The key to teaching Problem Solving in unfamiliar contexts in Maths is to begin as early as possible with all students. Give your students as much experience as possible as often as possible. This experience does not need to be for extended periods but should contain many short, sharp exercises in a great variety of contexts.
What we need to do is to build up our students' confidence by giving them lots of opportunities, rewarding them, not so much for a correct answer but for being involved in 'having a go'.
It is important that you, the teacher, must become a problem solver if you are to be an effective teacher of problem solving.
Here are themes/ideas that I use to develop this confidence:
• Adopt the premise that problems are 'easy'. Teach your students to start by looking for a simple approach.
• The word 'problem' has a negative connotation. Perhaps the word 'challenges' is a better way to speak about our 'problems' in unfamiliar situations.
• Mistakes are to be welcomed. They are learning experiences.
• Finding a dead end in an exercise should be regarded as a success not a failure. You have just proved you cannot do it that way.
• The seeds of the solution are always in the problem but often students overlook the obvious. So teach students how to interpret questions.
• I would often model verbally and on the board how I approach a problem and work towards a solution.
• I encourage my students to find and share different ways to solve the problems.
• Persistence is an important habit to develop. Include in your work, occasionally, long problems to solve. They don't need to be difficult but just have many steps.
• We need to give students greater opportunities to follow through with a problem. Give hints rather than solutions initially. Solutions should be given when all else has failed. Allow successful students to explain the solution to the class or group.
• Charles Lovitt, a well-known Australian researcher into Mathematics teaching, has a thesis that goes like this: "It's not the question that is important BUT how you ask the question." Mental Arithmetic in Middle and Junior High School is a great way to ask the same question in a variety of ways. I use it often.
Here are some simple examples of the one question:
• 7 plus 5
• Find the sum of 7 and 5
• What do I add to 7 to get 12?
• What do I subtract from 12 to get 7?
• What numbers have a sum of 12?
• What numbers, whose difference is 2, have a sum of 12?
One last point
I do all the problem solving 'challenges' I give first, before I give it to my class or I treat it as an unseen problem and do it for my class showing them that I am human and can be 'stumped' initially. Then, I treat it as a group exercise.
Rick Boyce has taught for over forty-five years, the last fifteen years as the Head of Mathematics in a large Australian school. There, he devoted much of his time developing ways to help students develop confidence in using their Mathematics in unfamiliar contexts. An eBook on problem solving to expanding this article and another with problem solving exercises will be available on our web site, soon