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New York: Springer. Murray, H.
Learning through problem solving. Newstead Eds. Stellenbosch: PME. Technology-based assessment of creativity in educational context: the case of divergent thinking and its relation to mathematical achievement. Pelczer, I. Problem posing strategies of mathematically gifted students.
Publication date: Topics: Mathematics -- Study and teaching. Publisher: Chicago: The Open Court Publishing Company; [etc, etc. Free kindle book and epub digitized and proofread by Project Gutenberg.
Leikin Ed. Tel Aviv: Center for Educational Technology. Polya, G. Silver, E. On Mathematical Problem Posing. For the Learning of Mathematics, 14 1 , Teaching Children Mathematics, 12 3 , An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27 5 , Sriraman, B.
An analysis of constructs within the professional and school realms. The Journal of Secondary Gifted Education, 17, 20— General Creativity.
Kerr Ed. New York: Sage Publications. Sternberg, R. The concept of creativity: Prospects and paradigms. Sternberg Ed. New York: Cambridge University Press. Educational Studies in Mathematics, 83 1 , Walter M. Problem posing and problem solving: an illustration of their interdependence. Mathematics Teacher, 70 1 , Wang, M. Theoretical physics has historically been, and remains, a rich source of inspiration.
Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically. Also provably unsolvable are so-called undecidable problems , such as the halting problem for Turing machines. Many abstract problems can be solved routinely, others have been solved with great effort, for some significant inroads have been made without having led yet to a full solution, and yet others have withstood all attempts, such as Goldbach's conjecture and the Collatz conjecture.
The all of mathematical new ideas which develop a new horizon on our imagination not correspond to the real world. Science is a way of seeking only new mathematics, if all of that correspond. So a mathematical problem that not relation to real problem is proposed or attempted to solve by mathematician. And it may be that interest of studying mathematics for the mathematician himself or herself made much than newness or difference on the value judgment of the mathematical work, if mathematics is a game.
Popper criticize such viewpoint that is able to accepted in the mathematics but not in other science subjects. Computers do not need to have a sense of the motivations of mathematicians in order to do what they do. The vitality of computer-checkable, symbol-based methodologies is not inherent to the rules alone, but rather depends on our imagination. Mathematics educators using problem solving for evaluation have an issue phrased by Alan H. The same issue was faced by Sylvestre Lacroix almost two centuries earlier:.
Such degradation of problems into exercises is characteristic of mathematics in history. High among the purposes stated for the study of arithmetic many authors of the time placed speed, memory and accuracy by mechanical rules. There was an emphasis in arithmetic on drill for perfection and automatic response at the expense of meaning and understanding.
In , a dissertation by Boulware is representative of the quest for the development of "meaning" in mental computation stirred by Brownwell , who urged that meaning and seeing sense in what is being learned should be the central focus of arithmetic instruction. Boulware's conception of mental computation is as follows: Mental arithmetic deals with number as a unified, consistent system, and not as an aggregate of unrelated facts.
It proceeds to the analysis of number combinations by processes of meaningful experiences with concrete numbers, reflective thinking in number situations, seeing relationships, and discovery of new facts as an outgrowth of known facts pp. In , in an article by Sister Josefina there seems to begin interest in mental computation and in the NCTM yearbook on computational skills there appears an article by Trafton where the need for including proficiency with estimation and mental arithmetic as goals for the study of computation is presented.
A good number of studies and articles about mental computation appeared in the period of the s e. Reys, R. With the increase of studies in cognitive skills and number sense e. Simon, ; Resnick, ; Silver, ; Schoenfeld, ; Greeno, ; Sowder, and more recent studies mentioned in this chapter, mental computation is suggested to be related to number sense, needed for computational estimation skills and considered a higher order thinking skill.
