Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.
Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.
How to Solve It
George Pólya was a great champion in the field of teachingeffective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.
George Pólya, circa 1973
- Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons ↵
In 1945, Pólya published the short book How to Solve It, which gave a four-step method for solving mathematical problems:
- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?
This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.
Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!
We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:
- What if the picture was different?
- What if the numbers were simpler?
- What if I just made up some numbers?
You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.
This brings us to the most important problem solving strategy of all:
Problem Solving Strategy 2 (Try Something!).
If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.
And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.
Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.
Problem Solving Strategy 3 (Draw a Picture).
Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?
Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?
After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.
Problem Solving Strategy 4 (Make Up Numbers).
Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!
You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.
Watch the solution only after you tried this strategy for yourself.
If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!
(Squares on a Chess Board)
How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)
Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!
Think / Pair / Share
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?
It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”
Problem Solving Strategy 5 (Try a Simpler Problem).
Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?
Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).
Problem Solving Strategy 6 (Work Systematically).
If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.
For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:
|size of board||# of 1 × 1 squares||# of 2 × 2 squares||# of 3 × 3 squares||# of 4 × 4 squares||…|
|1 by 1||1||0||0||0|
|2 by 2||4||1||0||0|
|3 by 3||9||4||1||0|
Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate).
Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!
For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.
Problem Solving Strategy 8 (Look for and Explain Patterns).
Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.
Think / Pair / Share
If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:
- Describe all of the patterns you see in the table.
- Can you explain and justify any of the patterns you see? How can you be sure they will continue?
- What calculation would you do to find the total number of squares on a 100 × 100 chess board?
(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)
This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. (Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)
Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)
Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?
Think / Pair / Share
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?
Problem Solving Strategy 9 (Find the Math, Remove the Context).
Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.
In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:
- What is the sum of all the numbers on the clock’s face?
- Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
- How do I know when I am done? When should I stop looking?
Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.
Problem Solving Strategy 10 (Check Your Assumptions).
When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?
In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:
Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.
3.1 Use a Problem-Solving Strategy
Translate “6 less than twice x” into an algebraic expression.
If you missed this problem, review Example 1.26.
Approach Word Problems with a Positive Attitude
“If you think you can… or think you can’t… you’re right.”—Henry Ford
The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?
How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?
Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?
When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.
Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure 3.3 and say them out loud.
Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!
Use a Problem-Solving Strategy for Word Problems
We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.
Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.
Use a Problem-Solving Strategy to Solve Word Problems.
- Step 1. Read the problem. Make sure all the words and ideas are understood.
- Step 2. Identify what we are looking for.
- Step 3. Name what we are looking for. Choose a variable to represent that quantity.
- Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
- Step 5. Solve the equation using good algebra techniques.
- Step 6. Check the answer in the problem and make sure it makes sense.
- Step 7. Answer the question with a complete sentence.
Pilar bought a purse on sale for $18, which is one-half of the original price. What was the original price of the purse?
Step 1. Read the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.
Step 2. Identify what you are looking for. Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!
- In this problem, the words “what was the original price of the purse” tell us what we need to find.
Step 3. Name what we are looking for. Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.
Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.
Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.
Step 5. Solve the equation using good algebraic techniques. Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.
Step 6. Check the answer in the problem to make sure it makes sense. We solved the equation and found that p = 36 , p = 36 , which means “the original price” was $36.
- Does $36 make sense in the problem? Yes, because 18 is one-half of 36, and the purse was on sale at half the original price.
Step 7. Answer the question with a complete sentence. The problem asked “What was the original price of the purse?”
If this were a homework exercise, our work might look like this:
Pilar bought a purse on sale for $18, which is one-half the original price. What was the original price of the purse?
Joaquin bought a bookcase on sale for $120, which was two-thirds of the original price. What was the original price of the bookcase?
Two-fifths of the songs in Mariel’s playlist are country. If there are 16 country songs, what is the total number of songs in the playlist?
Let’s try this approach with another example.
Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were 11 girls in the study group. How many boys were in the study group?
Problem Solving Strategies
On this page we discuss Problem Solving Strategies under three headings.
What Are Problem Solving Strategies?
Strategies are things that Pólya would have us choose in his second stage of problem solving and use in his third stage (What is Problem Solving?). In actual fact he called them heuristics. To Pólya they were things to try that he couldn’t guarantee would solve the problem but, of course, he sincerely hoped they would. So they are some sort of general ideas that might work for a number of problems. And then again they might not.
