What is a Problem?
A problem is a problem because you don’t know straight away how to do it. Let’s be clear at the start that we are talking about mathematical problems here. And let’s also say that this web-site is crawling with problems.
So what would stop you or one of the children 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.
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 will be able to understand the wording of a problem more quickly than one of their friends. You will be able to understand more problems than your children will simply because you are more experienced and have a larger vocabulary.
Some people too, will see what approach to take to a given problem more quickly than someone else. Sometimes a strategy is almost obvious. Sometimes too, it is far from obvious.
Of course, your mathematical knowledge is vital to solving problems. Clearly the more you know, the less questions will be problems. And on some days you’ll see how to put together the right strategy and the right maths more quickly than you will some other time.
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 children know the strategy to use on such questions, after all it is what you have just taught.
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 children. At the same time they should not be too much of a challenge. Children need to feel that they have a reasonable chance of solving the problem, either by themselves or in a group.
Because of this you need to look at the problems that we have posed and see whether or not you believe that they are appropriate for your class. You should also 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?
Mostly in this web site the problems are one-off problems. This means that they can be solved and that can be the end of it. There is no more to do. But we occasionally include problems that can be extended and generalised and these form the basis for an investigation (see What is an Investigation?) Actually though, most problems can be made into investigations. It's just a matter of finding the right generalisations or extensions.
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)
Four friends, John, Stephanie, Peter and Annie, all like watching TV but they all like different sorts of programmes. Using the clues below, decide what type of programme they each like. (Assume that each person only watches one type of programme.)
John’s best friend watches sport.
Stephanie likes comedy but once liked drama.
Annie used to like drama but she doesn’t any more.
John absolutely hates drama.
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?
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 children 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 Problems 2 and 3 by changing the names to those of children in your class. What could you do with Problem 4 to make it closer to the interests of your children?
By the way, if you have just taught your class how to do logic problems using a table, for instance, Problem 3 isn’t a problem for them. But it might be a useful problem to introduce them to the table strategy for logic problems.
As for Problem 6, it may be too difficult for all your class at first. Then why not let them tackle it at home and so involve their caregivers? Then over a week or so you could put the biggest product of the day on a noticeboard.
The aim of this web-site is to help you to provide learning experiences for your children not so that nothing is a problem for them but rather so that they are equipped to confidently tackle any problem that comes their way. What we are trying to do is to provide opportunities for children 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. One thing that we are not trying to do on this web-site is to teach pieces of mathematics such as division and fractions. That is someone else’s problem.