Development in Mathematics in Different Countries - Research Paper Example

What can other countries teach us when it comes to teaching mathematics? Critically discuss in relation to recent developments in mathematics education. Mathematics, as a subject, has managed to hold the interest of human civilization since time immemorial. The word ‘Mathematics’ has been derived from the Greek word ‘mathema’ which means ‘interested in learning’. Primary mathematics has been an integral part of the education system of the ancient civilizations. People made use of the subject for various purposes, right from keeping track of the solar and lunar movements to keeping track of their cattle and livestock. The objectives of teaching mathematics were many: a) To impart basic mathematical skills to students from a very early age b) The methodology of imparting mathematics education was designed in such a manner so that children could equip themselves to follow any line of business or craft c) It helped pupils to develop a basic skill of analyzing and reasoning d) Pupils were also encouraged to learn problem-solving techniques to solve problems that were not related to the day to day life. Mathematics education has evolved a lot with the changing objectives. When it comes to teaching mathematics, there has been lot of discussions and debates over the most effective way of teaching the subject. A good amount of research also has been done over the best ways of imparting mathematics education. But, that fails to provide the complete picture of the scenario. This is primarily due to the fact that a good number of random experiments have not been performed. This often leads to false claims but without enough or absolutely no justification. Hence, work is being done in this field to conduct more number of random experiments which will help in determining which methods work best in mathematics education. When it comes to teaching mathematics, emphasis should be given to the liberty to think and the freedom to explore the subject. Instead of just seeing whether a particular method works for a given problem or not, it is always considered a good practice to see why and under what conditions it works. A more experimental approach should be adopted in class to determine under which conditions the students are able to grasp things in a better way. It has been observed that if one tries to devise his/her own method to solve the problem and then applies the standard method of solving the problem, the student understands in a much better way and the chances of making mistakes are considerably lower. Hence if such teaching methodologies are encouraged in classrooms, pupils and students can benefit a lot from these practices. Similarly, research has also shown that doing regular assignments and home-works can also go a long way in improving the overall level of understanding and problem solving skills of the pupils. It is advisable that the assignments prepared for the students should be a mix of easy, tough and tougher problems. The overall length of the assignments should be small, so that students do not get bored while attempting them. It has been observed that students who are able to learn well in situations which involve some level of conceptual understanding can adjust to other situations as well. Also, group assignments should be encouraged. Rather than memorizing the formulae, a student should be given more liberty to experiment and devise his/her own method. There are a few approaches of teaching mathematics that are practiced widely in other countries. Different countries have different methodologies in place when it comes to teaching mathematics. We need to analyze the mathematical education system existing in these countries in order to understand these approaches. OVERVIEW OF THE EDUCATION SYSTEM IN TAIWAN In Taiwan, education plays a very important role in shaping up the country’s future. Taiwan has one of the best primary schooling systems in place. Reviews published, after successfully conducting surveys, have forced educationists and policy makers in England to look at Taiwan. There has been great concern among educationists about the declining standards of education in England. Several comparative studies have been done with respect to the existing education systems in England and other countries (Whetton, Twist and Ruddock, 2007). England has been following the spiral curriculum for many years. Due to this reason, many topics are revisited more than in other countries. Though text books suggest a frequency of teaching topics, sometimes re-visiting the topics is left entirely to the teacher. Teachers have the freedom to vary the lessons and the course of teaching. At times, teachers also include materials from other text books intended for other age groups. According to the Gatsby report, it was feared that the schools did not do much in terms of imparting mathematical skills much needed to find employment. It was emphasized that the low attainers could achieve more. The result of the