In a study by Reys, Reys and Hope they argued that the low mental computation performance reported in this study most likely reflected students' lack of opportunity to use mental techniques they constructed based on their own mathematical knowledge. The study of Reys, Reys, Nohda and Emori assessed attitude and computational preferences and mental computation performance of Japanese students in grades 2, 4, 6, and 8. A wide range of performance on mental computation was found with respect to all types of numbers and operations at each grade level.
The mode of presentation visual or oral was found to significantly affect performance levels, with visual items generally producing higher performance. The strategies used to do mental computation were limited, with most subjects using frequently a mental version of a learned algorithm. In a study by G. Thompson about the effect of systematic instruction in mental computation upon fourth grade students' arithmetic, problem-solving and computation ability, a significant difference favored the group taught mental computation, with girls improving more than boys.
According to Markovits and Sowder it would seem reasonable that if children were encouraged to explore numbers and relations through discussions of their own and their peers' invented strategies for mental computation, their intuitive understanding of numbers and number relations would be used and strengthened. Okamoto found that children's understanding of the whole number system seemed to be a good predictor of their performance on word problems. Cross-cultural research has identified a variety of mental computation strategies generated by students, e.
Sribner points out that individuals develop invented procedures suited to the particular requirements of their particular occupations. In a study on individuals who are highly skilled in mental arithmetic Stevens , forty-two different mental strategies were observed. Efficient, inefficient and unique strategies were identified for each of five groups grade 8.
Dowker describes in a study the strategies of 44 academic mathematicians on a set of computational estimation problems involving multiplication and division of a simple nature. Computational estimation was defined as making reasonable guesses as to approximate answers to arithmetic problems, without or before actually doing the calculation. Observing people's estimation strategies, Dowker suggests, may provide information not only about estimation itself, but also about people's more general understanding of mathematical concepts and relationships. From this perspective Dowker concludes that estimation is related to number sense.
Sowder who agrees with this position points out that computational estimation requires a certain facility with mental computation. In a study by Beishuizen , he investigated the extent to which an instructional approach in which students use of the hundreds board supported their acquisition of mental computation strategies. In the course of his analysis, he found it necessary to distinguish between two types of strategies for adding and subtracting quantities expressed as two digit numerals as follows:.
Beishuizen's analysis indicates that N10 strategies are more powerful, but that many weaker students used only strategies. The study's findings also suggest that instruction involving the hundreds board can have a positive influence on a student's acquisition of N10 strategies. Fuson and Briars and others have also identified these strategies.
Hope points out that because most written computational algorithms seem to require a different type of reasoning than mental algorithms, an early emphasis on written algorithms may discourage the development of the ability to calculate mentally. Lee recommends that perhaps it is time to investigate changing our traditional algorithms for addition and subtraction to left-to-right procedures. According to Reys et al.
Mental computation can be viewed from the behaviorist perspective as a basic skill that can be taught and practiced. But it can also be viewed from the constructivist view in which the process of inventing the strategy is as important as using it. In this way it can be considered a higher-order thinking skill Reys et al. Addition , subtraction and teaching strategies. The Curriculum and Evaluation Standards for School Mathematics NCTM recognizes that addition and subtraction computations remain an important part of the school mathematics curriculum and recommends that an emphasis be shifted to understanding of concepts.
Siegler indicated how important it is for children to have at least one accurate method of computation. In a study by Engelhardt and Usnick while no significant difference between second grade groups using or not using manipulatives was found, significant differences in the subtraction algorithm favored those taught addition with manipulatives. Usnick and Brown found no significant differences in achievement between the traditional sequence for teaching double-digit addition, involving nonregrouping and then regrouping, and the alternative, in which regrouping was introduced before non-regrouping examples in second graders.
Ohlsson, Ernst, and Rees used a computerized model to measure the relative difficulty of two different methods of subtraction, with either a conceptual or a procedural representation.
The results of the use of the model suggested that regrouping is more difficult to learn than an alternative augmented method, particularly in a conceptual representation, a result that contradicts current practice in American schools.