Creating a Reasonable Diversity of Problems (Theoretical Approach)

(Overall View) | (Example Oriented Approach) | (Theoretical Approach) | (Types of Problems) | (Collection of Problems)

About Problems

From now on we will usa a very broad idea of a problem, like it says:

A problem as a challenge to act is described as:

• Goals for action are influenced by urges, orientations for action (examples, knowledge, habits,...), qualities of the progress, check, reflection
• Contents for action may be mathematic terms, context, methods, learned actions (describing, choosing, recognizing, reasoning, memorizing, ...)
• Conditions for action, that are made up of objective components (degree of formalization, degree of complexity, degree of popularity, degree of implementation, ...) and subjective components.
  

A basic understanding of problem solving includes:

• The role of a teacher to create, modify and assign problems to a specific dynamic of students. Also note that his role is a supervisory position to the student role.
• The role of the student to find, manipulate and present a solution to an assigned problem.
  

The functions of problem-solving are:

• a method of acquiring knowledge and of expanding more skills
• an instrument to monitor the progress and results during the learning process
• the goal of capability

Key Concepts for Problem Solving

In this paragraph we will describe the diversity of problems under a structural aspect and discuss conceptional questions of working on problems in an academic learning situation.

Let's begin with the broad definition of problems, which involves diverse challenges in the learning process. You may imagine the different aspects of academic examination and any number of related scenarios from personal experience.

In the end, what is vital for the learning success, are the problems that students pose to themselves and how they deal with this so called "learning problem".

Usually you observe students in class who take ownership of the average problems suc-cessfully by asking additional questions and trying to grasp the intention of the teacher. However, it is frequently observed that the transformation of a problem into a learning problem fails completely or in parts.

Instead of dealing with the question, i.e. the course of motions or the text, students work schematically on parts of the given guidelines, or they turn to completely different things. In such a case, the given and the self-constructed learning problem are not identical. A match of those would be useful to meet the requirements of the curriculum as for the in-dividual achievement in successful learning. There is no automatism and it depends on different factors to initiate adequate learning problems. In this context it is wirth mentioning empiric results of surveys done by Jäger/Helmke of the MARKUS project. This study clearly show that good learning results require a good "classroom managment". This will not surprise an experienced staff. The first task of a teacher is to develop and initiate suitable and motivating challenges.

Caution! A goot atmosphere for productive studying is a very necessary but not solely sufficient element for the development of specific learning tasks.

Let us conclude the insight gained:

• The most beautiful problems are no good when they do not "hit home" in mutiple ways.
• Problems should be recognized by students as developmental and relevant for them, so to be accepted as manageable challenges.
• Diversity of problems alone is no guarantee for successful learning.
• Problems that give the student an opportunity for independent study offer greater possibilities for success in learning.

This is why a lot of attention is paid to so called "open problems" in all areas of technical didactics. Indeed the criteria "openness" in itself is not enough to initiate developmental and supportive student acitivity. Many open problems just do not challenge students, if it is just about completing many different tasks without giving any necessary structure and an outline of expectations. What is needed are problems appearing as hypothetical situations with suitable content for the target group, including intellectual, phsical or social activities.

Analysis by Franz E. Weinert show, which teaching and learning methods or rather activities are mainly relevant for what type of category.

Every goal in learning requires different lessons: (Weinert)

This chart is from the article "Die fünf Irrtümer der Schulreformer" by Franz E. Weinert translated into english.

So the message says: "Apply diversity of methods relevant to the goals."

It promptly applies to the construction of problems too. With only open problems it is not possible to reach all basic educational objectives, just like it is not possible to only rely on the project method. It is also, because time for academic learning is limited. Which are the most suitable tasks to reach certain goals, and how should the problems be dealt with during class to use the given chances for learning to full capacity?

Let us look at an example from geography, that you can create a diverse set of problems from:

"Wuaht kinds of processes are taking place inside a volcano? Magma, the liquid rock rises from deep within into the fireside. Is consists of gases, mainly water vapor and carbonic acid gas, that are under immense pressure. Through crannies and crevices those gases are on the rise just like in a soda can. This can happen explosively, so that ashes and bombs are thrown. After the escape of gas, the liquid magma rises in the crevices, or "vents". When it runs out of the funnel-shaped crater it is called lava. Like at the Äthna, volcanic eruptions can happen to all sides. That is why there are side craters."

The topic of problems holds great didactic challenges. One of them is introduced in the Example Oriented Approach.