Final Research Proposal Applied Mathematics

These numerical methods are very important to solve the numerical problems in research, teaching, and learning; moreover, I can use the numerical methods to write programs for computatio[r]

1 The Tran ESOL 5010 Research Proposal of Using Scientific Calculators to Teach and Learn Mathematical Problems Introduction Today, we use technology in many areas of life to support our work in the best ways Generally, that is the world of the internet, artificial intelligence, and computer software; particularly technology used to teach and learn some courses in school classrooms; specifically scientific calculators support students and teachers in learning and teaching As a result, educationalists go on to say that the use of technology in modern classrooms is very necessary for activities and is a standard to appreciate the success of learners and teachers Otherwise, technology supports as well as creates the relationship between theory and practice of activities; thus we can understand that technology will make to increase the effect of activities and to contribute to the success of the educational process We also know that many conferences and papers have had research of the use of technology in teaching and learning; typically the conferences such as the ATCM (Asian Technology Conference in Mathematics) and ICTCM (International Conference on Technology in Collegiate Mathematics) In fact, using technology in modern classrooms is a very important and helpful skill in professional education (Hooper & Rieber 1995) At high schools and universities, the following scientific calculators are used: graphics calculators, basic calculators, and financial calculators We see that scientific calculators are universally used in classrooms in many countries in the world with the following brands: Texas Instrument, Hewlett Packard, Casio, and Sharp Scientific calculators solved almost all of the mathematical problems in mathematics, physics, and chemistry Students use scientific calculators to calculate their mathematical problems containing algebraic problems, calculus, and statistics problems Moreover, teachers used graphics calculators to teach trigonometry in modern school classrooms (Kissane & Kemp 2009) Although many mathematicians presented specific mathematical problems, they always had the same ideas that scientific calculators will help students compute their problems quickly and exactly Students will save much time in learning, and the use of scientific calculators will make their calculations in the best ways Additionally, graphics calculators can plot graphs visually to help them intimately understand problems In the limit of this proposal, I would like to make some suggestions to instructors and students for the use of scientific calculators to teach and learn mathematical problems Literature Review The ability to use scientific calculators is an important skill in natural science courses It is necessary to note that the focus in many classrooms is on educational technology as compared to traditional classroom activities (Hooper & Rieber 1995) According to the authors, generally educational technology involves applying ideas and tools from different sources to create the best learning environments possible for students such as illustrated in Figure 1, the authors presented five steps in order to understand and compare both traditional and modern applications of technology in education, including Familiarization, Utilization, Integration, Reorientation, and Evolution; differently the technology will likely be misused or discarded (Rieber & Welliver,1989; Marcinkiewicz,1991) Figure (Hooper & Rieber 1995) Nevertheless, the use of scientific calculators has specific skills such as using graphics calculators to solve mathematical problems in trigonometry (Kissane & Kemp 2009) Both authors used the Casio fx- 9860G II and Casio Class Pad graphics calculator to solve and illustrate mathematical problems in trigonometry By the figure illustrations of problems, their own research further argued the relationship between the functions and graphs (see Figure 2), theory and practice, and statistical data and figures (see Figure 3) Figure (Kissane & Kemp 2009) Figure (Kissane & Kemp 2009) In other words, the graphing calculator activities are generalized and applied regularly in college textbooks to teach and learn the courses in college algebra (Kaufmann 2002) The novelty of the textbook was to present graphing calculator activities and show some ideas to make many calculations on the graphics calculators Actually, the textbook helped students to discover difficult algebraic problems by graphing calculator activities on the Texas Instrument – 84 graphics calculators, so the student’s learning environment and instructor’s teaching became be more attractive and visual Though there are some difficult problems in the skills of the use of scientific calculators, researchers always stated the graphing calculator activities in the best ways and wrote many guidebooks for the use of scientific calculators Furthermore, researchers developed a new branch in mathematics for scientific calculators and offered the computational algorithms to solve difficult problems in feasible ways (Nguyen 2007) In addition, researchers encouraged the development of applications of scientific calculators, so they used mathematical methods in applied mathematics in solving some problems by the Casio fx-570ES scientific calculator (Boon, Gaik & Han 2011) (see Figure 4) Figure (Boon, Gaik & Han 2011) Based on their research, I really encourage using scientific calculators to teach and learn natural science courses In fact, student can learn skills to use scientific calculators and apply skills to solve their mathematical problems quickly, but they not need to use calculators for all their simple problems For example, with simple problems such as “1+2, x 4, 10: 5, “ if students use calculators for these simple problems then technology will likely be misused or discarded (Rieber & Welliver 1989; Marcinkiewicz 1991) In this proposal I will investigate a method for students and instructors in learning and teaching Method The method for this research proposal will be referred from Boon, Gaik & Han’s article I will show a mathematical method in learning, teaching, and research to illustrate the