Lastly, students get to bring in to use the practical resources for example, spreadsheets, graphs and calculators, and to establish new mathematical ideas. According to DfEE, the National Curriculum for mathematics refuses to describe the aims, it delineates that mathematics implements students and pupils with a distinctly authorized set of equipments so as to comprehend and transform the world. These equipments are inclusive of logical reckoning, problem-resolution skills, as well as the ability to think in all the more theoretical way. Mathematics is significant in routinal life as it engrosses many forms of vocation, science and technology, drugs, finances, environment and development. Diversified mores have participated in the development and implementation of mathematics. In today's world, the subject goes above cultural restraints, and its significance is generally cognizant. As we have already discussed before that mathematics is a creative restraint, it is capable enough of stimulating the occurrences of delight and would speculate when a student resolves a problem for the very first time, finds out a more neat solution to the difficulty, or may discover sudden concealed associations (DfEE, 1999). Many countries and states have established innovative and even more precise mathematics curriculum concepts that outline their proposed curriculum. Whilst, some of the documentaries are supposed to be modules for the districts to exploit in forming the local curriculum requirements, the others are compulsory, spelling out the mathematics that all students within the states are supposed to study at scrupulous levels. All seem to serve as instructions for forming the annum state-wide level assessments. Section 3: Focus study: A study on an Aspect of Mathematics Teaching This section provides an idea about a profound concept for the design of school mathematics curriculum and norms. These principles are uncomplicated and concise, and are also meant to hold some practical significance to the communes which revise the standards of state mathematics. It is, however, realized that these principles are an augmented attempt which is unfinished in various ways. More to it, the communes of mathematics veterans who held possession of various outlooks on school mathematics, set up substantial accord about the initial principles to be. The virtues of mathematical pedagogy and the authority of mathematics in today's world crops up from the cumulative behavior of mathematics information where a small assimilation of straightforward actualities collaborated with suitable abstracts is brought into use so as to develop a sheath upon a sheath of considerably stylish mathematical knowledge. The core of mathematical education is the procedure of comprehending to each new-fangled sheath of knowledge and information, thereby, rigorously mastering that information and knowledge so as to be able to comprehend to the subsequent sheath. The principles are, thus, planned to endorse such mathematical pedagogy. Standards of School Mathematics The standards or norms of School Mathematics are as follows: 1. Whole number arithmetic and the place value system are found to be the basis for school mathematics with other mathematical filaments cropping up from this basis. In early levels, this basis ought to be the theme of most of the guidelines. 2. The guidelines should be thorough in a mathematical way in a level-suitable trend, where all the terms and expressions should be described with a mathematical dialect which is accurate. Moreover, the key theorems and formulae should be verified in all possible ways. 3. At every level, the mathematics set of courses or curriculum requires to be meticulously centralized on a small number of titles. Most of the mathematics guidelines should be devoted to the establishment of thoughtful mastery of the essential topics by means of calculation, problem-resolving and logical reckoning. 4. Multitudinous pupils should be educated about the mathematical knowledge and reckoning capabilities which are required to succeed in the university. The planning of students with regards to a Bachelor's degree should opt for a more challenging mathematics path in the high school itself which sets them up to accentuate calculus when they go in to the college (Paper, 2004). The Use of Equipment/ICT in Mathematics Teaching The preamble of Information and communication Technology in the pedagogy of mathematics under the education of Science and Mathematics has been augmented by the Ministry of Education since 1993 (Keong et. al, 2005). The augmenting observations have found out that teachers do not wholly bring in to use these amenities in their teaching. With particular opinion polls conducted in order to analyze the constraints which prevent the incorporation and acceptance of information and communication technology or the ICT in teaching mathematics, six key constraints were recognized out of which lack