The selection and stimulation of talented students in mathematics is an important challenging problem. Mathematical Olympiads give a standard tool to achieve this goal. Recent studies, for example [3], give some concrete enlightening on the impact of Mathematical Olympiads on the professional career of a mathematician later on.
Creativity is another candidate predictor of life achievements [4]. Scientists developing new ideas or knowledge that may have profound impact on society usually are connected with the so called “big-C” creativity [5]. There is also a lot of creativity in everyday life as people try to solve problems at work and at home or on the road in between (“little-c” creativity).
Math Clubs – doing math together 69 We shall not discuss in details the research on the general notion of creativity and shall recall that creativity is determined by the development of original ideas that are useful or influential. The reader can see [6] for more information and references.
It is well known that problem generation has the potential to stimulate the creativity. As students are encouraged to raise questions and pose problems of their own, rather than to merely solve standard problems following standard algorithms, they take a new and more active role in their own learning. This argument is developed in [7].
Our main goal is to present some results concerning the impact of some problem posing activities in Mathematical circles (clubs) formed by motivated teachers and high school students and the development of pupils’ creativity as a group. Since the group creativity and problem generation are studied separately in the literature, it seems natural to consider their correlation and effective interaction. To study this problem we analyze problems posed and solved by different Math Clubs formed by students and teachers in Tuscany, created with the partnership of the Department of Mathematics at Pisa University.
One can accept the challenge to be involved in problem posing activities in order to acquire a deeper understanding of arguments presented in standard lectures in class.
The creation of new ideas derived from any given topic is also a possible attractive point. Some new problems can be elaborated during classroom activities, starting from the motivation of pupils to pose questions. Our point is slightly different and based on the idea that the real world is a source of many mathematical questions and problems.
Therefore, our initial point seems probably unrealistic, since it is difficult to believe that pupils and their teachers can create new deep mathematical problems starting from concrete argument or activity in the real world. The examples below show that concrete models and units can be constructed and implemented in a class setting.
However, some natural questions arise when one tries to connect (group) creativity and problem generation.
• Is it possible to evaluate or measure the novelty of the problems posed and the creativity of students and teachers involved?
• Can one expect that a high quality preparation and good results in solving mathematical problems imply abilities to create new and stimulating mathematical problems?
• What is the best choice for group activities on problem posing – short term appointments (a couple of hours competitions or lectures) or longer term activities?
Here we try to give some partial answers of the above questions based on a concrete data analysis.
Our approach is based on the following:
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a) implementing a group method of teaching and studying: small groups of secondary school teachers and pupils work together;
b) work on a specific subject (in our case – the 400th anniversary of the Galileo Galilei, astronomy and navigation) in the context of studying the world around us (e.g. preparing math models and math problems on the subject),
The math clubs formed by math teachers and their high school students started their activities after 2005 in Tuscany, Italy, in connection with the Comenius project
"Meeting in Mathematics". In 2009 there were ten clubs – four from Italy (Acutangoli, Brescia, La Squadra del Forte, DINI4, DINI1), three from Bulgaria (Aprilov, PMG1-Burgas, Dimitrovgrad), one from Russia (Saint Petersburg) and one from Japan (Japan). The purpose of our approach has been on the one hand to improve the relations between the secondary schools and the universities in Tuscany, and on the other – to improve the quality of math teaching by proposing new attractive models of work with pupils. To involve a larger group of teachers and pupils we organized the out-of-school activities within a larger and interdisciplinary event – the "Unsung Hero".
It is important to note that the competitiveness was not the dominant element in the organization of the work of the math clubs. The key evaluation criteria were the creativity and the novelty of mathematical ideas, and the capacity in identifying deep mathematical problems in the world around us.
Below we present examples of problems proposed and submitted (together with their solutions) by the math clubs.
As a first step in spherical trigonometry it was suggested to start with the notion of the segments and triangles on a sphere.
Given two points A and B on the sphere with centre O and radius R one can call a
"segment AB on the sphere" the arc connecting A and B and lying on the plane AOB.
Naturally, one can make the interpretation that this is the shortest curve or geodesic connecting A and B on the sphere. The angle between segments AB and AC on the sphere can be defined in an appropriate way. So the general starting point suggested was to study how some basic properties of the plane geometry can be “translated” in corresponding statements for the geometry on the sphere. Here are some of the possible statements or problems proposed by different math clubs.
Problem 1. Let A, B and C be points on the sphere with centre O and radius R, and AB and AC be "segments on the sphere". Then the "angle" between them (with vertex A) is defined by the angle between the tangent lines to the arcs AB and AC.
Find a triangle ABC on the sphere, such that the sum of "angles" with vertices A, B and C is 270°.
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Fig. 1. Spherical triangles ABC and A B C ' ' '
Another example of the joint work of pupils and teachers in the math clubs is the Moon Satellite problem:
Problem 2. The THETA satellite of the GoogleMoon society is equipped with digital camera taking images with an angle of view of 20°. To make photos of high quality the satellite must circle around the Moon at a constant altitude (equal to the Moon radius R=1738 km) its digital camera being oriented all the time to the center of the Moon. Find the minimal length of the satellite’s path such that the whole surface of the Moon could be covered by images of the digital camera.