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.
There is a lot going on in these units. The students are doing ‘real’ mathematics here. The units have been set up to help students think like mathematicians. Surprisingly this is possible. What is more surprising is that you will find that the students who take to this most readily are not necessarily the ones that are good at the traditional mathematics. But you shouldn’t expect miracles right from the start. Actually we would suggest that you don’t start on problem solving units until you have tried some of the extended problem solving lessons that we have talked about above. And when you do start, don’t push the students too hard at first. Give them a lot of help and accept whatever they say and try to use it.
Hopefully most students should be able to get answers to the problems (as opposed to solutions containing proofs/justifications). It might be helpful if, before you gave the unit to the class you tried the problem and got an idea of how far you can go. It might even be useful to do these units as a team, combining with another teacher in the school.
The development to Problem Solving units is a natural one. If you remember the history of problem solving on this site you will recall that we started first with one-off lessons based on a single problem. Over the last year or so, the new lessons have often been produced in groups, at various Levels, around a single theme or idea. These grouped lessons allow students with a range of abilities to tackle very similar problems but achieve success at a range of depths. They also allow for the more able students to go further in a problem. With the production of problem solving units, there is still the opportunity for the more able students to go deeper into a problem but at the same time, all students will have the opportunity to get an overall view of the way mathematics works and is created.
We have already talked about mathematics and how it is related to problem solving so we won’t go any further into that here. Suffice to say, that one of the aspects of mathematics that one-off problem solving lessons have been unable to stimulate has been the areas of extension and generalization.
Basically a generalisation of a problem is a problem that contains the original as a special case. For instance, recall the Stamp Problem Level 4. This asks what amounts of postage can be made with only 3c and 5c stamps. There are several ways to generalize this problem. First we could replace the 5c stamps by t cents stamps. Then we would want to know what amounts of postage could be made with 3c and tc stamps. Second we could replace the 3c stamps by s cent stamps as well and ask what amounts of postage could be made form sc and tc stamps. In the first case the original problem is a special case in that when we have solved the problem for 3c and tc, we only have to put t = 5 and we have the solution to the original problem. In the second generalization we have to put s = 3 and t = 5 to get the original problem.
It turns out that for t not divisible by 3, the first generalization leads to us finding that we can make any amount of postage from 2(t – 1) on. As it happens with the original problem, we can show that any amount from 8c on can be produced. Now when we put t = 5, 2(t – 1) = 8. So we indeed do get the original problem solved as a special case of the first generalization.
When we have s and t stamps, with s and t having no factors in common, we can make any amount of postage from (s – 1)(t – 1) on. Now thinking abut the original problem and putting s = 3 and t = 5, we see that (s – 1)(t – 1) = 2 x 4 = 8 – the original answer.
Just as an aside here, the second generalization is a generalization of the first generalization. When we put s = 3 in (s – 1)(t – 1) we get 2(t – 1).
Then there are extensions. An extension of a problem is one that is some way based on the original while not being a generalization of the original. If we go back to the stamp problem, an extension might ask what amounts of postage can be made from 4c and 7c stamps. Another extension might ask what postage can be made from 2c, 3c and 7c stamps. In the first case we just change the denominations of the stamps being used. In the second case we do that and we introduce a third denomination.
These aspects of mathematics were introduced to some extent in the grouped problem solving lessons but we hope that they will become clearer to more students in the problem solving units. So we hope that the new units will give more students a better idea of what mathematics as a discipline tries to do; what mathematicians try to do when they do research in the subject.
One of the major ideas in mathematics is finding patterns. At the stage when we have guessed a pattern it is called a conjecture. This is just a guess (but, of course, mathematicians are too sophisticated, all-knowing, and important to guess!). If we are able to find a proof to a conjecture (i.e., we can justify it), then we have proved a theorem. If we can’t find a proof we may find an example which defies the conjecture. Such an example is called a counter-example.
There is an idea in some of our proofs that might be worth commenting on. This is ‘if and only if’. When you see a statement like X is true if and only if Y, this means that if X is true, then so is Y AND if Y is true then so is X. In mathematics we look for this situation all the time. When we find it we are happy because it gives us a characterisation – two things are equivalent. Generally we are happy because one of these things is hard to check and the other isn’t. So once we have established a characterisation, we only have to check the easier thing.
The final word we want to use here is scaffolding. This is a pedagogical device to help students. The idea is that, like a building scaffold, the teacher produces a ‘device’ to help students reach to a new piece of knowledge. You might find the Ministry of Education Report No. 587, OPEN Plan for Teaching Mathematical Problem Solving, by Holton, Anderson and Thomas (1997) useful as an introduction to this. They suggest that to help students with problem solving there are a range of generic questions that might be useful. We list these below. (The headings represent the stage in the problem that they might be used.)
What are the important ideas here?
Can you rephrase the problem in your own words?
What is this asking us to find?
What information is given?
What conditions apply?
Anyone want to guess the answer?
Anyone seen a problem like this before?
What strategy could we use to get started?
Which one of these ideas should we pursue?
While students are working
Tell me what you are doing?
Why did you think of that?
Why are you doing this?
What are you going to do with the result once you have it?
Why do you think that that stage is reasonable?
Why is that idea better than that one?
You’ve been trying that idea for 5 minutes. Are you getting anywhere with it?
Do you really understand what the problem is about?
Can you justify that step?
Are you convinced that that bit is correct?
Can you find a counterexample?
After students are finished
Have you answered the problem?
Have you considered all the cases?
Have you checked your solution?
Does it look reasonable?
Is there another solution?
Could you explain your answer to the class?
Is there another way to solve the problem?
Could you generalise the problem?
Can you extend the problem to cover different situations?
Can you make up another similar problem?
Naturally there are a range of specific questions that you can use too but these will depend on the problem and the students’ position in the problem.
One thing about the use of scaffolding is that it has a two-fold importance. First it is to help students over a current hurdle and second it will hopefully be internalised by the student and used on a later problem. To enable the student to pick up and use scaffolding of their own, it is worth pointing out to students from time to time what you are doing.
Finally, by producing problem solving units we hope to make the point that problem solving is in no way different from any of the other units that we have on the site either in importance, philosophy or pedagogy. Problem solving is equally important as any other aspect of the subject. In fact it may be considered to be more important in that solving problems is both one of the main aims of mathematics and one of the main aims of education. In this area of the curriculum, there is less scope for rote learning of basic facts and more scope for the higher order cognitive skills of ideas, heuristics (strategies), logical thinking and originality.
We hope that you enjoy and find useful this new approach to problem solving. We’d be grateful for your comments on what we are producing and particularly we’d like your comments on how your classes are responding to the units. We would especially like you to let us know if you have found any neat extensions, generalisations, conjecture, proofs, etc., that we haven’t included here (email: firstname.lastname@example.org). We’ll definitely add them to the site with due acknowledgements.