IAEA, IAEP research projects fuelled the debate that the education standards in England were declining. It was found that pacific-rim countries like Taiwan performed better than North America and Europe (Whetton, Twist and Ruddock, 2007). Performance wise, it was found that the English students perform relatively poorly when compared to the Taiwanese counterparts even though English students have a good understanding of data handling and geometry (Whetton, Twist and Ruddock, 2007). The country adopts whole-class teaching methods (in the same way as England) while imparting education to students, thereby treating them as equal and that is a very notable feature. On the other hand, there are stronger links between performance and social class in countries like England. Teaching is a respected profession in Taiwan and teachers earn 20% higher salary than civil servants. Also, their earnings are not taxed. The entry into the teaching profession is highly competitive and only the best students are selected (Vulliamy, 1998). Among other Asian countries, growth of expenditure on the development of education was more notable in Taiwan. Hence, the teaching force was upgraded very rapidly. Primary school teachers were trained for a three year period before they could actually start teaching. However, now the training of the primary school teachers is taken care of by the teachers’ colleges which offer an equivalent of a bachelor’s degree. In addition to this, teachers are provided with facilities to upgrade their academic qualification by the way of night school courses. The salaries of teachers are fixed after taking their academic qualifications into account. Also, half the fee amount is paid by the local authorities. It is also ensured that the number of teachers allotted is 1.5 times the number of classes in a particular school (Vulliamy, 1998). The role of assessments cannot be ignored. They provide an important tool in understanding the level of understanding of the pupil. The National Tests in England have been instrumental is shaping up the curriculum in England. These days Education system in Taiwan is becoming increasingly competitive. There is a widespread use of private cramming lessons outside school which primarily aim at enabling students to secure good marks. Due to this, several students break down under the pressure of performing in the exams (Vulliamy, 1998). Several teachers and educationists also complain that whole-class teaching has its own disadvantages. According to teachers in Taiwan, ‘whole-class’ teaching does not help the lowest attainers in a class, who form approximately 20% of the class population (Vulliamy, 1998). Especially when learning complex mathematical topics, group activities are much better when compared to whole-class teaching. OVERVIEW OF THE SINGAPORE MATHS EDUCATION SYSTEM The first primary mathematics curriculum was brought into existence by the ‘Curriculum Planning & Development Institute of Singapore. As this was the first curriculum that was developed, more importance was given to the content and problem solving was not given much importance. Slowly, the need of problem solving was realized and this was incorporated in the syllabus later on (Ruddock, 1998). Problem solving ideally requires students to apply the knowledge gained to more complex situations. However, now the academic content has been reduced to provide teachers with a chance to incorporate more mental approach in the study of mathematics (like thinking skills and counting skills). The usage of technology is also being encouraged in imparting mathematics education. The content has been reduced also to lessen the burden on students by not making it compulsory for them to learn topics that do not require any thinking at all. Also, topics that are taken care of in other subjects are not taught in the Singapore mathematics education system. If any topic has already been taught in a particular year, the next year will involve a more detailed and different approach to deal with that topic unlike the spiral curriculum existing in England which involves a topic that is revisited more than once over a span of time. Hence, it can be said that students move from an easy to complex level of studying any given topic in mathematics. A lot of emphasis is given on the order in which the topics are taught in Singapore. Hence repetition of topics can be avoided. This reduces efforts on the part of both the educators and the students and a good amount of time can be saved for something more constructive. Emphasis on word problems is given so as to improve the overall level of understanding. Care is taken to ensure that at all levels the students are carefully guided. The emphasis on problem solving is of interest and it is considered important. For this reason, assistance is provided in terms of pictorial representations, charts, graphs etc. This is quite different from most of the other countries where more emphasis is laid down on memorizing formulae and applying them in various hands-on