As speaking in riddles isn’t likely to be of much assistance to you, let’s get down to some examples. There are a number of common strategies that children of primary age can use to help them solve problems. We discuss below several that will be of value for problems on this web-site and in books on problem solving. In this site we have linked the problem solving lessons to the following groupings of problem solving strategies. As the site develops we may add some more but we have tried to keep things simple for now.
Common Problem Solving Strategies
- Guess (this includes guess and check, guess and improve)
- Act It Out (act it out and use equipment)
- Draw (this includes drawing pictures and diagrams)
- Make a List (this includes making a table)
- Think (this includes using skills you know already)
We have provided a copymaster for these strategies so that you can make posters and display them in your classroom. It consists of a page per strategy with space provided to insert the name of any problem that you come across that uses that particular strategy (Act it out, Draw, Guess, Make a List). We have found that this kind of poster provides good revision for children. It also establishes links across curriculum areas. Through these links, children can see that mathematics is not only connected by skills but also by processes.
An In-Depth Look At Strategies
We now look at each of the following strategies and discuss them in some depth. You will see that each strategy we have in our list is really only a summary of two or more others.
This stands for two strategies, guess and check and guess and improve.
Guess and check is one of the simplest strategies. Anyone can guess an answer. If they can also check that the guess fits the conditions of the problem, then they have mastered guess and check. This is a strategy that would certainly work on the Farmyard problem but it could take a lot of time and a lot of computation.
Because it is such a simple strategy to use, you may have difficulty weaning some children away from guess and check. If you are not careful, they may try to use it all the time. As problems get more difficult, other strategies become more important and more effective. However, sometimes when children are completely stuck, guessing and checking will provide a useful way to start and explore a problem. Hopefully that exploration will lead to a more efficient strategy and then to a solution.
Guess and improve is slightly more sophisticated than guess and check. The idea is that you use your first incorrect guess to make an improved next guess. You can see it in action in the Farmyard problem. In relatively straightforward problems like that, it is often fairly easy to see how to improve the last guess. In some problems though, where there are more variables, it may not be clear at first which way to change the guessing.
2 Act It Out
We put two strategies together here because they are closely related. These are Act it Out and Use Equipment.
Young children especially, enjoy using Act it Out. Children themselves take the role of things in the problem. In the Farmyard problem, the children might take the role of the animals though it is unlikely that you would have 87 children in your class! But if there are not enough children you might be able to press gang the odd teddy or two.
There are pros and cons for this strategy. It is an effective strategy for demonstration purposes in front of the whole class. On the other hand, it can also be cumbersome when used by groups, especially if a largish number of students is involved. We have, however, found it a useful strategy when students have had trouble coming to grips with a problem.
The on-looking children may be more interested in acting it out because other children are involved. Sometimes, though, the children acting out the problem may get less out of the exercise than the children watching. This is because the participants are so engrossed in the mechanics of what they are doing that they don’t see through to the underlying mathematics. However, because these children are concentrating on what they are doing, they may in fact get more out of it and remember it longer than the others, so there are pros and cons here.
Use Equipment is a strategy related to Act it Out. Generally speaking, any object that can be used in some way to represent the situation the children are trying to solve, is equipment. This includes children themselves, hence the link between Act it Out and Use Equipment.
One of the difficulties with using equipment is keeping track of the solution. Actually the same thing is true for acting it out. The children need to be encouraged to keep track of their working as they manipulate the equipment.
In our experience, children need to be encouraged and helped to use equipment. Many children seem to prefer to draw. This may be because it gives them a better representation of the problem in hand. Also, if they’re a little older, they may feel that using equipment is only 'for babies'. Since there are problems where using equipment is a better strategy than drawing, you should encourage children’s use of equipment by modelling its use yourself from time to time.
It is fairly clear that a picture has to be used in the strategy Draw a Picture. But the picture need not be too elaborate. It should only contain enough detail to solve the problem. Hence a rough circle with two marks is quite sufficient for chickens and a blob plus four marks will do for pigs. There is no need for elaborate drawings showing beak, feathers, curly tails, etc., in full colour. Some children will need to be encouraged not to over-elaborate their drawings (and so have time to attempt the problem). But all children should be encouraged to use this strategy at some point because it helps children ‘see’ the problem and it can develop into quite a sophisticated strategy later.
It’s hard to know where Drawing a Picture ends and Drawing a Diagram begins. You might think of a diagram as anything that you can draw which isn’t a picture. But where do you draw the line between a picture and a diagram? As you can see with the chickens and pigs, discussed above, regular picture drawing develops into drawing a diagram.
Venn diagrams and tree diagrams are particular types of diagrams that we use so often they have been given names in their own right.
