Instructor: Dr. Péter JUHÁSZ
Course Prerequisits: none
In the center of the traditional education of mathematics we see the teacher standing at the teachers' podium revealing the truth and endlessly explaining various topics. They see their great moments in making difficult things easy. It does not bother them that in most cases the children find their questions strange and their answers are of no interest to the kids, and also that their efforts result in little or no fun. By .thinking. he or she means that the students should think about what was told them.
Instead of this we have a vision about students in schools being autonomous thinkers and creative human beings. We do not need obedient slaves, rather companions who take part in the great adventure of the human mind called mathematics. In the center of the classes should be the kids pursuing answers for questions they are interested in and happy to reach their goal. The role of the teacher is still very important, but they are not in the center, they are rather pulling strings from the background. They plan classes so that it is always clear which questions should be answered, why these questions are interesting, and the teacher is careful to give the students the right tools well in advance so they can successfully tackle the problems.
This method slowly but steadily will find its way into education, and not only in mathematics. The designers of this course strongly believe that sooner or later schools will become places where kids enjoy spending time because they study interesting things and use their own brains for thinking.
Students of this course will take this route. Initially they will explore the world of mathematics as .brave pioneers., which means they will take the role of kids-in-a-classroom. But the ultimate goal is to enable them to teach this way, so we will switch the role of being a child with the teachers-to-be and discuss the inner workings of what we do. We also show them the similarities and differences of their and the actual kids' reactions.
The choice of mathematics subject areas we develop will depend on student.s preference and background. Special emphasis will be put on pursuing paths laid out by student.s questions.
To summarize of the philosophy of this course: mathematics is not the private affair of a selected few, but the fun of discovering and autonomous thinking is a right of every kid. Nevertheless, as most of our experience by dealing with gifted students, this course will mainly focus on methods and problems that can be used to educate gifted children.
Course StructureClasses will be very similar to a problem solving seminar. But there is a significant difference: during the course the students will assume two roles.
The first role is the role of the kids. They will meet problems given to gifted children. These problems will be presented similarly to the way we present them to kids: group work, individual work, team play, quick questions, etc.
The other role is the role of the teacher. We will discuss what should be taken care of when giving the kids a problem, what kind of factors should be taken into account and what the potential caveats are. What are the typical solutions we expect from the kids and what are the most probable wrong answers, proofs and constructions.
This means that the classes will be very similar to the programs for gifted kids occasionally interrupted by didactic explanations.
- Creating connected problems or problem sheets which lead to the desired goal while possibly many of the kids will solve most of the problems by their own.
- Creating a web of ideas inside the kids' head, often placing the methods or ideas required to solve the problem unnoticed, so that in the deciding moment everything should be together to make the discovery possible.
- Parallel presentation of several threads of problems similar to the several melodic voices in polyphonic music. What are the ingredients of such a polyphonic construction to sound great and to make the process of solving these problems a pleasurable experience?
- What to do if despite all our efforts the discovery is simply not happening? And what if some kids find it too easy what is just right for the others?
- How to ask sensible and interesting questions after solving a problem and how to make this an internal ability and ambition in children?
How to come up with good problems?
We will demonstrate several classroom and work situations. Individual work, group work, team play, quick questions and discussions.
The above mentioned objectives will be mainly approached by using the first role. Students will solve problems, which are connected to each other, and more than one thread of problems will run parallel. Students need a lot of idea to solve them, and after solving a problem they have to ask good questions. They have to guess the typical answers and solutions of children.
We will discuss the following topics: proofs of impossibility, mathematical induction, interesting constructions, two-player-games, special real functions, etc.
The actual problems of the course cannot be decided in advance, and the set of problems can be changed between and during semesters. There are two important reasons for this:
It is part of the method that we adapt to the abilities, interests and possibilities defined by the group.
The course is closely connected to another course with the goal to introduce students to the camps also by visiting. The material of this course will be in sync with the material of the camp to be visited, so students can study how the problems they've met during the course are applied in real life situations.
The aim of this course is to introduce students to the basics of making children discover mathematics. Parallel with the course objectives the goals are the following:
Participants will be able to solve problems which have been designed for talented high school students.
Participants will have a detailed conception what a thread of problems means and how to create an own one.
Participants will be able to give appropriate hints to children if the discovery is not happening.
Participants will have conception about how to ask good question after solving a problem, and how to teach asking students.
It would be also an important goal that students will be able to recognize gifted students and work with them. Among the several possibilities to engage these kids we will put most of the emphasis on demonstrating the work in our special camps for gifted children.
Mid-term exam. This examination contains problems we discussed at the previous lessons. Students do not need own ideas in problem solving, they have to show that they understood the problems and their solutions, and they can write down the solutions correctly. 25%
Final exam. This examination contains new problems, but these are similar to those we discussed at the lessons. Students need own ideas in problem solving, they have to show that they can write down the solutions of unknown problems correctly. 35%
Reading: John Mason: Effective Questioning and Responding in the Mathematics Classroom. Corresponding to this paper students have to write an essay about effective questioning and to work out a list of problems and solutions. The problems have to be organised into well-thought-out structure considering basic principles of discovery learning and the content of the above mentioned article. 30%
Students have to keep journal on what they are learning at the lessons. Every week indicates an entry in the journal. These entries may tell what they like at the lessons of the week, what they find surprising or what they do not agree. 10%