activities. Technically obsolete topics have been removed from the curriculum because these topics are no longer relevant to the present day. Also, the difficulty level of any topic is kept in mind and topics that are too difficult for any given level have also been removed from the present and existing mathematics curriculum. In a survey that was conducted, it was found that the students in England used calculators and schemes more often to solve complex problems. Teaching strategies like mental mathematics, pencil and paper algorithms are often used. According to the same survey it was also found that English students received less homework when compared to their counterparts in Singapore and other countries. Since the year 2000, a lot has been done in terms of the methods to impart mathematics education, the assignments that are given to students and the examinations conducted to assess the level of understanding of the students. OVERVIEW OF MATHEMATICS EDUCATION IN GERMANY Education in Germany is given due importance. Education is the responsibility of the nation. Even though the educational structure differs from state to state, the basic structure is quite the same in some ways. However, there is no national curriculum for mathematics, unlike countries like England. The guidelines are laid down which run from the start of primary education. The basic Educational system is categorized into elementary level, secondary level of schooling. Apart from these, vocational and technical schools also exist. People who leave the educational system before 18 are supposed to attend vocational schools or technical schools (Ruddock, 1998). The vocational schools impart the basic knowledge with regards to changing objectives of the vocational training. The primary motive of vocational schools is to prepare students for employment. On the other hand, the technical schools impart education to the students in various technical fields provided they have some amount of experience in the related technical field. Educational institutions in Germany are a part of the state machinery. Hence, most of the employees including teachers and administrative staff are state employees. As the education system is regulated by the state, a number of regulations exist in terms of the syllabus, the number of assessments to be conducted, and various other notations to be followed in mathematics education. The basic idea that revolves around teaching mathematics is that the subject is a method or a tool to solve the problems concerning science and other problems. Emphasis is given on the need of proving, deducing, algorithms and mentally solving. Some emphasis is also given on the history of mathematics. Overall, it is a very good approach as the basic reason of studying mathematics is clearly stated. The content of mathematics education is well formed. Students are imparted knowledge right from basic mathematical principles of addition and subtraction, geometrical shapes and patterns to calculus, vector space and analytical geometry. Another interesting point of difference in the education systems existing in Germany and England is that official approval is required for textbooks in Germany unlike England (Ruddock, 1998). Books in Germany must be in accordance with the education acts and constitution. Efforts are being taken to incorporate computer based teaching in the normal teaching methodology. With the ever decreasing cost of technology, very soon this is going to be a reality. The availability of technology has made problem solving quite easier. Hence, more importance is being given to understanding the procedure and the logic behind solving the problem. The use of technology in imparting mathematics education helps in better visualization of the problem. The German education system has managed to change the teaching methodologies in mathematics. Hence, mathematics does not remain the subject of the privileged or the gifted. The recent changes that have been incorporated in teaching mathematics are a) Teaching mathematics involves a lot of demonstration using models so as to boost the level of understanding of the students b) Co-operation within the teaching staff teaching different subjects can actually improve the teaching methodology. Newer ways of teaching could be identified this way. c) Students are encouraged to give newer ideas and thus contribute to the overall development of mathematics. The educators in Germany are also given a lot of independence in terms of his/her style of teaching. The methods to mark the assignments, class work totally rest with the teacher. Some states have state examinations designed for students, when they have finished their schooling. In Germany, there are a number of national level mathematical competitions that are held on an yearly basis. Students of all ages are given a chance to showcase their mathematical skills. Also, there are research oriented competitions with a branch in mathematics, where young researchers come up with innovative ideas and based on the decision