It’s probably worth saying at this point that acting it out, drawing a picture, drawing a diagram, and using equipment, may just be disguises for guessing and checking or even guessing and improving. Just watch children use these strategies and see if this is indeed the case.
4 Make a list
Making Organised Lists and Tables are two aspects of working systematically. Most children start off recording their problem solving efforts in a very haphazard way. Often there is a little calculation or whatever in this corner, and another one over there, and another one just here. It helps children to bring a logical and systematic development to their mathematics if they begin to organise things systematically as they go. This even applies to their explorations.
There are a number of ways of using Make a Table. These range from tables of numbers to help solve problems like the Farmyard, to the sort of tables with ticks and crosses that are often used in logic problems. Tables can also be an efficient way of finding number patterns.
When an Organised List is being used, it should be arranged in such a way that there is some natural order implicit in its construction. For example, shopping lists are generally not organised. They usually grow haphazardly as you think of each item. A little thought might make them organised. Putting all the meat together, all the vegetables together, and all the drinks together, could do this for you. Even more organisation could be forced by putting all the meat items in alphabetical order, and so on. Someone we know lists the items on her list in the order that they appear on her route through the supermarket.
In many ways we are using this strategy category as a catch-all. This is partly because these strategies are not usually used on their own but in combination with other strategies.
The strategies that we want to mention here are Being Systematic, Keeping Track, Looking For Patterns, Use Symmetry and Working Backwards and Use Known Skills.
Being Systematic, Keeping Track, Looking For Patterns and Using Symmetry are different from the strategies we have talked about above in that they are over-arching strategies. In all problem solving, and indeed in all mathematics, you need to keep these strategies in mind.
Being systematic may mean making a table or an organised list but it can also mean keeping your working in some order so that it is easy to follow when you have to go back over it. It means that you should work logically as you go along and make sure you don’t miss any steps in an argument. And it also means following an idea for a while to see where it leads, rather than jumping about all over the place chasing lots of possible ideas.
It is very important to keep track of your work. We have seen several groups of children acting out a problem and having trouble at the end simply because they had not kept track of what they were doing. So keeping track is particularly important with Act it Out and Using Equipment. But it is important in many other situations too. Children have to know where they have been and where they are going or they will get hopelessly muddled. This begins to be more significant as the problems get more difficult and involve more and more steps.
In many ways looking for patterns is what mathematics is all about. We want to know how things are connected and how things work and this is made easier if we can find patterns. Patterns make things easier because they tell us how a group of objects acts in the same way. Once we see a pattern we have much more control over what we are doing.
Using symmetry helps us to reduce the difficulty level of a problem. Playing Noughts and crosses, for instance, you will have realised that there are three and not nine ways to put the first symbol down. This immediately reduces the number of possibilities for the game and makes it easier to analyse. This sort of argument comes up all the time and should be grabbed with glee when you see it.
Finally working backwards is a standard strategy that only seems to have restricted use. However, it’s a powerful tool when it can be used. In the kind of problems we will be using in this web-site, it will be most often of value when we are looking at games. It frequently turns out to be worth looking at what happens at the end of a game and then work backward to the beginning, in order to see what moves are best.
Then we come to use known skills. This isn't usually listed in most lists of problem solving strategies but as we have gone through the problems in this web site, we have found it to be quite common. The trick here is to see which skills that you know can be applied to the problem in hand.
One example of this type is Fertiliser (Measurement, level 4). In this problem, the problem solver has to know the formula for the area of a rectangle to be able to use the data of the problem.
This strategy is related to the first step of problem solving when the problem solver thinks 'have I seen a problem like this before?' Being able to relate a word problem to some previously acquired skill is not easy but it is extremely important.
Uses of Strategies
Different strategies have different uses. We’ll illustrate this by means of a problem.
The Farmyard Problem: In the farmyard there are some pigs and some chickens. In fact there are 87 animals and 266 legs. How many pigs are there in the farmyard?
Some strategies help you to understand a problem. Let’s kick off with one of those. Guess and check. Let’s guess that there are 80 pigs. If there are they will account for 320 legs. Clearly we’ve over-guessed the number of pigs. So maybe there are only 60 pigs. Now 60 pigs would have 240 legs. That would leave us with 16 legs to be found from the chickens. It takes 8 chickens to produce 16 legs. But 60 pigs plus 8 chickens is only 68 animals so we have landed nearly 20 animals short.
Obviously we haven’t solved the problem yet but we have now come to grips with some of the important aspects of the problem. We know that there are 87 animals and so the number of pigs plus the number of chickens must add up to 87. We also know that we have to use the fact that pigs have four legs and chickens two, and that there have to be 266 legs altogether.
