Educational objective for all students: how to solve problems - how can it be achieved?

Intuitively successful problem solvers are characterized by highly capable agile minds (flexibility, ability to transfer), within a certain realm of knowledge and action.
Agility is a quality thinking process, besides consciousness, orderliness, independence, activity (see Lompscher/Hasdorf).

Such qualities thinking processes are contextual and can be practiced to some degree. This means that a lack of agility regarding mathematical expertise can be "compensated" at least partly.

Central idea:

The procedures of extraordinary agile problem solvers can be analyzed and described. This results in heuristic principles, rules and strategies or means. Polya [1949] articulated such heuristic means. If it is possible to acquire and apply these, similar effects are achievable just as they are for all intuitive problem solvers. Class has to do with deliberate learning of useful heuristics, so they can be applied more and more subconsciously.

It is just like learning how to drive:

At first the gears are being worked in a highly concentrated manner and a little clumsily, but soon the operations are going smoothly without a lot of thinking. Heuristics are rational tools, one needs to know them and apply them flexibly. How can this be achieved?

_{Polya, George [1949]: Schule des Denkens. Vom Lösen mathematischer Probleme. Francke: Tübingen und Basel}

Requirements:

Creativity allowed

Requirements, conditions or chances to allow or require creativity permitting its input from a professional and educational point of view.

Opportunities to be creative during math class are offered by posing developmental appropriate and development supportive learning expectations, based on related objectives and contents of math class, in combination with a welcoming creative learning environment.

Opposite: Learning situations where no creativity is required, are phases of working through algorithm formula (regulations) (calculating the value of a term, constructing according to a formula ...). Such learning situations are valid too, but should not dominate in class.

Atmosphere

• It is okay to be wrong. Scrap paper is also allowed...
  (there are phases of no grading and time to think during class - enabling fearless learning)
• Every spoken idea is taken seriously
  (two way respect of students and teacher, prejudice-free work)
• Different procedures are tolerated and accepted - even in tests
• The need of students to orient themselves is met by clear communication of objectives and grading scales within a set frame for independent learning, neglecting procedures that are too detailed or prevent independent thinking and action.
• ... (core values of a "good" class...)

Objectives and contents of mathematical class

• Balanced relation of formal and praxis oriented competencies
• Linked teaching structure to prevent "island knowledge"
• Passing on of various mathematical terms, contexts and procedures just as methods and techniques, which are useful for working on a module (modular topic-oriented curriculum structure vs. a onedimensional systematical theory-oriented structure) to chose from a broad source.

Types of learning requirements

• choice problems
  There is a possibility to choose from various problems of one topic with several levels of difficulty and if necessary different ways of being written.
• "open" problems
  as learning requirements can be mastered by many different styles and with different results (like writing a story about a distance-time-diagram, comparing several tasks and their solutions by various aspects) especially also:
  
  Making up problems
  
  as learning requirement, out of which more detailed, expanded, supplemented or deepened problems can be developed by appropriate change/ variation / addition of a given problem (blossom-model of problem variation), are especially useful with inner-mathematical problems.
  
  In the form of problem situations, mathematical problems need to be developed (funnel-model of problem variations) for example with applicational or modelling problems.
• issues with the character of a problem in the "zone of the next development"
  (i.e. riddles or: "Find as many as possible different methods on how the width of a river can be determined with dipstick, protractor and measuring tape")

The requirements for these opportunities to be taken (avoiding over or under-challenging) are: enabling students step-by-step to assess their own capability realistically (knowing their own strength and weaknesses) and taking on responsibility for learning on their own.

Wanting to be creative

Challenges and nice conditions for the development of the willingness to learn, to buy into the objectives and develop ownership of the objectives -
best by intrinsic motivation (interest in the matter itself)

On top of the mentioned and applicable conditions (of a learning environment that welcomes creativity and is neither over- nor under-challenging - especially giving freedom of choice or some scopes of freedom of choice and correlating objectives and contexts of math class), the following aspects showed to increase interest:

Atmosphere

• Praise and acknowledgement of special achievements, of unusual new options to solve a problem, and support of healthy competition

Objectives and contents of mathematic class

• Mathematics can be experienced in its development; contexts and important coherences become transparent.
• age-appropriate choice of topics in regard to the changing interests in free time activities, life and experiences of students (without bending to it - students also have to be confronted with matters that are not of interest for them - yet...)

Kinds of educational requirements

• The problem contains something striking, makes students stand in awe or gets them to wonder: How does it come to that ...? Is that always applicable? Is that even right?
• The question is posed in such a manner that students are ascribed competencies. i.e. How would you decide? Give some advice for his decision!
  Is there another way or another opportunity?
  

Four phases of problem solving by Polya

"A mathematical problem can sometimes be
as entertaining as a crossword,
and concentrated intellectual activity can be as much a
desired exercise as a quick tennis match."

About Polya

George Polya a hungarian-american mathematician was born 13^th of Dec. 1887 and died 7^th of Sep. 1985. A professor at the ETH Zürich he worked in several areas of mathematics, especially areas of problem solving.