of the jury, the winner is selected. Hence, it can clearly be seen that the German education system is highly evolved and gives ample number of opportunities to students to show case their skills. KEY ISSUES IN MATHEMATICS EDUCATION According to one of the interim reports from the Primary Review of the education system, measures of achievement are only a fragment of large international surveys. A large amount of contextual data ranging from the students’ attitude and their backgrounds to the teachers’ experience and qualifications is collected. More recently, even parents’ views are taken into consideration. As a result of the surveys conducted world wide, the major issues in mathematics education are found to be: a) Emphasis on fundamental principles should be given. Whatever be the difficulty level of the problem the solving techniques revolve around addition, subtraction, division and multiplication. Hence, from the primary level itself, more emphasis should be given on fundamental and basic principles. According to a recent report, data on students' understanding were obtained from examinations as part of the normal process of schooling, and from content-based interviews with individual students withdrawn from class. In the interviews, explanations of students’ gave an idea of what they were trying to do as they worked on the test papers, thus gaining valuable insights into their reasoning. Data was collected from 3000 students in 34 schools; lessons were observed, teachers talked about their programs, textbooks were analyzed, and teaching interventions were arranged. This resulted in the conclusion that basic principles should be emphasized on for proper and better understanding. b) Another key issue that mathematics education faces is that it should be imparted in a more effective manner. Charts, handouts, diagrams, live models can actually go a long way in enhancing the understanding of the students. c) Students sometimes face great anxiety when solving mathematics problems. This can be due to several reasons. The fear of being rebuked and working under a stipulated time frame can sometimes lead to great anxiety among students. Hence, newer and more effective teaching methods necessarily need to be less lecture oriented and more discussion oriented. It must be understood that everybody is capable of learning, but the speed of learning and the method of learning may vary from person to person. Hence, an effort should be made to introduce newer teaching methods so that each student can easily adapt to the newer methods of imparting mathematics education. PART2 A MATHEMATICS FILE The mathematical activity that is taken for this part is: Consecutive numbers Some numbers can be expressed as the sum of consecutive numbers. The solution to the above problem is as follows: All the odd numbers reflect this property. Numbers such as 3, 5, 7, 9….and so on, exhibit this property. For example, 3= 1+2; 5= 2+3; 7= 3+4…….. and so on. Observation1: When we add up two consecutive numbers e.g. 1+2, 2+3, 3+4…… the series we obtain is of the form 3, 5, 7, 9.. Hence the common difference is 2. Observation2: When we add up 3 consecutive numbers e.g. 1+2+3, 2+3+4, 3+4+5 … and so on, the series we obtain is 6, 9, 12……… Hence the common difference is 3 Observation3: When we add up 4 consecutive numbers e.g. 1+2+3+4, 2+3+4+5, 3+4+5+6 … and so on, the series we obtain is 10, 14, 18……… Hence the common difference is 4. Deduction: Hence whenever, we add ‘n’ consecutive numbers the difference is ‘n’. The above problem unveils another mystery about numbers. The steps that were used to solve this particular problem were: a) Understanding the content of the problem is as important as solving the problem itself. Half the problem is solved if one is able to understand the problem itself. At times, students find a great deal of difficulty in understanding written problems. Therefore a student needs to make sure whether he/ she has understood the problem right. Often experimenting and observing helps to understand the problem. The first part of solving the given problem involved identification of those numbers which exhibited the property of being expressed as a sum of consecutive numbers. b) The series of natural numbers was taken, starting from 1, 2, 3… and so on. Each number was broken up so that it could be expressed as the sum of more than one number. For example, 1 cannot be expressed as the sum of any other natural numbers. Moving to 2, it can be expressed as 1+1. Similarly moving to 3, it can be expressed as 3=1+1+1 or 3=1+2. In this case, 1 and 2 happen to be consecutive numbers. Therefore, 3 could be expressed as the sum of two consecutive numbers. But interestingly, when 4 was taken, it was observed that it can be expressed in the following manner: 4=1+1+1+1 or 4=1+3. Hence 4 cannot be a part of the family of those numbers which show this property of being expressed as the sum of consecutive numbers. Therefore, 4 was excluded from this family. Moving to 5, it can again be expressed in the