Problem Solving Seminar for Future Physics Teachers

The paper presents a seminar for future physics teachers focused on problem solving and creativity in problem solving. In this seminar, students are encouraged to find effective strategies for various problems on their own, to use multiple representations, and to be aware of barriers as well as possible clues to solving problems. Students and their teachers reflect together on the goal of each activity or problem, and look for ways to bring them into the regular classroom. The seminar is aimed not only at developing students’ problem-solving competencies, but more importantly to draw them towards thinking like a future teacher.


Introduction
Creativity, reasoning and open problem solving are among the main goals of education in the twentyfirst century learning [1].It is important for future teachers to think about how to develop these competencies in pupils.Therefore, we prepared a seminar on problem solving for future physics teachers.The inspiration for it came from the two-train problem described in [2], which focuses on dynamic visualization.

Seminar on problem solving
The problem solving seminar is intended for first year undergraduate students of physics teaching.It is held weekly in the first semester for one class period (45 minutes), with each class period focusing on one activity.
Students in this seminar are encouraged to find useful strategies for solving various problems, to use different ways of thinking, and to think like future teachers.At the end of each activity, there is a reflection session during which students, under the guidance of the teachers, reflect on the aim of the activity and look for ways to transfer similar problems into regular teaching.This reflection is very important for the students.

Examples of problems
The problems that we assign to future physics teachers at the seminar are the same as those that can be assigned to lower secondary school pupils.The mathematics in all activities does not exceed the knowledge of children aged 13!
In this chapter we present several selected problems from the seminar.

The nine-dot problem
The nine-dot problem is a well-known well-structured problem [3], whose instruction is as follows: Nine dots are arranged in a square and the dots must be connected by four straight lines drawn without lifting the pen from the paper and without any line being repeated (see figure 1).
Although formally it is an easy task (it has a clear assignment, a clearly described solution method, a stated goal and limited options for solving it), in a psychological sense it is very difficult.The problem requires "thinking outside the box" -a creative, not immediately obvious approach that helps solvers see the problem from a new perspective.

The swimming pool problem
The following problem focuses on using and working with different types of graphs in physics.
Instruction: A swimmer swims in a 50-meter-long pool, one length of the pool takes him 1 minute.The swimmer swims a total of four pools.Draw a graph that corresponds to this movement.
In a group of future physics teachers, several different graphs emerge as a solution to this problem.The two most common ones, which also appear in the solutions of lower secondary school pupils, are position-time and distance-time graphs (see figure 2).We continue to work with these two graphs in the seminar.
After drawing these graphs and discussing their differences, students are asked to complete the next task: For each graph, write three questions that the graph would answer well.
We list here some examples of such possible questions.Readers can create more questions themselves: a) Distance-time graph How many meters did the swimmer swim in 3.5 minutes?How long does it take a swimmer to swim 125 meters?b) Position-time graph How many pools did the swimmer swim in 3 minutes?How far from the start is the swimmer in 2.5 minutes?A follow-up task for students might be: Add the movement of a slower swimmer to both graphs.Here we leave it up to the readers to create their own questions, which the graphs can now answer.
This activity is useful for students to understand that different types of graphs are suitable for answering different questions.At the same time, this activity can make students realize that there can be multiple correct answers to a single question -an important lesson for future physics teachers.

The Traffic Jam activity
In addition to math and physics problems, we also include movement activities in the seminar.One of these activities is the Traffic Jam [4] -a cooperative activity in which the task is to move two groups of students facing each other from one end of the line to the other (see figure 3).Although the Traffic Jam activity seems simple at first, it usually takes several attempts before the students succeed in solving it.It involves some serious problem solving, critical thinking and group communication.We also discuss with the students after dealing with this activity that such an activity in the classroom can help the teacher to find out among other things how the pupils react, how they communicate with each other, who the leader is.Moreover, students appreciate such physical activities because they can get out of their chairs and stretch after a busy day.

Students feedback on the seminar
At the end of the semester we ask students for feedback on the problem solving seminar.Students can write anything that comes to their mind; their answers are anonymous.We usually get mostly positive feedback.Some of students' responses regarding what they gained or learned from the seminar are listed below. It's not wrong that I don't solve something right away, even if it seems simple to others.
 Everyone looks at a problem differently and may have a different approach to solving it.It is therefore necessary to be able to accept someone else's approach. For me as a future teacher, this is a great inspiration to teach differently than I was used to from my school. Always try to understand how others think and respect alternative ways of thinking. I've often been stuck, and it's a good thing.It facilitates the idea of empathizing with pupils and helping them develop. It was Wow! and it shifted my thinking.

Conclusion
The seminar for future physics teachers described in this paper is aimed at solving (not only physics) problems.Students are encouraged to look for efficient ways to solve the problems, to use different representations of knowledge (calculation, diagram, graph, picture, etc.), or to be aware of obstacles and possible aids in solving problems.The seminar helps to develop students' problem solving competences and creativity.
Students undergo the seminar in the first semester, which is very difficult for them after the transition from upper secondary school.In a way, it is a kind of break from college math and physics, but at the same time, in this seminar, students have to think intensively and solve unusual problems.It is also the first seminar in their studies where they meet the didactic side of teaching physics, look a little under the hood of cognitive psychology, and address how to communicate with their future pupils.Students appreciate the seminar and find it meaningful and useful for both their studies and their future careers.

Figure 1 .
Figure 1.The nine-dot problem and its solution.This initial problem can be followed up with other tasks.Their instructions are given below, the solution is left to the readers: a) Draw again a pattern of nine dots arranged in a square, but this time instead of dots, draw nine small circles.Connect all nine circles with one stroke of three straight lines.(Hint: You have circles, not dots!) b) Let us go back to the pattern of nine dots in a square.Connect all dots with two lines with only one stroke.(Hint: Read the instruction carefully.)c) Draw the basic figure again.Now you have to connect all nine points with one straight line in one stroke.There are several different solutions, try to find more.Thanks to this activity, students realize that similar problems can serve to break down various barriers at school (communication barrier, fear of failure, breaking out of ruts, etc.).

Figure 2 .
Figure 2. Graphs of motion for the swimming pool problem (left: distance-time graph, right: positiontime graph).

Figure 3 .
Figure 3.The position of the students at the beginning of the Traffic Jam game.