Fostering Mathematically Talented Children (PriMa)

At the university of Hamburg we work with mathematically gifted primary grade pupils (8-10 years old) since autumn 1999[1]. It is an enrichment program as at school the pupils participate in their regular lessons and come to us in their leisure time.

There are about 12.000 third-graders in Hamburg. Therefore, one of our research questions is ”How to create a process of talent searching to build up a group of 50 children?”.

Due to the age of the children, we have to deliberate what kind of forms we can use to select the children. In our opinion it is not enough to test exclusively intelligence. For that reason, we created a process orientated procedure which contains some lessons per weekend in order to prepare the children for the following maths test and to help them to decide by themselves whether they like our problems or not. Our tasks are much more complex than those in school: during this weekend, the children work on four tasks.

Our research about IQ>130 and results in our math tests shows a correlation of .34 between the IQ-tests and the math-tests. We can conclude that high abilities in mathematics can occur in children who are not overall gifted and overall gifted children not always are successful in our math test. Most of the best children are as good in the maths test as in the rehearsal lesson as well as in the results of the intelligence test. But there are always children who are very good in mathematics and show only an average or an above average intelligence. There are also children who perform very good results in the maths test but only partly in the intelligence test. It is not the case that all children with very good results in the maths test also show good language capabilities.

 

Possible reasons

1. The problems we give the children are not of the same character as the questions in intelligence tests. From analyzing our problems we find comparable components like those in intelligence tests, like building analogies, finding patterns, inductive thinking, … but only from an abstract point of view. Sternberg showed that ”the performance components of inductive reasoning … are quite general across test formats typically found in intelligence tests” (Sternberg 1986, S. 226). Research like that of of Waldmann and Weinert (1990) leads them to the opinion that the capability to use such components depends on the character of the problem.

2. Due to the complexity of a problem, differences between performance on a math task and results of an intelligence test might arise.

We developed maths problems in order to offer the opportunity to develop various patterns of action, as Kießwetter (1985) calls them, which are useful especially for solving mathematical problems. These problems are usually not practiced in school because of their high complexity.

About our problems:

We suppose that constructing the learning environment is the biggest challenge for teachers when working with mathematically gifted children. In our courses, a special mathematical knowledge is not important for the children. We use an enrichment program in that sense that we try to develop problems which can be solved on the base of the knowledge demanded by the official curricula for children of this age. However nevertheless, the children are demanded because the given problems are complex in several respects:

  • Many steps have to be worked out.
  • Much information represented with low redundancy has to be processed.
  • The represented problems are real ones which means: children of this age do not know any ways of solving before.

The problems we present are open in several respects:

  • The solution process is open:

There are several ways to find the solution, demanding or less demanding ones. This is a very important aspect due to the different experiences and capabilities of the children. Therefore, we construct our material that way that every participating child can be successful in solving at least parts of the questions. Because they often need quite a long time to work on the problems, it is necessary for all of them to be successful. We hope that  this strategy creates not only  an initiative motivation but also a motivation during the solution process, in order to stimulate patience and stamina over a longer period of time.

  • Various patterns can be found

As we ask the children to look for patterns and structures, their answers can be different. Looking for mathematical structures and patterns is one of the important approach to reduce the abundance of information.

  • Finding of connected problems to develop problem fields

From this perspective the end of a problem is open too. The children can learn to ask further. They learn, that finding an answer only means to find just one out of several possibilities to solve a problem.  A first step in this direction is to pose them connected problems after solving a question. By this they can see analogies between the posed questions. Nevertheless, the questions are posed in a way that they can be solved isolated. We are interested whether the children use analogies and what kind of help supports the children to recognize them.

Our intention is to develop materials that allow children to behave like little researchers. Mathematics in this sense is a live and permanently developing science. It is natural that children of primary school age do not yet carry out theory formation. However, they do steps in this direction. Some of our problems are not only used in the primary school project but also in lower and upper secondary level groups within the framework of a promotion project of the William Stern Society under the guidance of Prof. Kießwetter. In these groups some small theory formatting processes can be observed.

Opening problems is a method for fostering mathematically gifted children which for example are also found in Hashimoto and Becker (1999) and (Sheffield 1999). One important question concerns the kind of opening the problems. As the children we work with are still very young and have no experience with work techniques like the control of the procedure or patterns of action (Kießwetter 1985) like sorting the available materials for recognising patterns, and there are borders regarding their working memory. Thus, it is necessary to help the children not to loose control during their solution process. One aspect therefore is, that they experience first successes very quickly during the process; and another aspect is that we control their first steps in order to lead them to the mathematical core of the problem.

Goals

Developing the following capabilities

  • mathematical thinking
  • Problemsolving
  • Metacognitive components
  • “Staying power” and dealing with frustration (Frustrationstoleranz und Durchhaltevermögen)

Working with a problem

First we show them how to deal with the problems, to pose supporting questions for finding solutions or asking them to go deeper into a problem. Beside this, we have to take into account the differences concerning their experiences with problem solving. And there are also differences in the children’s capabilities. Apart from this, the lessons at university are demanding for every child concerning social and emotional aspects. Many children are very good in maths at school so that they are used to solve a task at once and to be successful and also to be the best. Therefore, to work with other children who perhaps work quicker, who are similarly bright, as well as to work at an unfamiliar place with unknown teachers is sometimes hard for the children. Not every child who seems to be able to participate in our project  is able to cope with these conditions. We try to minimize these kinds of problems for the children by an elaborate organisation that gives the children the feeling that at least the external conditions are reliable.

References

Hashimoto, Y. / Becker, J. (1999). The open approach to teaching mathematics – mathematics in the classroom: Japan. Developing mathematically promising students. L. J. Sheffield. Reston, Virginia, National Council of Teachers of Mathematics: 101-119.

Kießwetter, K. (1985). ”Die Förderung von mathematisch besonders begabten und interessierten  SchĂŒlern – ein bislang vernachlĂ€ssigtes sonderpĂ€dagogisches Problem.” Mathematisch-naturwissenschaftlicher Unterricht 38. Jg., Heft 5: 300-306.

Sheffield, L. J. (1999). Serving the Needs of the Mathematically Promising. Developing Mathematically Promising Students. L. J. Sheffield. Reston, Virginia, National Council of Teachers of Mathematics: 43-55.

Sternberg, R. J. (1986). A triarchic theory of intellectual giftedness. Conceptions of giftedness. R. J. Sternberg and J. E. Davidson. Cambridge. New York, Cambridge University Press: 223-243.

Waldmann, M. / Weinert, F. E. (1990). Intelligenz und Denken. Perspektiven der Hochbegabungsforschung. Göttingen, Verlag fĂŒr Psychologie Dr. C. J. Hogrefe.


[1] Our project can be viewed as a continuation of a William-Stern-Society project for fostering mathematical gifted pupils of the secondary level Prof. Dr. Kießwetter and his group runs since more than 20 years.

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