following way: 5=1+4, 5=2+3. Considering the second possibility, it can be clearly seen that 5 can be expressed as the sum of two consecutive numbers. Hence, 5 can be included in the family of numbers having this property. Moving to 6, it can again be observed that it cannot be expressed as the sum of two consecutive numbers. On the other hand, 7 again exhibits this property, so 7 can be included in this particular family. Hence, numbers which can be expressed as the sum of two consecutive numbers are 3, 5, 7,9,….and so on. c) Now if we add 1+2+3, we get 6. If we add 2+3+4, we get 9. Similarly 3+4+5=12. Hence, a series can be formed which looks like 6, 9, 12……. In this case, it can very well be observed that the common difference is 3. d) If we extend the discussion to four consecutive numbers, i.e. 1+2+3+4=10. Similarly, 2+3+4+5=14. Hence, it can be deduced from the above discussion that the common difference in this case is 4. e) Hence, in general it can be said that whenever we are adding ‘n’ consecutive numbers, the common difference is ‘n’. Understanding and analyzing the pattern of the series that is obtained as the result of adding up the consecutive numbers. The process of adding numbers helps improve the basic numerical skills. A lot of emphasis is given on mental calculation techniques and these techniques were effectively used to analyze the pattern of the series. When we analyze this problem, we can very clearly observe how one result set can be extended to find the solution to the other problems that are of the similar nature. This problem involved very basic and fundamental thinking. As we have clearly seen, sticking to such basic principles not only helps in finding out the solution, but also improves one’s level of understanding. Similarly, solving this problem would have been almost impossible without understanding the stated objectives of the problem. Once the stated objectives were understood, it became much easier to solve the problem. Then an approach had to be designed that could tackle this problem. So it becomes very important to identify that particular approach that suits the type of problem. Breaking the problem down can help us achieve this more accurately. To complete this problem, a series of random experiments were supposed to be done. Starting with 1, it was observed that it could not be broken down into other natural numbers. However, with 2 it was observed that it could be expressed as the sum of other natural numbers. However, 2 did not meet our requirements and had to be excluded from the list. This specific approach was extended to other numbers and the result set was obtained. Having worked with two consecutive numbers, it became relatively easier to work with three consecutive numbers. The approach was more or less the same. However, the startling difference that was observed is that the common difference in the earlier case was 2, but in this case, it was 3. To confirm, one or more random experiments were conducted on numbers which could be expressed as the sum of 4 consecutive numbers. In this case, it was found that the common difference turns out to be 4 as predicted. Finally it was deducted that when ‘n’ consecutive numbers are added, the common difference is ‘n’. A Mathematical Autobiography: Mathematics has always been a subject that I have loved. I like doing school mathematics a lot. Solving problems without using a calculator can be quite challenging, but this is what I really like doing. Numbers have always fascinated me. Hence, to me Mathematics is more of a game and less of a subject. I love it when people do appreciate me for my quick problem solving skills. My father always encouraged me to be a good problem solver. This enabled me to solve problems quicker and with a greater accuracy. As time passed, I started taking interest in several areas of mathematics apart from addition, subtraction, multiplication and division. I realized how mathematics can be quite helpful in learning and understanding other subjects. Whenever I learn new things, I start learning right from the scratch. I always love to work in groups, I think working in a group will yield positive results always, since there is always someone to comment on my work and give different ideas on how to approach and solve a given problem This assessment was quite challenging and I thoroughly enjoyed doing this. The first problem required a lot of analysis from my side. It was really challenging to figure out the number of hand shakes that would be taking place in the room. Initially I got another answer, but slowly I realized that a person will shake hands only once. This helped a lot in getting the answer to the first problem. Similarly, the second problem was very challenging too. The other problems were quite simple and I managed to solve them pretty quickly. My current interests include solving Sudoku, mathematical puzzles and other number games. The reason why I was really interested in this project is that I wish to hone my skills in problem solving. I know this