He divided the processes of solving problems into four phases and developed a manual for problem solvers based on it. This manual basically is a question catalogue, presenting various possible directions of thinking at any point of working through a problem. If following these guidelines - so following Polya - the chances of solving the problem are high.

With this manual Polya expects problem solvers to have an agile mind - problem solvers are supposed to ask themselves some questions to draw nearer to a solution.

1^st Principle: Understanding the problem

• What is unknown? What is given? What is the condition?
• Is it possible to fulfill the conditions? Is the condition sufficient to determine the unknown? Or is it not enough? or undefined? or contradictory?
• Draw a picture! Introduce an appropriate label!
• Separate the various parts of the condition
• Can you write it down?

Polya emphasizes strongly how problem solvers have to clearly see the problem or task. During this phase no plan is to be made yet - nor the beginning of a solution. As PISA-studies have shown, this part is of crucial importance in solving the problem.

Polyas questions can be helpful - which is his concern - in gaining a better understanding of the problem stated. He expects the consideration already in this first phase, if the problem has a solution, it is obviously solvable or undefined. Students often give this basic point no thought.

Requirement to draw a figure is very important, because visualization techniques are valuable when working through a problem. We will go about it in more detail: The simple task to draw ´any` figure is too general, students need specific help with what such a figure could look like. Figures are used over and over again by the experienced problem solvers. That is why several different so called informative figures are recommended in the problem-solving-module, to teach your students.

2^nd Principle: Devise a plan

• Did you come across this problem before? Or did you see a similar problem just in a different shape or form?
  
• Do you know a related problem? Do you know a theorem that could be of help?
  
• Look at the unknown! Try to recall a familiar problem, with the same or a similar unknown.
• Here is a problem, related to yours and solved before. Can you use it? Can you use its result? Can you use its method? Would you introduce any instruction element, so you could apply it?
  
• Can you express the problem differently? Can you vary or change your problem too? Go back to the definition!
  
• If you cannot solve the problem at hand, try solving another related problem. Can you find a problem related to yours but more easily accessible? A more general one? A more specific problem? An analog problem? Can you solve a subproblem or side problem? Maintain only parts of the condition and drop the others; how far can the unknown be assessed now, how can I modify it? Can something useful be concluded from the data? Can you come up with different data that is helpful in defining the unknown? Can you change the unknown or the data, if necessary both, so that the new unknown and the data are closer to each other?
  
• Did you use all the data? Did you use the whole condition? Did you
  take all essential terms into account that the problem contained?

Polya accumulates many questions in this second phase that are definitely helpful when working on a target-oriented and systematical plan. Modern problem solving prefers individual strategies instead of one topic.

3^rd Principle: Carry out the plan

• As you are following through on a plan for solution, check every step. Can you see clearly that this step is right? Can you prove that it is right?

Polya expects self-check here. On the one hand meaning the check of correct mathematics (right calculation, proper rearrangements, or the like ...). On the other hand - just as important - a check, that every step is carrying out the plan already made. This ensures that nothing is done carelessly, ensuring purposeful and ordered implementation of the plan.

Students orientating themselves on the questions and guidelines of this phase, should not be misled or stop too often with a part solution, forgetting that the problem is not already solved.

4^th Principle: Review / extend

• Can you check the result? Can you check the proof?
• Can you derive the result in various ways? Can you see it first sight?
• Can you apply the result or method to any other problem?
• Which strategies of solving helped with the solution of the problem?
• Which new procedures could be recognized with the solved problem?
• Which option of solving suites this problem best?

This fourth phase was articulated by Polya for two reasons: For one to more conventionally utilize the validity of the problems solution. Next to utilize the gained experience of working through a problem for future problems. Such a review - or phase of reflection on top of that is - scientifically proven - crucial for lasting learning effects among students.

4. A model of class:

According to Polya it is useful to word heuristic strategies as questions. These questions however should not be given to the students or learned by heart. These or similar questions have to be acquired and verbalized from their own experience in solving problems.

The learning of heuristics can be done by the following four steps:

1^st Stage

The students are getting used to certain heuristic procedures and typical questions step-by-step. When giving hints the teacher uses question strategies of each heuristic consistently, not teaching them as a special topic in class.

Example: principle of analogy

Teachers hints:

• What did we do in similar situations?
• What does seem familiar to you in this problem?
• Compare recent problems (and their solutions) - what are similari-
  ties?

2^nd Stage

The explicit strategies that need to be learned are developed and introduced based on appropriate questions depending on the school level.

The strategy gets a name and is indicated by typical questions. The sample problem works as a link to aid the memorization. Samples are collected where the strategy just recognized has been applied intuitively in the past.

3^rd Stage

During these (short) phases of practice with problems of various difficulty, it is expected to independently apply the new strategy. The contexts of problems vary step-by-step.

The individual preferences of certain strategies and the diversity to apply the newly learned strategy are to be talked about and made aware of.

4^th Stage

The phases of practice are geared toward a gradual subconscious flexible application of the strategy.

The new strategy is put into place among the general images of how to solve problems - in a general model of solving problems (according to Polya).