Problem Solving Information
This page provides a summary of the six sections above. More detailed information can be accessed from each of the 'More Information' headings.
A problem is a question that motivates you to search for a solution. This implies first that you want or need to solve the problem and second that you have to search for a way to find a solution. Whether a question is a problem or an exercise depends on the prior knowledge of the problem solver.
In this web-site you will find problems for all Strands and for Levels 1 to 6. An example of a problem at each of these Levels is given in What is a Problem?
Problem solving was part of the Mathematical Processes Strand of the New Zealand Mathematics Curriculum (1992). While problem solving is not a category within the New Zealand Curriculum (2007), the key competency of Thinking includes problem solving.
We believe that by solving problems students get a much better feel for what mathematics is all about, what it can do and how it does it.
In this web-site every time we use the term ‘problem solving’ we mean mathematical problem solving. And mathematical problem solving is about finding solutions and not just answers to mathematical problems. We say that
method + answer = solution.
We believe that there are four steps that you need to go through in solving most mathematical problems. These are:
- Understand and explore the problem;
- Find a strategy;
- Use the strategy to solve the problem;
- Look back and reflect on the solution.
Of course, as we point out, problem solving can be more complicated than this.
The selection and use of strategies is a part of the process of problem solving. An understanding of specific problem solving strategies helps make problems clearer, simpler and more manageable. It also helps students develop better problem solving skills.
In this web-site you will find problems for the following problem solving strategies:
- Guess (this includes guess and check, guess and improve);
- Act it Out (this includes using equipment);
- Draw (this includes drawing pictures and diagrams)
- Make a List (this includes making a table)
- Think (this includes using skills you already know)
Of course, you also need to use other problem solving skills as you solve problems. We consider being systematic, keeping track, looking for patterns and working backwards.
There are many benefits to teaching problem solving. These include:
- It bases students’ mathematical development on their current knowledge;
- It is an interesting and enjoyable way to learn mathematics;
- It is a way to learn new mathematics with greater understanding;
- It produces positive attitudes towards mathematics;
- It makes the student a junior mathematician;
- It teaches thinking, flexibility and creativity;
- It encourages co-operative skills;
- It is a useful way to practice mathematical skills learned by other means;
- It is similar to the approach used in other curriculum activities.
When implementing problem solving in your mathematics programme, you need to consider:
Problem Solving Units
This area of the problem solving web site is taking a new direction. This is the result of the evolution of the lessons that have been written for problem solving, a recognition that problem solving plays a similar role to the non-process Strands, and a reflection of the basic nature of mathematics and the way it is created.
For the immediate future, new Problem Solving material will be added to the site in unit form. Like the material in Number, Algebra, Statistics, and Geometry, problem solving ideas will be presented in units that are approximately equal to 5 lessons.
This page contains an annotated bibliography of books that contain problems and ideas that can immediately be used in your classroom. We are happy to add it the list any books that you have found useful. Please e-mail us the details.
- Lesson Structure
We suggest a 3-stage format that consists of introducing the problem, group work and a reporting back or sharing phase.
- The Role of the Teacher: Good Questions to Ask
As the teacher's role is that of a facilitator we suggest a framework of questions that you can use during the different stages of the lesson.
- The Year Plan
All teachers need to consider how they are going to show coverage of the curriculum within their mathematics programme. We illustrate how you can include problem solving within your term plans.