Some strategies are methods of solution in themselves. For instance, take Guess and Improve. Supposed we guessed 60 pigs for a total of 240 legs. Now 60 pigs imply 27 chickens, and that gives another 54 legs. Altogether then we’d have 294 legs at this point.
Unfortunately we know that there are only 266 legs. So we’ve guessed too high. As pigs have more legs than hens, we need to reduce the guess of 60 pigs. How about reducing the number of pigs to 50? That means 37 chickens and so 200 + 74 = 274 legs.
We’re still too high. Now 40 pigs and 47 hens gives 160 + 94 = 254 legs. We’ve now got too few legs so we need to guess more pigs.
You should be able to see now how to oscillate backwards and forwards until you hit on the right number of pigs. So guess and improve is a method of solution that you can use on a number of problems.
Some strategies can give you an idea of how you might tackle a problem. Making a Table illustrates this point. We’ll put a few values in and see what happens.
|pigs||chickens||pigs legs||chickens’ legs||total||difference|
From the table we can see that every time we change the number of pigs by one, we change the number of legs by two. This means that in our last guess in the table, we are five pigs away from the right answer. Then there have to be 46 pigs.
Some strategies help us to see general patterns so that we can make conjectures. Some strategies help us to see how to justify conjectures. And some strategies do other jobs. We’ll develop these ideas on the uses of strategies as this web-site grows.
What Strategies Can Be Used At What Levels
In the work we have done over the last few years, it seems that children are able to tackle and use more strategies as they continue with problem solving. They are also able to use them to a deeper level. We have observed the following strategies being used in the stated Levels.
- Draw a Diagram
- Act it Out
- Use Equipment
- Guess and Improve
- Make a Table
- Make an Organised List
It is important to say here that the research has not been exhaustive. Possibly younger children can effectively use other strategies. However, we feel confident that most children at a given Curriculum Level can use the strategies listed at that Level above. As problem solving becomes more common in primary schools, we would expect some of the more difficult strategies to come into use at lower Levels.
Strategies can develop in at least two ways. First children’s ability to use strategies develops with experience and practice. We mentioned that above. Second, strategies themselves can become more abstract and complex. It’s this development that we want to discuss here with a few examples.
Not all children may follow this development precisely. Some children may skip various stages. Further, when a completely novel problem presents itself, children may revert to an earlier stage of a strategy during the solution of the problem.
Draw: Earlier on we talked about drawing a picture and drawing a diagram. Children often start out by giving a very precise representation of the problem in hand. As they see that it is not necessary to add all the detail or colour, their pictures become more symbolic and only the essential features are retained. Hence we get a blob for a pig’s body and four short lines for its legs. Then children seem to realise that relationships between objects can be demonstrated by line drawings. The objects may be reduced to dots or letters. More precise diagrams may be required in geometrical problems but diagrams are useful in a great many problems with no geometrical content.
The simple "draw a picture" eventually develops into a wide variety of drawings that enable children, and adults, to solve a vast array of problems.
Guess: Moving from guess and check to guess and improve, is an obvious development of a simple strategy. Guess and check may work well in some problems but guess and improve is a simple development of guess and check.
But guess and check can develop into a sophisticated procedure that 5-year-old children couldn’t begin to recognise. At a higher level, but still in the primary school, children are able to guess patterns from data they have been given or they produce themselves. If they are to be sure that their guess is correct, then they have to justify the pattern in some way. This is just another way of checking.
All research mathematicians use guess and check. Their guesses are called "conjectures". Their checks are "proofs". A checked guess becomes a "theorem". Problem solving is very close to mathematical research. The way that research mathematicians work is precisely the Pólya four stage method (What is Problem Solving?). The only difference between problem solving and research is that in school, someone (the teacher) knows the solution to the problem. In research no one knows the solution, so checking solutions becomes more important.
So you see that a very simple strategy like guess and check can develop to a very deep level.
How to Solve Math Problems
This article was co-authored by Daron Cam. Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College.
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Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall. Keep reading to learn about some of these math problem solving strategies.
Using and Applying Mathematics
Aspects of using and applying reflect skills that can be developed through problem solving. For example:
In planning and executing a problem, problem solvers may need to:
- select and use appropriate and efficient techniques and strategies to solve problems
- identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting particular approaches
- break down a complex calculation problem into simpler steps before attempting a solution and justify their choice of methods
- make mental estimates of the answers to calculations
- present answers to sensible levels of accuracy understand how errors are compounded in certain calculations.
