- Define a problem
- Determine what is problem-solving
- Identify types of problems
- State the importance of teaching problem-solving
- Determine the goals of a problem solving oriented curriculum
- Identify factors affecting the development of problem-solving skills and implications for classroom instructions
- Identify problem-solving strategies/approaches to problem-solving
- Create different modes in which to present problems
- Integrate problem-solving across the Mathematics curriculum
It is an easily-recognizable problem when students question the reason for learning mathematics skills that they feel will not be used beyond the classroom, into their adult lives. Math teaches analytical skills and therefore allows teachers to build the crucial step of acquiring analytical skills into the school curriculum. Children accept that they must learn math far more readily than they would if a teacher sat them down and told them that they were about to start the process of learning analytical skills!
To acquire analytical skills it is necessary to enhance cognitive learning. Organizing information logically, developing the memory, learning to reason and interpret and therefore solve problems, all guide a student to a better understanding of any problem, mathematical or otherwise. This is a major goal when building problem solving strategies into the school curriculum.
Often, one of the most challenging difficulties is actually understanding a problem. It is difficult to devise problem-solving strategies when a person does not even understand the problem. Hence, something that sounds so simple and obvious - understanding the problem - needs to be clearly defined.
Different psychologists and child-development specialists devise different ways of enhancing the learning process. Jean Piaget has his four stages of cognitive development (which start in very young children) and George Polya developed his problem-solving model in about 1945. Piaget's model allows parents and carers to recognize and encourage early learning without any direct tools but rather developing pre-existing ideals. These then make the process easier for a school-going child to develop effectively.
Of course, there are many factors that will affect the secure entrenchment of cognitive learning. A child's environment and exposure to situations that allow him or her to develop skills is crucial. Children who are not permitted to reach decisions by themselves may be compromised in their development of cognitive or analytical thinking. Learning styles are also very important. Logical thinkers need to understand each situation in their own minds whereas visual learners may need concrete examples before they can conceptualize and understand.
Polya's methods, as previously stated, are direct and simple but sometimes overlooked when problems appear to be complex.
- Understand the problem. Split it into easier parts, looking for solutions to each part, not just the whole.
- Devise a plan - for example, make a table; look for a pattern, make an equation, solve a similar, less complex problem
- Carry out the plan- apply the findings in the table to the actual problem
- Review the results.
Polya's methods can be applied in all circumstances - distance, speed and time, sum of numbers, etc.
- A man is 9 times the age of his son but in 3 years he will only be 5 times his son's age. What is the man's age?
1. We need to establish the father's age through a process of manipulating numbers. In this case if x= the son's age, the father is 9x.
2. The plan is to make an equation based on given information - basically translating it into "math language."
In 3 years the son will be 3+x. The father will be 9x+3
The father will be 5 times his son's age in 3 years, thus:
3. Once the equation has been made, solve for the unknown.
4. Reviewing the results means, we can establish that the father is 27 years old.
Technology in the teaching of Mathematics. Show how each is used as a teaching aid and the advantages and disadvantages of each