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Regula falsi in lower secondary school education II
121-142Views:155The aim of this paper is to investigate the pupils' word problem solving strategies in lower secondary school education. Students prior experiences with solving word problems by arithmetic methods can create serious difficulties in the transition from arithmetic to algebra. The arithmetical methods are mainly based on manipulation with numbers. When pupils are faced with the methods of algebra they often have difficulty in formulating algebraic equations to represent the information given in word problems. Their troubles are manifested in the meaning they give to the unknown, their interpretation what an equation is, and the methods they choose to set up and solve equations. Therefore they mainly use arithmetical and numerical checking methods to solve word problems. In this situation it is necessary to introduce alternative methods which make the transition from arithmetic to algebra more smooth. In the following we will give a detailed presentation of the false position method. In our opinion this method is useful in the lower secondary school educational processes, especially to reduce the great number of random trial-and-error problem solving attempts among the lower secondary school pupils. We will also show the results of some problem solving activities among grade 6-8 pupils. We analysed their problem solving strategies and we compared our findings with the results of other research works.
Subject Classification: 97-03, 97-11, 97B10, 97B50, 97D40, 97F10, 97H10, 97H20, 97H30, 97N10, 97N20
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Maximum and minimum problems in secondary school education
81-98Views:85The aim of this paper is to offer some possible ways of solving extreme value problems by elementary methods with which the generally available method of differential calculus can be avoided. We line up some problems which can be solved by the usage of these elementary methods in secondary school education. The importance of the extremum problems is ignored in the regular curriculum; however they are in the main stream of competition problems – therefore they are useful tools in the selection and development of talented students. The extremum problem-solving by elementary methods means the replacement of the methods of differential calculus (which are quite stereotyped) by the elementary methods collected from different fields of Mathematics, such as elementary inequalities between geometric, arithmetic and square means, the codomain of the quadratic and trigonometric functions, etc. In the first part we show some patterns that students can imitate in solving similar problems. These patterns could also provide some ideas for Hungarian teachers on how to introduce this topic in their practice. In the second part we discuss the results of a survey carried out in two secondary schools and we formulate our conclusion concerning the improvement of students' performance in solving these kind of problems. -
Heuristic arguments and rigorous proofs in secondary school education
167-184Views:83In this paper we are going to discuss some possible applications of the mechanical method, especially the lever principle, in order to formulate heuristic conjectures related to the volume of three-dimensional solids. In the secondary school educational processes the heuristic arguments are no less important than the rigorous mathematical proofs. Between the ancient Greek mathematicians Archimedes was the first who made heuristic conjectures with the methods of Mechanics and proved them with the rigorous rules of Mathematics, in a period, when the methods of integration were not known. For a present day mathematician (or a secondary school mathematics teacher) the tools of the definite integral calculus are available in order to calculate the volume of three dimensional bodies, such as paraboloids, ellipsoids, segments of a sphere or segments of an ellipsoid. But in the secondary school educational process, it is also interesting to make heuristic conjectures by the use of the Archimedean method. It can be understood easily, but it is beyond the normal secondary school curriculum, so we recommend it only to the most talented students or to the secondary schools with advanced mathematical teaching programme. -
Solving word problems - a crucial step in lower secondary school education
47-68Views:87Algebra is considered one of the most important parts of Mathematics teaching and learning, because it lays the foundations of abstract thinking as well as reasoning abilities among the lower secondary school pupils who have just transited from the world of numbers and computations to the area of equalities, signs, symbols and letters. The present article focuses on the fact that how the transition from arithmetic to algebra can be made more smooth. We have concentrated our experiments towards the approach of algebraic reasoning and its utilities in filling the gap between arithmetic and beginning algebra in lower secondary school education.We also underline the importance of another approach in overcoming the challenges in the transition from arithmetic to algebra, to enhance and make algebraic learning more effective, with special considerations to word problem-solving processes. In our opinion, we have to go through three phases in the introducing of algebra in Grade 7 Mathematics education: Regula Falsi method (based only on numerical calculations); functional approach to algebra (which combines the numerical computation with letter-symbolic manipulation); and writing equations to word problems. The conclusions of the present article would be helpful to Mathematics teachers for applying themselves to develop the pupils’ interest in word problem-solving processes during algebra teaching classroom activities.
Subject Classification: 97B10, 97C30, 97C50, 97D10, 97D40
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Regula falsi in lower secondary school education
169-194Views:99The aim of this paper is to offer some possible ways of solving word problems in lower secondary school education. Many studies have shown that pupils in lower secondary school education (age 13-14) encounter difficulties with learning algebra. Therefore they mainly use arithmetical and numerical checking methods to solve word problems. By numerical checking methods we mean guess-and-check and trial-anderror. We will give a detailed presentation of the false position method. In our opinion this method is useful in the loweer secondary school educational processes, especially to reduce the great number of random trial-and-error problem solving attempts among the primary school pupils. We will also show the results of some problem solving activities among 19 grade 8 pupils at our school. We analysed their problem solving strategies and compared our findings with the results of other research works. -
Transition from arithmetic to algebra in primary school education
225-248Views:104The main aim of this paper is to report a study that explores the thinking strategies and the most frequent errors of Hungarian grade 5-8 students in solving some problems involving arithmetical first-degree equations. The present study also aims at identifying the main arithmetical strategies attempted to solve a problem that can be solved algebraically. The analysis focuses on the shifts from arithmetic computations to algebraic thinking and procedures. Our second aim was to identify the main difficulties which students face when they have to deal with mathematical word problems. The errors made by students were categorized by stages in the problem solving process. The students' written works were analyzed seeking for patterns and regularities concerning both of the methods used by the students and the errors which occured in the problem solving process. In this paper, three prominent error types and their causes are discussed.