Mathematical Knowledge, the Analytic Method, and Naturalism

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried to naturalize mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge

A Sequential Homotopy Method for Mathematical Programming Problems

We propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract Hilbert spaces under the Guignard constraint qualification. The method is equivalent to performing projected backward Euler timestepping on a projected gradient/antigradient flow of the augmented Lagrangian. The projected backward Euler equations can be interpreted as the necessary optimality conditions of a primal-dual proximal regularization of the original problem. The regularized problems are always feasible, satisfy a strong constraint qualification guaranteeing uniqueness of Lagrange multipliers, yield unique primal solutions provided that the stepsize is sufficiently small, and can be solved by a continuation in the stepsize. We show that equilibria of the projected gradient/antigradient flow and critical points of the optimization problem are identical, provide sufficient conditions for the existence of global flow solutions, and show that critical points with emanating descent curves cannot be asymptotically stable equilibria of the projected gradient/antigradient flow, practically eradicating convergence to saddle points and maxima. The sequential homotopy method can be used to globalize any locally convergent optimization method that can be used in a homotopy framework. We demonstrate its efficiency for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth Newton approach for the regularized subproblems.Comment: 27 pages, 6 figure

Developing Students Ability To Write Mathematical Proof By Polya Method

Both writing and reading a proof is equally not easy. Some mathematicians attested that students found difficulties in mathematical proving. Mathematics and mathematics education experts like Jones (1997, 2001), Weber (2001), and Smith (2006) found that difficulty in proof writing is due to: lack of theorem and concept understanding, lack of proving ability, and there is a teaching-learning process that unites with the subject. So, it is truly needed that class of writing proof in order to help to generate students’ ability to do mathematical proving. Polya method is going to be that purpose. Keywords: mathematics proof, Polya metho

Sunspot analysis and prediction

Evaluation of several procedures that apply rarely used and exotic mathematical functions yields a unique method of application of common trigonometric functions. These functions appear to produce results in the development of a mathematical model capable of describing all available sunspot data

Developing Clean Technology through Approximate Solutions of Mathematical Models

In this paper, the role of mathematical modeling in the development of clean technology has been considered. One method each for obtaining approximate solutions of mathematical models by ordinary differential equations and partial differential equations respectively arising from the modeling of systems and physical phenomena has been considered. The construction of continuous hybrid methods for the numerical approximation of the solutions of initial value problems of ordinary differential equations as well as homotopy analysis method, an approximate analytical method, for the solution of nonlinear partial differential equations are discussed

Effects of the Jigsaw and Teams Game Tournament (TGT) Cooperative Learning on the Learning Motivation and Mathematical Skills of Junior High School Students

This study aims to: 1) describe the effectiveness of the jigsaw and TGT cooperative learning in the learning motivation and mathematical skills of junior high school (JHS) students, and 2) investigate the significant difference in the learning motivation and mathematical skills between the JHS students learning through the jigsaw cooperative learning and those learning through the TGT cooperative learning. This study was a quasi-experimental study using the non-equivalent pretest and posttest group design. This study involved two experimental classes. The research population comprised Year VII students of SMP Pembangunan Piyungan and the research sample consisted of two classes selected from all Year VII groups, with Year VII.A receiving a treatment of the jigsaw cooperative learning and Year VII.B receiving a treatment of the TGT cooperative learning. The instruments consisted of a test, i.e. a mathematical skill test, and a non-test, i.e. a questionnaire of mathematics learning motivation. To investigate the effectiveness of the jigsaw and TGT cooperative learning in the learning motivation and mathematical skills of JHS students, the data were analyzed using the one sample test. To investigate the significant difference in the learning motivation and mathematical skills between the students learning through the jigsaw cooperative learning and those learning through the TGT cooperative learning, the data were analyzed using the T2 Hotelling. To compare the effectiveness of the jigsaw and TGT cooperative learning in the learning motivation and mathematical skills of the students, the data were analyzed using the t-test. The normality was tested using the univariate approach, namely the Kolmogorov Smirnov, the homogeneity using the Box' M test, and the equivalence of the variance-covariance matrix using the Levene's test. The results of the study show that: 1) the jigsaw cooperative learning is effective for the JHS students’ mathematical skills and mathematics learning motivation; 2) the TGT cooperative learning is effective for the JHS students’ mathematical skills and mathematics learning motivation; 3) there is a difference in the effectiveness of the jigsaw and TGT cooperative learning in the JHS students’ mathematical skills and mathematics learning motivation; 4) the jigsaw cooperative learning method is more effective than the TGT cooperative learning method for the JHS students’ mathematics learning motivation; and 5) the jigsaw cooperative learning method is more effective than the TGT cooperative learning method for the JHS students’ mathematical skills. Keyword: Cooperative Learning, Jigsaw, Teams Game Turnament, Learning Motivation, Mathematics Skill