We discuss "What is a problem?" under three headings: definition, examples, and problems that interest students.
Definition
A problem is a problem because you don’t know straight away how to do it.
The strange thing about problems is that what is a problem for one person is not necessarily a problem for someone else. This is because no two people have the same set of experiences. Hence one person may be able to understand the wording of a problem more quickly than one of their friends. You will most likely be able to understand more problems than your students will simply because you are more experienced and have a larger vocabulary.
Now not all mathematical questions are problems. For a start, a question that relates to the latest mathematics that you have taught in class is not a problem in the sense that we will use the word. Your students already know the strategy to use on such questions.
What’s more, although many problems are word problems, it is not the case that all questions with words, are problems. In the same way, a question without words, or with only a few, might still be a problem.
Problems have to be pitched at an appropriate level. They should provide a challenge for students. At the same time they should not be too much of a challenge. Students need to feel that they have a reasonable chance of solving the problem, either by themselves or in a group.
You need to look at the problems that intend to use and ensure that they are appropriate for your class. You should check to see that all of the members of the class are able to get something from the problem. Is there a point where the weaker students can stop but still gain from having tackled the problem? Is there something to extend the more able students? Is there a method that all students can use to solve the problem? Is there a more sophisticated method that will challenge some of your students?
So what would stop you or one of the students in your class from doing a mathematical problem? Well, first there may be something about the wording that you don’t understand. Then second, you may not see how to get started. There may be no obvious strategy for you to use. Third, you may not know the right piece of mathematics to use. And fourth, you may know the right strategy and the right mathematics but you may not be using them correctly or you may not be able to see how to put them together to come up with a solution.
Examples
To help you get a better idea of what is a problem, and for whom it is a problem, here are some examples of problems. We think that Problem 1 is appropriate for Level 1, that Problem 2 is appropriate for Level 2, Problem 3 for Level 3 and so on.
Problem 1: Measle Spots (Level 1)
Poor Pam has measles. She has one spot on her chin, one spot on each leg, one spot on each arm and one spot on her tummy. How many measles spots does Pam have?
The next morning, Pam wakes up with even more spots! Now she has two on her chin, two on each arm and each leg, and two on her tummy. How many spots does she have now?
Problem 2: Tapes (Level 2)
Rosey and Ratu were hunting around in the family car. They each collected together all the tapes that they could find. That night Rosey and Ratu sorted and counted the tapes. They found that
- when they counted by fours they had three left over
- when they counted by fives they had none left over
- when they counted by threes they had none left over.
Their father knew they had less than 18 tapes. How many tapes had they collected?
Problem 3: TV Programmes (Level 3)
The animals in a barn have a total of 26 legs and 10 heads.
If there are only sheep and chickens in the barn, how many of each are there?
Problem 4: Towers (Level 4)
Tom likes to build towers. He has a collection of black cubes and white cubes. Putting different cubes on top of one another forms a tower. If the height of a tower is the number of cubes used in that tower,
- how many different towers can be made which are of height one?
- how many different towers can be made which are of height two?
- how many different towers can be made which are of height three?
- how many different towers can towers be built for any particular height?
Problem 5: Tennis (Level 5)
In a round robin tennis championship, 20 people are to play each other. How many games need to be played?
The organisers decide that that's too many games and so instead they use a knock-out competition. How many games are played under this system?
Problem 6: Numbers (Level 6)
Various whole numbers add together to make 2001. What is the biggest possible product of these whole numbers?
Problems that interest students
Another aspect of problems is their intrinsic interest. In the classroom a problem should be something that interests the students and something that they definitely want or need to solve. You can make problems more attractive for students by putting them in contexts that interest them and by using their names for the characters in the problem.
You can probably see how to make the above six problems better suited for your class. For instance, if your Level 1 class has a thing about big cats, then you might change measles spots to spots on a leopard. It’s very easy to change Problem 2 by changing the names to those of students in your class. What could you do with the other problems to make them closer to the interests of your students?
The aim of this website is to help you to provide learning experiences for your students so that they are equipped to confidently tackle any problem that comes their way. To do this they will need to see how to interpret the question; to choose and employ suitable strategies; and to use the strategies and mathematics that they know, to solve problems.