of time in institute schedule for assignments that involved ICT topped the list. Secondly, inadequate teacher training advantages for the ICT assignments and insufficient technical assistance for these projects was a matter of concern. Moreover, lack of knowledge about the ways to incorporate ICT so as to add to the curriculum and problems in the incorporation and usage of different ICT implements in a single chapter and inaccessibility of resources at home for the students to make use of the essential educational implements are yet another constraints. According to Ernest, new technologies like the micro-computer and assessments such as the GCSE seek fordifferent patterns of teaching and at no time in the chronology of mathematics teaching have there been transformations on such a broad and rapid growth of knowledge (Ernest, 1989). The usage of ICT in teaching mathematics can prove to be a boon in making the teaching procedure substantially efficacious, thereby, enhancing the capabilities of students in comprehending to the basic concepts. However, the implementation of its effective usage in teaching involves a number of problems for the reason that quite a few constraints may crop up. Yet, the nature of mathematics has transformed substantially because of the ease of use of ICT for the reason that the procedures of modeling, estimation, hypothesis and information have become considerably significant. As a result, ICT can help the students to access and construe information, test the dependability and accuracy, and interact with others by presenting information. The key uses of ICT in the teaching of mathematics stalk from calculators, databases and spreadsheets, dynamic geometry, internet, word processing and programming (National Curriculum, 2009). The Use of Misconceptions in Mathematics Teaching According to Hai and Yusuf, the skills which are essential for recognizing and analyzing the errors and faults made by students are required by teachers at all grades. If the students get successful in dealing with the mathematical intricacies in their schooling, the one precondition is the effective mastery of the basic concepts in their primary mathematics (Hai and Yusuf, 2008). Regardless of the best attempts of the teachers, students still hold some mathematical misconceptions. In view of Gunstone et al, only a few of the studies have tried to transform student misconceptions in mathematics where the common methods have been the Cognitive Clashes and Predict-Observeexplain teaching sequence (Gunstone et al, 1992). According to Prescott, a review of the literatures on cognitive clashes suggests that it would be most successful when the students are made aware of the misconceptions and discussion in a key constituent of the teaching as well as learning process. Also, the teachers are cognizant of their own misconceptions as well as those of the pupils (Prescott, 2004). Learner Attitudes and Mathematics With the society becoming increasingly cognizant of the desire of life-long comprehension of mathematics, institutions come across with a much more diversified clientele that that to which they may have become accustomed to. Students are often older, and the ones who do not have taken college-prep courses in high school have undergone quite a few losses. Meticulously in mathematics, we can find students facing loss for it is supposed that the mathematics learning is increasing where certain required facts and concepts must be accumulated before learning can actually augment. Nonetheless, these procedures can only gauge the student's present grade of mathematical achievement and do not take in to account the student's own insight of the achievement, or themselves as mathematics learner. These procedures do not consider the student's anticipations for success at mathematical chores and do not scrutinize the casual attribution for success and failure at such tasks. In teaching and studying mathematics in a theme-oriented manner, the instructions are organized around thematic projects for the reason that according to Freeman and Sokoloff, themes appear as the organizers of the mathematical curriculum, and concepts and strategies are trained aaround a centralized topic which is supposed to provide meaning and way to the learning procedure (Freeman and Sokoloff, 1995). Methodology The aim of this methodology is to comprehend a sense of pride in the targets and achievements in the mathematics subject which would help the students in exemplifying the aims and attainments. Some general questions which can be helpful to be able to formulate some kind of answers for at all levels can be helpful. The questions for teachers of mathematics at all grades are basically involved with the limit to which the training of mathematicians should implement professional discourse to. Some of the questions and issues for the mathematicians are as follows: 1. What is the significance of mathematics and in what contexts, and