During problem solving, solvers need to communicate their mathematics for example by:
- discussing their work and explaining their reasoning using a range of mathematical language and notation
- using a variety of strategies and diagrams for establishing algebraic or graphical representations of a problem and its solution
- moving from one form of representation to another to get different perspectives on the problem
- presenting and interpreting solutions in the context of the original problem
- using notation and symbols correctly and consistently within a given problem
- examining critically, improve, then justifying their choice of mathematical presentation
- presenting a concise, reasoned argument.
Problem solvers need to reason mathematically including through:
- exploring, identifying, and using pattern and symmetry in algebraic contexts, investigating whether a particular case may be generalised further and understanding the importance of a counter-example identifying exceptional cases
- understanding the difference between a practical demonstration and a proof
- showing step-by-step deduction in solving a problem deriving proofs using short chains of deductive reasoning
- recognising the significance of stating constraints and assumptions when deducing results
- recognising the limitations of any assumptions that are made and the effect that varying the assumptions may have on the solution to a problem.
Explicitly Model Mathematics Concepts/Skills and Problem Solving Strategies
What is the purpose of Explicitly Modeling Mathematics Concepts/Skills and Problem Solving Strategies?
The purpose of explicitly modeling mathematics concepts/skills and problem solving strategies is twofold. First, explicit modeling of a target mathematics concept/skill provides students a clear and accessible format for initially acquiring an understanding of the mathematics concept/skill. Explicit modeling by you provides students with a clear, accurate, multi-sensory model of the skill or concept. Students must first be able to access the attributes of a concept/skill before they can be expected to understand it and be able to use it in meaningful ways. Explicit teacher modeling does just that. Second, by explicitly modeling effective strategies for approaching particular problem solving situations, you provide students a process for becoming independent learners and problem solvers. While peers can sometimes be effective models for students, students with special needs require a well qualified teacher to provide such modeling, at least in the initial phases of instruction.
What is Explicit Modeling?
Explicit modeling involves well-prepared teachers employing a variety of instructional techniques to illuminate the key attributes of any given mathematics concept/skill. In a sense, you serve as a "bridge of learning" for your student, an accessible bridge between the student and the particular mathematics concept/skill they are learning:
The level of teacher support you provide your students depends on how much of a learning bridge they need. In particular, students with learning problems need a well-established learning bridge (teacher model). They learn most effectively when their teacher provides clear and multi-sensory models of a mathematics concept/skill during math instruction.
What are some important considerations when implementing Explicit Modeling?
Instructional techniques that provide students such an accessible learning bridge:
- The teacher purposefully sets the stage for understanding by identifying what students will learn (visually and auditorily), providing opportunities for students to link what they already know (e.g. prerequisite concepts/skills they have already mastered, prior real-life experiences they have had, areas of interest based on your students' age, culture, ethnicity, etc.), and discussing with students how what they are going to learn has relevance/meaning for their immediate lives. To learn more about how to build meaningful student connections when introducing a new mathematics concept/skill visit the MathVIDS website (Descriptions of Instructional Strategies, Instructional Strategies By List, Building Meaningful Student Connections).
- Teacher both describes and models the math skill/concept.
- Teacher breaks math concept/skill into learnable parts/steps. Think about the concept/skill and break it down into 3-4 features or parts.
- Teacher clearly describes features of the math concept or steps in performing math skill using visual examples.
- Teacher describes/models using multi-sensory techniques. Use as many "input" pathways as possible for any given concept/skill including auditory, visual, tactile, and kinesthetic means. For example, when modeling how to compare values of different fractions to determine "greater than," you might verbalize each step of the process for comparing fractions while pointing to each step written on chart paper (auditory and visual), represent each fraction using fraction circle pieces, running your finger around the perimeter of each piece, laying one fraction piece over the other one and running your finger along the space not covered up by the fraction of lesser value/area "thinking aloud" by saying your thoughts aloud as you examine each fraction piece (visual, kinesthetic, auditory), verbalizing your answer and why you determined why one fraction was greater than the other, and having students run their fingers along the same fraction pieces and uncovered space (auditory, visual, tactile, kinesthetic).
- Teacher provides both examples and non-examples of the mathematics concept/skill. For example, in the above example, you might compare two different fractions using same process but place the fraction of greater value/area on top of the fraction of lesser value/area. Then prompt student thinking of why this is not an example of "greater than."
