I am a teacher in Croatia. In my country teaching is primarily directed towards an average student. Gifted students have special needs, and I think they deserve special and individual treatment. It is not easy to prepare all necesary for such diferentiation in one classroom. Giftedness is something a child can develop and build, so it is our obligation to give them the means to do so. I will try to share my way of creating a stimulative and creative environment for those students.
Working with potentially gifted students in Croatia still does not have a solid framework. Therefore, many of us experiment and try to find a model of work that will meet the needs of the students we work with. So I have designed models of work and projects that will meet the needs of my gifted students, but also the desires of other students who attend additional math classes. My goal is to inspire them to do Math. With this article, I will share my ideas and experiences.
Many researchers have dealt with giftedness. Gardner, Renzzulli, Sternberg, Zhang, Gagne, Vlahovic-Stetic, and Cvetkovic-Lay are just some of them. Their work, conclusions, and research methods differ from each other, but most of them have come to similar conclusions.
I will try to summarize their description of giftedness in a few lines:
- Giftedness is not characterized solely by high IQ
- The environment is key to realizing giftedness
- Giftedness consists of cognitive and non-cognitive factors
- Identification is necessary, and it can be used to confirm and verify theories.
Gifted students superiority combine:
- metacognitive skills
- learning skills
- thinking skills
- declarative and procedural knowledge
- motivation (as the main driver)
- faster progress
- achieve a higher level of expertise than average
- creation / creativity.
For them, intrinsic motivation and enjoyment of challenges (focus), setting high internal standards (perfectionism) and difficult relationships with peers (introversion or poor socialization) are specific. Gifted people quickly recognize the problem that needs to be solved, spontaneously find a whole range of solutions, know how to set priorities, conduct a quality selection of information, easily decide what resources they need, and analyze the problem and the process of solving it. Most gifted people have a field of interest in which they develop their talent.
According to Huzjak (2006), Gagné’s model of talent development explains how talent is combined with a chance (luck), environment, motivation, and personality. Areas of talent can be school (academic), art, sports, technology. Gagné also cites “catalysts” that enable the development of giftedness into talent: these are openness to new experience (it means curiosity, but also tolerance of uncertainty), a positive self-image, autonomy, and resilience to stress.
Within the legal framework, the gifted are cared for. However, what should we do when such a child is sitting in our classroom? Most of us find ourselves in front of something new and unexplored. I say new because gifted students are very different from each other and we cannot approach them uniformly. For me, this is a special challenge. Mathematics and logic are often the strong sides of the gifted. Most of them are very early directed by their teachers to additional mathematics classes.
The school where I work identifies potentially gifted students at the end of 3rd grade. Through the 4th grade, they work in special groups. When they reach higher grades, this type of education is not provided through the system. Support is mostly reduced to two volunteer-organized and realized Saturdays in which their creativity and love for science are encouraged, as well as one school-trip a year.
Since I learned a lot about the gifted, I couldn’t allow my extra math classes to be just a “drill” and preparation for the competition. In almost every generation of students who attend additional classes, there are those who love mathematics but do not want to go to the competition.
I conduct additional classes in two parts. One is preparation for the competition and the other is divided into three levels: math games, collaborative learning, and projects.
In mathematics teaching, games are a mean of practice for typical procedures in math. Conducted research shows that such educational games have a positive effect on the speed and quality of learning. However, in additional math classes, I use so-called “math games”. Math games are specific because they:
- include thinking skills,
- do not include luck,
- can be played for leisure (do not require specific mathematical knowledge),
- one game lasts a reasonably short time,
- most often these are games for two players,
- do not require special equipment,
- are suitable for all ages
- a good knowledge of school mathematics is not of special value
It is intuitively clear that math games are important because they develop many abilities. Math games are important because players during the game ask questions that have a mathematical basis: “How is this played?”, “What is the best way to play?”, “What is the winning strategy?”, “What if??”, ” What is the chance that… ..? “. Their mathematical background is interpretation, optimization, analysis, variations, and probability.
Students, faced with such new situations, form attitudes (opinions) that can be related to mathematical ideas, for example: “this game is like…” (isomorphism), “I can win if…” (abstract opinion / special case), “this is true in all these games….” (Generalization), “I can show you that this is worth…”(proving), “I remember a game like this….” (Special case).
The advantage of these games is that they can be quickly presented and explained to students, and they can be tested immediately. After a few “games” played, we can analyze with students the strategies. See pictures of me and my students playing the game.