will be quite challenging, but I will still give my best shot and work hard. Mathematical problems 1. If everyone in your group shakes hand with everyone else. How many handshakes are there? The number of handshakes in this case would be n (n-1)/2. These are the decisions we need to make: Let us assume that there are three people in the group A, B, C. The handshakes are done in this form: A B C A-B B-C A-C 2 handshakes 1 handshake total: 3 handshakes The explanation is as follows: 1)A shakes hands with B and C 2) B shakes hands with C .Hence for C no independent calculations are required because A and C or B and C or A and B shake hands only once. Hence number of people: 3 Hence total number of handshakes: 2+ 1+ 0 =3 This whole problem could be generalized like this: A B C D …………………….n people n-1 n-2 n-3 n-4 0 Therefore, totally there are (n-1) + (n-2) +…… (n-n) = n.n-(1+2+3+4+…..+n) = n.n-(n+1)n/2=n(n-1)/2 handshakes. 2. Insert the numbers 1-9 in this square so a) All rows add up to 15 6 7 2 4 3 8 1 5 9 Before doing this problem, we have to keep in mind that the numbers in each of the rows cannot be interchanged with other numbers of the other rows. Hence, in any particular row, there are 6 ways of arranging numbers. Hence, in this way the arrangements possible would be 6x6x6. But we can change the position of the rows also. Hence, for every row, there are two ways of arranging the other rows. e.g. 6 7 2 6 7 2 4 3 8 1 5 9 1 5 9 4 3 8 Hence totally the number of ways of arranging this particular set up is: 6x6x6x6=1296 And there is one more set up: 6 1 8 7 5 3 2 9 4 In this case also, it would be 1296. Hence, adding them up together, we get= 2*1296 = 2592 b) Try to rearrange all the digits so that all the rows and columns add up to 15. How many ways can you do this? The following arrangements are possible. Since the requirement is that the rows and the columns individually should add up to 15, but there is no restriction on the diagonals. Hence, for every arrangement shown above, there will be 6 possibilities including each of the arrangements. E.g. 8 1 6 3 5 7 4 9 2 3 5 7 4 9 2 8 1 7 4 9 2 8 1 6 3 5 7… Totally there are 48 arrangements. c) Finally find a way so that all the rows, columns and diagonals add up to 15. In this case, there are only 8 possibilities of creating the squares. Under these circumstances there are eight possibilities building a square:  3) Some numbers can be expressed as the sum of consecutive numbers. Exactly which numbers have this property? All the odd numbers reflect this property. When we add up two consecutive numbers e.g. 1+2, 2+3…… the series we obtain is of the form 3, 5, 7, 9.. Hence the common difference is 2. When we add up 3 consecutive numbers e.g. 1+2+3, 2+3+4 … and so on, the series we obtain is 6, 9, 12……… Hence the common difference is 3 Hence whenever, we add ‘n’ consecutive numbers the difference is ‘n’ 4) How many squares on a chess board? There are 64 squares on a chess board. If there is a 2x2 board, number of squares are 4. Hence for nxn board the number of squares is nxn. Hence, when n=8, number of squares=64. 5) Here is a 3x3 cube made out of smaller 1x1 cubes. If the 3x3 cube was painted red how many of the 1x1 cubes would be painted on one face, two faces and three faces? Number of cubes which have only one face painted: 6 Number of cubes which have only 2 faces painted: 12 Number of cubes which have only 3 faces painted: 8 Number of cubes which have 4 faces painted: 0 6) If the digits of my present age are reversed then I get the age of my son. If one year ago, my age was twice that of my son, find my present age. Let my present age be xy, therefore according to the given problem my son’s age would be yx. One year back, my age was xy-1 and my son’s age was yx-1. . Only, these combinations can be possible, 12 & 21, 13 &31 and so on….. Now, xy-1= 2(yx-1). The equation is, xy -1 = 2yx – 2; xy – 2yx = -1. Hence, the only possibilities which satisfy the given equations are: 37 & 73. Hence, the father must be 73 and the son must be 37. My experiences I thoroughly enjoyed doing this assessment. It involved a lot of understanding, analyzing the problem, retrospection on my part whether I am dong the problem right or not. I kept asking myself whether there were other possibilities of doing the same problem, but in a different manner. It involved a complete and exhaustive study of the fundamental principles of mathematics. These types of assessments can boost the level of understanding and confidence levels of the student. Hence the student is able to think and analyze in a much better way. And slowly he can take his problem solving skills up by notches. Bibliography Ruddock, Graham (1998) Mathematics in the School Curriculum: an International Perspective Vulliamy Graham (1998) Teacher Development in Primary Schools: Some other lessons from Taiwan. University of York, United Kingdom Whetton Chris, Twist Liz, Ruddock Graham (2007) STANDARDS IN ENGLISH PRIMARY EDUCATION: The international evidence. PRIMARY REVIEW INTERIM REPORTS Read More