- Explicitly cue students to essential attributes of the mathematics concept/skill you model. For example, when associating the written fraction to the fraction pieces and their respective values, color code the numerator and denominator in ways that represent the meaning of the fraction pieces they use. Cue students to the color-coding and what each color represents. Then demonstrate how each written fraction relates to the "whole' circle:
2/4 = 2 of four equal pieces
- Teacher engages students in learning through demonstrating enthusiasm, through maintaining a lively pace, through periodically questioning students, and through checking for student understanding. Explicit modeling is not meant to be a passive learning experience for students. On the contrary, it is critical to involve students as you model.
- Scaffold your direction as students begin to demonstrate understanding through questions you ask.
- After modeling several examples and non-examples, begin to have your students demonstrate a few steps of the process.
- As students demonstrate greater understanding, ask them to complete more and more of the process.
- When students demonstrate complete understanding, have various students "teach" you by modeling the entire process.
- Play a game where you and your students try to "catch" each other making a mistake or leaving out a step in the process.
How do I implement Explicit Modeling?
- Select the appropriate level of understanding to model the concept/skill or problem solving strategy (concrete, representational, abstract).
- Ensure that your students have the prerequisite skills to perform the skill or use the problem solving strategy.
- Break down the concept/skill or problem solving strategy into logical and learnable parts (Ask yourself, "What do I do and what do I think as I perform the skill?"). The strategies you can link to from this site are already broken down into steps.
- Provide a meaningful context for the concept/skill or problem solving strategy (e.g. word or story problem suited to the age and interests of your students. Invite parents/family members of your students or members of the community who work in an area that can be meaningfully applied to the concept/skill or strategy and ask them to show how they use the concept/skill/strategy in their work.
- Provide visual, auditory, kinesthetic (movement), and tactile means for illustrating important aspects of the concept/skill (e.g. visually display word problem and equation, orally cue students by varying vocal intonations, point, circle, highlight computation signs or important information in story problems).
- "Think aloud" as you illustrate each feature or step of the concept/skill/strategy (e.g. say aloud what you are thinking as you problem-solve so students can better "visualize" the metacognitive aspects of understanding or doing the concept/skill/strategy).
- Link each step of the problem solving process (e.g. restate what you did in the previous step, what you are going to do in the next step, and why the next step is important to the previous step).
- Periodically check student understanding with questions, remodeling steps when there is confusion.
- Maintain a lively pace while being conscious of student information processing difficulties (e.g. need additional time to process questions).
- Model a concept/skill at least three times.
To see a teachers using Explicit Modeling click on the link below:
How does Explicitly Modeling Mathematics Concepts/Skills and Problem Solving Strategies help students who have learning problems?
- Teacher as model makes the concept/skill clear and learnable.
- High level of teacher support and direction enables student to make meaningful cognitive connections.
- Provides students who have attention problems, processing problems, memory retrieval problems, and metacognitive difficulties an accessible "learning map" to the concept/skill/strategy.
- Links between parts/steps are directly made, making confusion and misunderstanding less likely.
- Multi-sensory cueing provides students multiple modes to process and thereby learn information.
- Teaching students effective problem solving strategies provides them a means for solving problems independently and assists them to develop their metacognitive awareness.
What Mathematics Problem Solving Strategies can I teach my students?
Mathematics problem solving strategies that have research support or that have been field tested with students can be accessed by clicking on the link below. These strategies are organized according to mathematics concept/skill area. Each strategy is described and an example of how each strategy can be used is also provided. Additional information on teaching mathematics strategies can also be found at this site:
What are additional resources I can use to help me implement Explicitly Modeling Mathematics Concepts/Skills and Problem Solving Strategies?
Using Questioning to Stimulate Mathematical Thinking
Good questioning techniques have long being regarded as a fundamental tool of effective teachers. Unfortunately, research shows that 93% of teacher questions are "lower order" knowledge based questions focusing on recall of facts (Daines, 1986). Clearly this is not the right type of questioning to stimulate the mathematical thinking that can arise from engagement in open problems and investigations. Many Primary teachers have already developed considerable skill in good questioning in curriculum areas such as Literacy and History and social studies, but do not transfer these skills to Mathematics. Teachers' instincts often tell them that they should use investigational mathematics more often in their teaching, but are sometimes disappointed with the outcomes when they try it. There are two common reasons for this. One is that the children are inexperienced in this approach and find it difficult to accept responsibility for the decision making required and need a lot of practise to develop organised or systematic approaches. The other reason is that the teachers have yet to develop a questioning style that guides, supports and stimulates the children without removing the responsibility for problem-solving process from the children.