why 2. What is the way it goes about its job 3. What is the nature of mathematics as compared to other subjects 4. What kind of research does mathematics involve and what are its key aims 5. What do you mean by good mathematics There are a number of causes apart from the authority to consider the above issues. Professors suggest that the aim of considering these questions was to get the students to imitate on the procedures of mathematics. On one hand, the feature of this imitation is that it results in the idea of value estimation, which is a facility that humans have and which is not in a way in which we can interact with each other. There is yet another cause for consideration to the about issues which is through a comparison of aspects of education in art. For the reason that it is well known a fact that education in art and design is substantially to the front of science in accentuating an interest and self-sufficiency of students, hence, it is worth considering how these mathematicians construe to things. This course of design crops up various aims which are: to educate the students regarding good design; to hearten independence and creativity; and to provide students with an array of practical capabilities so that they can easily implement the principles of good design in their vocations. The major aim is to investigate the mathematics teacher's beliefs and conceptions about mathematics. However, these beliefs cannot be examined in solitude. There is a need to develop the beliefs of teachers about their schools so that a comparison could be inculcated between their views about cultural and school mathematics. The main methods we can use to obtain relevant information on the research questions are: Questionnaires, interviews and observations. This study is not just about the beliefs and practices of teachers but also about the investigation of the influence of the perceptions of teachers on students. According to Thompson, the future directions in teachers' beliefs and conceptions should be relevant enough to provide consideration to questions regarding the relationship between teacher conceptions and student conceptions and how do they communicate. As a result, the questionnaire presented above can be used in this case (Thompson, 1992). Moreover, interviews can be included in the methods for the reason that they provide the researcher with an opportunity to examine the teacher belief inclinations. If at all, the replies from the teachers to the questionnaire signal some contradictions, the researcher may look forward to clarify them in the interview. Also, interviews provide the respondents with various opportunities to respond in numerous ways. Along with questionnaires and interviews, observations serve as in integral part of the research because it is quite significant to keep an eye on the teacher practices. Again, as per Thompson's views, all examination of teacher beliefs should make use of verbal statistics and should also bring in to use the observational statistics on the instructional practices or mathematical behavior (Thompson, 1992). This can be done with recording the observed lessons so that the observations appear to be developed, albeit an observational routine would be brought in to use as an instruction to uphold the consistency in the main points the researcher seeks for in the subjects that are scrutinized. Section 4: Conclusions Ernest, in one of his papers, argues that the beliefs of mathematics teachers possess a powerful influence on the practice of teaching since, during their transition in to practice, there are two aspects which influence these beliefs: the limitations and opportunities of the social context of teaching, and the grade of the teacher's insight. The social context lucidly limits the teacher's independence of choice and implementation, restraining the realms of the teacher's independence. Higher-level thoughts such as self-assessment regarding the implementation of convictions in to practice are a key constituent of the freedom in teaching. With due consideration to all the factors can we justify the intricate notion of the self-sufficiency involved in mathematical teaching (Ernest, 2008). Ernest, in yet another writing work speaks about the absolutist outlook of mathematics which is an entity of knowledge whose actuality appears to everyone as essential and unambiguous (Ernest, 1991). This can be construed in certain assumptions which are supposed to be self-apparent. Without a doubt, absolutists construe to the fact that mathematics is totally autonomous with regards to human-kind as it performs in its own administration, binding the galaxy together. This consistency has proved to be one of the most authoritative appeals of mathematics. Mathematics is taken as a definition which does not enable the questioning of basic principles of the subject and by which they are verified. The association between studying mathematics in formally educational