problems which are presented in the introduction and literature review section Furthermore, in the educational process, I need to have the methods to materialize and analyze the problems; one of the methods is the Newton-Raphson method This method is applied to solve mathematical problems; typically it will help students in solving linear equations by using the built-in derivative function in the Casio fx-570ES This method is also used by the scientific calculators in special ways as well as helps students and teachers to discover new problems of technology in the educational process Mathematically, in numerical analysis, the Newton-Raphson method is one of the best approximation methods to find the roots of the linear equations The Newton-Raphson method is given as the following general form xn+1 = xn − f ( xn ) , f ' (xn ) with i = 0,1,2 where f ( x ) is the linear equation, f ' ( x ) is the derivative of the linear equation f ( x ) , and this method is illustrated by the below figure Figure In fact, I can apply the numerical methods in applied mathematics to determine the approximate roots By the experience of the use of calculators, I can use the numerical methods on the scientific and graphics calculators For example, by using the Newton- Raphson method, I can find the approximate root of the equation f(x) = x^2 - 6x+2 In the traditional way, I can use the Newton- Raphson formula above to solve this problem on paper by hand, but I really apply technology in combination with the Newton-Raphson formula, and using the Casio fx-570ES calculator to find an approximate root of the equation By analyzing and calculating the functions on the calculator, I found an approximate root for the equation f(x) = x^2- 6x+2 is x= 0.3542 with four decimal places From using the Newton-Raphson method on the calculator, I will carry out the other methods such as the Euler method, Lagrange method, the Simson method to evaluate the approximate values in different mathematical problems Obviously, these methodologies are meaningful to teach and learn the mathematical courses in algebra and in my study; additionally I can apply the methods and combine the use of calculators to increase the effect of activities in the educational process These numerical methods are very important to solve the numerical problems in research, teaching, and learning; moreover, I can use the numerical methods to write programs for computational functions on the scientific calculators, and approaches to research depend on these methods which vary considerably both within and between theory and applied mathematics Results and Discussion As stated in the numerical method above, I will obtain the approximate roots of the equation to see the process of the computation of the Newton- Raphson method Through applying the Newton- Raphson formula and the functions of the Casio fx- 570ES scientific calculator, I can evaluate the roots of the equation According to the result of solving the equation f(x) = x^2 - 6x+2, I found a table of the approximate roots of this equation x[1] = 0.250000000000000 x[2] = 0.352284376457591 x[3] = 0.354248331161185 x[4] = 0.354248689003011 x[5] = 0.354248688935397 x[6] = 0.354248688935409 x[7] = 0.354248688935409 x[8] = 0.354248688935409 x[9] = 0.354248688935409 From the data in the table above, I can see from the first root through the ninth root of this equation which changed the decimal numbers Actually, I can conclude the roots will have the best approximate root if the roots of this equation converged; typically the roots of this equation converged to x = 0.354248688935409, and by the loops of the Newton-Raphson method, I see the next roots are more accurate than the previous roots In this method, I need to choose an initial root to find approximate roots, and I have two ways to determine the roots of equations, including the traditional way such as the calculation is written on paper by hand, and technological way such as the calculation is made on scientific calculators In fact, I need to understand the numerical methods clearly and apply these methods to solve problems on scientific calculators Moreover, with the use of technology, I can find new application ways to use the functions of scientific calculators Though these numerical methods are usually written on computers by programming language, using programming languages on computers is a difficult problem for students, so I can use scientific calculators in simple ways to discover and write programs quickly Accordingly, that is an important field of research to develop new methods in my field of study, and it will encourage students in their study Finally, the application of the numerical methods can be limited to students in colleges, teachers, and researchers It can not use to teach and learn numerical methods for students in high school At high schools, students apply only the normal functions and simple methods References Boon, L.K., Gaik, T.K & Han, C.T (2011) Solving Non-Linear Equation by Newton-Raphson Method using Built-in Derivative Function in Casio fx570ES Calculator Proceeding of the 16th Asian Technology Conference in Mathematics Blacksburg, VA: Mathematics and Technology, LLC The Tran (2009) Using Casio fx – 500/570MS Scientific Calculators to solve mathematical problems in Vietnam school Proceeding of the 14th Asian Technology Conference in Mathematics Blacksburg, VA: Mathematics and Technology, LLC, 315-324 Kissane, B., & Kemp, M (2009) Teaching and Learning Trigonometry with Technology Proceeding of the 14th Asian Technology Conference in Mathematics Blacksburg, VA: Mathematics and Technology, LLC, 315-324 10 The Tran (2007) Using Casio fx – 500/570MS Scientific Calculators to solve mathematical problems Hanoi, Vietnam: VietnamCalculator Company Kaufmann, E J (2002) College Algebra (5th ed.).Pacific Grove, CA: Thomson Learning, Inc Hooper, S., & Rieber, L P (1995) Teaching with technology In A C Ornstein (Ed.), Teaching: Theory into practice, (pp 154-170) Needham Heights, MA: Allyn and Bacon Marcinkiewicz, H (1991) The Relationships of Selected Personological Variables To the Use of Available Microcomputers by Elementary School Teachers Doctoral dissertation, The Pennsylvania State University Rieber, L., & Welliver, P (1989) Infusing Educational Technology into Mainstream Educational Computing International Journal of Instructional