Types of Questions
Within the context of open-ended mathematical tasks, it is useful to group questions into four main categories (Badham, 1994). These questions can be used be the teacher to guide the children through investigations while stimulating their mathematical thinking and gathering information about their knowledge and strategies.
1. Starter questions
These take the form of open-ended questions which focus the children's thinking in a general direction and give them a starting point. Examples:
How could you sort these.
How many ways can you find to . ?
What happens when we . ?
What can be made from.
How many different . can be found?
2. Questions to stimulate mathematical thinking
These questions assist children to focus on particular strategies and help them to see patterns and relationships. This aids the formation of a strong conceptual network. The questions can serve as a prompt when children become 'stuck'. (Teachers are often tempted to turn these questions into instructions, which is far less likely to stimulate thinking and removes responsibility for the investigation from the child).
What is the same?
What is different?
Can you group these . in some way?
Can you see a pattern?
How can this pattern help you find an answer?
What do think comes next? Why?
Is there a way to record what you've found that might help us see more patterns?
What would happen if.
3. Assessment questions
Questions such as these ask children to explain what they are doing or how they arrived at a solution. They allow the teacher to see how the children are thinking, what they understand and what level they are operating at. Obviously they are best asked after the children have had time to make progress with the problem, to record some findings and perhaps achieved at least one solution.
What have you discovered?
How did you find that out?
Why do you think that?
What made you decide to do it that way?
4. Final discussion questions
These questions draw together the efforts of the class and prompt sharing and comparison of strategies and solutions. This is a vital phase in the mathematical thinking processes. It provides further opportunity for reflection and realisation of mathematical ideas and relationships. It encourages children to evaluate their work.
Who has the same answer/ pattern/ grouping as this?
Who has a different solution?
Are everybody's results the same?
Have we found all the possibilities?
How do we know?
Have you thought of another way this could be done?
Do you think we have found the best solution?
Levels of Mathematical Thinking
Another way to categorise questions is according to the level of thinking they are likely to stimulate, using a hierarchy such as Bloom's taxonomy (Bloom, 1956). Bloom classified thinking into six levels: Memory (the least rigorous), Comprehension, Application, Analysis, Synthesis and Evaluation (requiring the highest level of thinking). Sanders (1966) separated the Comprehension level into two categories, Translation and Interpretation, to create a seven level taxonomy which is quite useful in mathematics. As you will see as you read through the summary below, this hierarchy is compatible with the four categories of questions already discussed.
1. Memory: The student recalls or memorises information
2. Translation: The student changes information into a different symbolic form or language
3. Interpretation: The student discovers relationships among facts, generalisations, definitions, values and skills
4. Application: The student solves a life-like problem that requires identification of the issue and selection and use of appropriate generalisations and skills
5. Analysis: The student solves a problem in the light of conscious knowledge of the parts of the form of thinking.
6. Synthesis: The student solves a problem that requires original, creative thinking
7. Evaluation: The student makes a judgement of good or bad, right or wrong, according to the standards he values.
Combining the Categories
The two ways of categorising types of questions overlap and support each other.
For example, the questions:
Can you see a pattern?
How can this pattern help you find an answer? relate to Interpretation, and
What have you discovered?
How did you find that out?
Why do you think that? require Analysis, and
Have we found all the possibilities?
How do we know?
Have you thought of another way this could be done?
Do you think we have found the best solution? encourage Evaluation.
In the process of working with teachers on this topic, a table was developed which provides examples of generic questions that can be used to guide children through a mathematical investigation, and at the same time prompt higher levels of thinking.
You may also find Jennie Pennant's article Developing a Classroom Culture That Supports a Problem-solving Approach to Mathematics useful, which includes a section on questioning.
Badham, V. (1994) What's the Question?. Pamphlet 23. Primary Association for Mathematics (Australia)
Badham, V. (1996). Developing Mathematical Thinking Through Investigations. PAMphlet 31. Primary Association for Mathematics (Australia)
Bloom, B. (1956). Taxonomy of Educational Objectives Handbook 1: Cognitive Domain. New York: David Mackay
Dains, D. (1986). Are Teachers Asking the Right Questions? Education 1, 4 p. 368-374.
Sanders, N. (1966). Classroom Questions: What Kind? New York: Harper and Row.
3 Strategies to Conquer Math Word Problems
Guest post by Kady Dupre
Here’s a word problem for you:
Miss Friday’s class does a daily word problem. Ten of her students are great at word problems involving addition, and only 7 seem to understand subtraction word problems. Five of her students are bored with the easy problems. Thirteen students are still struggling with basic math facts and 3 have trouble reading the word problems at all. How many of her students are engaged and learning?