I always offer different games to students at the beginning of the school year. I classify them into three groups:
a) Strategic: Chomp, Hex, Dots and boxes, Go
b) Combinatorial: Set, Sudoku, Hanoi Towers
c) Tiling and space: HIQU puzzle (Pa´s T-puzzle), Soma cube, Pentomino, IQ puzzler, Tangram, Origami… …
I have been collecting games for years, I have some in their original form, and some made according to models. According to Cvetković-Lay (2002), games evoke satisfaction in students, and develop logical-mathematical and visual-spatial intelligence, combinatorics and competitiveness.
2. COOPERATIVE LEARNING
Collaborative learning as a model of work is very popular. Summer science schools and camps operate on the principle of knowledge exchange among young people. Older students prepare and conduct workshops or lectures on a specific topic under the mentorship of a professor. This model has proven to be very effective and valuable for both groups. Younger people learn more easily from colleagues than from professors, because communication is more direct and different. Older students, in addition to feeling important, have the opportunity to transfer their knowledge and practice their teaching skills and collaboration with younger ones. Older students, in order to be able to transfer their knowledge to younger ones, must know, understand, know how to apply, know how to analyze and judge the area they are talking about and teaching.
Several times a year, I prepare senior students to do workshops with younger students. For such occasions, we prepare math problems, quizzes, puzzles, origami, math games, and enigmatic papers. It is an opportunity for younger students to see the fun side of mathematics, collaborate with older colleagues who are often their role models, and also meet teachers who will work with them from 5th grade on. It is an opportunity for older students to transfer their knowledge and teach younger ones.
Children who get to know each other in this way connect, and collaborate in the years to come, becoming friends and collaborators on various new projects that are often not related to mathematics. This allows them to connect in a way they wouldn’t get the chance otherwise.
It is especially interesting to follow how students who participated in workshops in the 4th grade where they first encountered such methods, become conductors of similar workshops a few years later.
They say that there are many fantastic ideas in this world, but very few are realized. Realization is a very difficult and demanding step and everyone, even the gifted ones, often give up when they see how much work needs to be done and built into their idea in order to create something.
According to Sternberg (1985 according to Vlahović-Štetić, 2005): analytical talent is manifested in the ability to analyzed and abstract thinking. It is measurable by tests. Creative talent emphasizes synthesis, integration, the ability to formulate new ideas, and connect seemingly unrelated content. Giftedness implies the ability to adapt to new situations and to shape the environment according to one’s own needs. I have experienced all these kinds of talents with the students I have worked with so far. Some generations have been particularly creative and persistent enough to realize their ideas. Some have not.
With two groups of gifted students, I realized two big projects that stretched over several months from idea to realization. Under my mentorship, the students created two board games which were also assessed by the professional judges, at the mathematics festival in Pula, as high-quality materials.
“Who am I?” It is a board game for 1-25 players that connects 25 math problems and historical data related to the city of Karlovac. The player draws the card on which the task is located. The task is solved classically on paper. When the player finds a solution, he has to look it up on the board where all the solutions are. The solutions are years, well-known people from Karlovac, statistical data, or some interesting things related to the history of the city of Karlovac. Under the solution card, the player will find a brief interpretation of the number he was given. Then, he covers his solution with the task card and starts paving the game board. Solutions arranged according to a given rule reveal a solution – and that is the image of the Karlovac star.
“Save Karolina” is a game whose plot is located in the old town of Dubovac. Princess Carolina is in the castle. The castle is guarded by the terrible dragon Drago. This game is conceived as a competition, in the knowledge and skill of solving mathematical problems of four players. The players in this game will be knights competing for the princess’s hand, by saving her from the dangerous dragon guarding the Dubovac castle. The knights are arranged on the game board so that they each come their own way. They come from Sisak, Ogulin, Ozalj, and Slunj. Each of them must go through 10 steps. Knights cross-steps by solving math problems. They can choose between 3 difficulty levels: task for 1 point, 2 points or 3 points. Each point is one step.
In conclusion: the way I work with gifted children varies from generation to generation. The pace of work is never the same. I adapt to the needs and desires of students. I don’t always do everything as I planned, nor every year equally. Realization depends on the students I work with. This model of work requires a lot of research, preparation, creativity and goodwill, which unfortunately does not fit into our working hours. Yet such work fills me with joy and satisfaction and it makes that important distinction between occupation and vocation.
Author: Tea Borković
Tea Borković is a mathematics and physics teacher at OŠ Grabrik in Karlovac, Croatia. She is specially dedicated to gifted education and has produced two board games with her gifted students. Her interests are the application of mathematics in everyday life and board games with a mathematical background.