Developing problem-solving thinking should not be the task of only mathematics. It should be the task also of informatics. Students' problem-solving abilities can be greatly enhanced by solving consciously structured, interrelated programming tasks. To create a well-structured series of tasks it is necessary, that we understand and become aware of our own system of concepts. As a teacher, we need to be aware of the concepts that are really important and how they relate to each other. Our aim is to help our students to develop connections and correspondences within their concepts that will facilitate the integration of new concepts into their concept map. In this paper we deal with this concept map and the issues that arise with it.
MSC: 68
William James and György Pólya also said [
The steps of Pólya’s problem-solving model [
Problem solving in mathematics is very similar to algorithmic program solving.
As proof of this we show one of the algorithm problem solving strategies [
S2. Restate the problem S3. Write out examples (input-output) S4. Break the problem into its component parts S5. Make an algorithm S6. Step through your example data with algorithm S7. Code it up S8. Make tests
S9. Refactor It would be easy to compare these steps with the steps of Pólya’s problem solving model. Their similarity can be seen very well if we compare the helping questions for each step that guide us through the problem-solving process.
Of course, not only Pólya, but also others have developed problem-solving methods
and problem-solving models. György Kontra gave a detailed review of these [
At the international level there are studies about the similarity and diference
between algorithmic and mathematical problem-solving which have been previously
published [
Perhaps the most striking diference is that while in mathematics it is our job
to perform calculations, in the case of algorithmic problem solving the machine
does the calculations for us. The advantage of this is that we can solve complex
tasks that require a lot of computing. However, compared to solving mathematical
problems, we have a harder task with algorithmic problem solving, since after
solving the problem we have to write out the solution for the users with correct
syntax [
Problem solving methods used in programming are often used in real life. Knowing these problem solving methods and applying them to small problems will help us to have a toolbox that we can work on when we have to solve a bigger problem. It is similar to the reason why we learn and teach mathematics, but it focuses on other types of problem solving in mathematics than in programming. By programming, we can solve problems that we would not be able to calculate manually.
Designing a problem solving process, decomposing problems into sub-problems, applying small problems to larger cases, recursion, thinking backwards are all very important techniques that can we easily apply in our everyday practice. The easiest way to illustrate these techniques is to illustrate them through programming.
For teaching programming and problem-solving thinking correctly and eficiently, and for forming up of the suitable concept structure it is necessary to think over the concepts and knowledge of the topic.
György Pólya used the terminology of guided discovery to build the desired
mathematical concept structure. He worked with consciously, purposefully selected
and structured sets of exercises to help students to reach the required concepts
and contexts [
To create well-structured series of tasks it is necessary, that we understand and become aware of our own system of concepts. As a teacher, we need to be aware of the concepts that are really important and how they relate to each other. Our aim is to help our students to develop connections and correspondences within their concepts that will facilitate the integration of new concepts into their concept map.
One of the basic assumptions of cognitive psychology is that students largely
“construct” new knowledge themselves. They not only add new information to their
existing knowledge repository, but they also link new knowledge to their
existing knowledge [
From this point of view, informatics and programming are fortunate, because we
can make some relations with programming tasks to many other prior knowledge.
“In the Netherlands, informatics has been defined as a new generation discipline,
because it is linked with Mathematics, Physics, Engineering, Linguistics,
Philosophy, Psychology, Economy, Business, and Social Science in general” [
To understand our own concept structure, we first need to gather concepts and then systematize them. When we collect the concepts, we must realize that the topic of making algorithms and programming uses many concepts, but in many cases the relations between the concepts are not definite. There is not, or only very dificult to determine the “correct” order of the teaching of the concepts.
Relationships between concepts, which appear below, do not represent the order in which they should be taught. They refer only to relationships between individual concepts. Because the “correct” order of teaching these concepts depends on many factors, and because the concepts are interrelated at many points to each other, so we present only a “general situation”. Thus, it is possible that a connection is not marked.
The whole concept structure is quite complex, so we will first illustrate a simplified version of the whole structure and then its smaller units. After that we show the complete concept structure by denoting the relationships between the smaller units.
A simplified version of the concept structure is shown in Figure 1. Here, the closely related terms are denoted by a collective name.
At the beginning of the topic making algorithms and programming, we almost always have to use concepts of instruction, command, algorithm, and some related concepts. They are illustrated in Figure 2.
The concept of sequence is a good example of the fact that not all concepts are named during the teaching process and thus in the students’ own conceptual structure. Most programming students probably do not hear this concept, but they still need to know what it means.
When we solve algorithmic and programming tasks we will soon need to work with data and perform operations with that. To store data we have to use diferent types of variables and their declaration. Related to this we use input and output operations. These concepts are interconnected at several points, as illustrated in Figure 3.
If we need to work with larger amounts or more complex data, we also need to use diferent data structures (Figure 4). In relation to these concepts the depth, the environment and the age of group in which these are taught is particularly important. Because it depends on these factors which data structures we teach and which we do not.
One of the most important parts of the concept structure is the concepts related to conditions, selection and repetition (Figure 5). Understanding these is a particularly important and indispensable part of learning and teaching this topic, because algorithms and programs are built mainly on these control structures.
We also have to deal with troubleshooting and testing when we would like to check and run our algorithms and programs. Linking concepts (Figure 6) and debugging methods are highly dependent on the programming environment and the depth of teaching the topic. Nevertheless a minimal level of troubleshooting is required regardless of the environment. Debugging cannot really be linked to previous parts of the concept structure, because debugging and testing may be required at all levels of making algorithms and programming.
By combining the above mentioned concepts and the smaller parts of the concept structure, we can create the “complete” conceptual structure of the topic of algorithmic programming (Figure 7).
It contains a few concepts that have not yet been discussed. These concepts have been left out of the discussion of smaller sections because they are not so closely related to them, or because they are not necessarily discussed when we teach the basics of the topic making algorithm and programming, or because they are not present in all frequently taught programming environments. Examples of this are operations with diferent types of variables, global and local variables, and recursion.
The concept of control structures also appears here for the first time. This can be omitted from the learning process because it is not important to know the concept of control structures but the goal is to make the students aware of the concept and usage of control structures.
Based on this “general version” of conceptual structure of the topic of making algorithms and programming, it can be seen that there are very complex relationships between the concepts.
Therefore there is not, or only very dificult to determine the “correct” order of the teaching of these concepts. This “correct” order depends on many factors. One of these is the context in which students are learning a programming language and where they meet with these concepts. Another important factor is the time, that we can spend on teaching the topic, the students’ or group’s previous knowledge, and their age-specific characteristics (such as how abstractly they can think). Similarly, learners’ interests can play an important role, because it can influence which areas they can relate the new concepts to.
We are planning to develop several versions of the concept structure, taking into account the specialties of each programming environment and programming language.