vicinity and studying it in the work-place condition is the hub of a great deal of research as it is the more non-specific intricacy of transition. It is usually agreed that the formal skills which are studied in school do not always prove to be significant, nor do they send out to other conditions. However, this does not relate to the fact that other capabilities would prove to be substantially significant and would send out in a better way, but it means that knowledge and practice are necessarily placed and more relevance is required to be given to the practices within which the knowledge is given a shape. The more a class imitated the ways of working and ways of being with appropriate inclusion of equipment use, power relations, interactions which are significant to the work place, the more significant the learning would be (Watson, 2005). The aims of mathematics education are suppressed by political and economic forced which are over and over again articulated through the medium of public estimation which is given a shape largely by the insight of actual mathematics instead of the one which is known only to a few. Many documents delineate the visions of secondary education which is transformed with the help of successful implementation of sets of recommendations. However, the recommendations must be implemented in the real world which comprises of an ordinary citizen and an average teacher and not the rarified world of expertise and paramount professional skills. The attitudes, knowledge and values of the community as students or teachers describe the environment in which the educational transformation is sought. The students of middle as well as high school in today's world are born into a world of technology. The implementation of technology during mathematics is natural for them and the exclusion of these devices from their routinal studies is to set apart their experiences from their life. The objective in preparing teachers for the future is to make sure that their classrooms are inclusive of all the basic amenities with technology par excellence which will be a commonplace for the upcoming generation of mathematics learners, as a result, making sure that the mathematicians, educators as well as the citizens of tomorrow feel an accord between their arena of mathematics and the world in which they reside. References 1. Davis, P. J. and Hersh, R. 1980, The Mathematical Experience, Boston: Birkhauser. 2. Ernest, P. 1986. Social and Political Values, Mathematics Teaching, No. 116. 3. Ernest, P. (Ed) 1989, Mathematics Teaching - The State of the Art, London: Falmer Press. 4. Ernest, P. 2000, Why Learn Maths, edited by John White and Steve Bramall, London: London University Institute of Education. Ernest, P. 2008, The Impact Of Beliefs On The Teaching Of Mathematics, Exeter University < http://www.people.ex.ac.uk/PErnest/impact.htm> [Accessed: 2 October 2008] 5. Freeman, C., and Sokoloff, H. J. 1995, Children learn to make a better world: Exploring themes. Childhood Education, 73. 6. Gunstone, R. F., Gray, C. M. R., & Searle, P. 1992, Some long-term effects of long-term uninformed conceptual change. Science Education, 76. 7. Hai, See Kin and Hjh Jamilah Yusuf. 2008, Analysis of Mathematical Errors in Primary Schools. University Brunei Darussalam, Negara Brunei Darussalam. 8. Keong et al. 2005, A Study on the Use of ICT in Mathematics Teaching. Malaysian Online Journal of Instructional Technology (MOJIT), Vol. 2, No. 3: 43-51. 9. Kister, J. 2002, The Mathematical Reviews Database: Past, Present and Future. Mathematical Reviews, Ann Arbor (March 18, 2002). 10. Koehler, M. S., & Grouws, D. A. 1992, Mathematics teaching practices and their effects. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. New York: Macmillan. 11. National Curriculum. 2009, Mathematics key stage 4. Qualifications and Curriculum Authority, London. 12. Oldknow, Adrian. 2004, ICT enhancing mathematics teaching. The Mathematical Association, BETT 2004 (9th January). 13. Paper. 2004, What is Important in School Mathematics. Mathematics Standards Study Group. 14. Prescott, A. E. 2004, Student understanding and learning about projectile motion in senior high school. Unpublished doctoral thesis, Macquarie University, Sydney. 15. Smith, A. 2004, Making Mathematics Count: The report of Professor Adrian Smith's Inquiry into Post-14 Mathematics Education. London: The Stationery Office. 16. The Georgia Framework. 1996, Georgia Framework for Learning Mathematics and Science. University of Georgia, National Research Council. 17. Watson, Anne. 2005, Maths 14-19: Its Nature, Significance, Concepts and Modes of Engagement. Aims, Learning and Curriculum Series, Discussion Paper 13 (18 May 2005). 18. Williams, R. 1961, The Long Revolution. London: Penguin Books. Appendix A Source: (Grouws and Koehler, 1992) Appendix B 2. Graphical Calculator: Source: (Oldknow, 2004) Appendix C 3. Geometer Sketchpad: Source: (Oldknow, 2004) Read More