Here’s a better question: “How do you grow confident and effective problem solvers?”
Why Students Struggle with Math Word Problems
Students struggle with math word problems for many reasons, but three of the biggest I’ve encountered include:
Issue #1: Student Confidence
For many students, just looking at a word problem leads to anxiety. No one can think clearly with a sense of dread or fear of failure looming!
Issue #2: Flexible Thinking
Many kids are taught to solve word problems methodically, with a prescriptive step-by-step plan using key words that don’t always work. Plans are great, but not when students use them as a crutch rather than a tool. Today’s standardized tests and real-world applications require creative thinking and flexibility with strategies.
Issue #3: Differentiation
Teachers want students to excel quickly and often push too fast, too soon. In the case of word problems, you have to go slow to go fast. Just like in Guided Reading, you’ll want to give lots of practice with “just-right” problems and provide guided practice with problems just-above the students’ level.
3 Problem Solving Strategies
The solution is to conquer math word problems with engaging classroom strategies that counteract the above issues!
1. Teach a Problem-Solving Routine
Kids (and adults) are notoriously impulsive problem solvers. Many students see a word problem and want to immediately snatch out those numbers and “do something” with them. When I was in elementary school, this was actually a pretty reliable strategy! But today, kids are asked to solve much more complex problems, often with tricky wording or intentional distractors.
Grow flexible thinkers and build confidence by teaching a routine. A problem solving routine simply encourages students to slow down and think before and after solving. I’ve seen lots of effective routines but my favorites always include a “before, during, and after” mindset.
To make the problem solving routine meaningful and effective:
- Use it often (daily if possible)
- Incorporate “Turn & Teach” (Students orally explain their thinking and process to a partner.)
- Allow for “Strategy Share” after solving (Selected students explain their method and thinking.)
2. Differentiate Word Problems
No, this doesn’t mean to write a different word problem for every student! This can be as simple as adjusting the numbers in a problem or removing distractors for struggling students. Scaffolding word problems will grow confidence and improve problem solving skills by gradually increasing the level difficulty as the child is ready. This is especially effective when you are trying to teach students different structures of word problems to go with a certain operation.
For example, comparison subtraction problems are very challenging for some students. By starting with a simple version, you allow students to focus on the problem itself, rather than becoming intimidated or frustrated.
I’ve had great success in using scaffolded problems with my guided math groups. After solving the easier problem, students realize that it’s not that tricky and are ready to take on the tougher ones!
3. Compare Problems Side-by-Side
To develop flexible thinking, nothing is more powerful than analyzing and comparing word problems. Start by using problems that have similar stories and numbers, but different problem structures. Encourage conversation, use visual representations, and have students explain the difference in structure and operation. Here’s an example showing student work on two similar problems about monkeys. Click here to download a blank copy of these problems. My freebie includes several variations to help you differentiate.
Use these three strategies to get kids thinking and talking about their problem solving strategies while building that “oh-so-important” confidence, and you CAN conquer math word problems!
Kady Dupre has worked as a classroom teacher, instructional coach, and intervention teacher in elementary grades. She loves creating learning resources for students and teachers. She authors Teacher Trap, a blog aimed at sharing her challenges, successes, and insights as a teacher.
Importance of Problem Solving Skills
In math class with almost every problem that is presented there is some sort of method that is followed that places the student at the solution, however, there is not always one single method that leads to the answer. There can be many different solution paths that allow someone to reach the answer to a problem but every person looks at a problem in a different way, which is why some people may choose one method over another. By teaching students this discipline of solving problems the students will be better equipped to reach their goals in the future because they will learn that there are different ways to approach a problem and if the "problem solver" gets stuck they can try to look at the problem from a different angle and attempt a different method to reach a solution.
Another idea that is too often overlooked is that being a problem solver is not an ability it is a a character trait and a mind-set because it takes a person who is motivated intrinsically to go out and solve a problem. However, math teachers have the ability to shape the minds of their students to become problem-
solving minds. Problem solvers often take control of their own learning and persevere when faced with adversity. Everyday a student asks why they are learning something in math class and want to know when they will ever use it again. While there are many concepts that can be applied to math, every concept requires some sort of problem solving which allows the students to learn to think like a problem solver, which is something that can be applied to any aspect of life.
Members of society that lack problem solving skills are not as driven, if they run into an obstacle during the course of obtaining a goal he or she may simply give up rather than try to look at a problem from a different angle or they may not even realize that there could be another way to achieve the goal they believe they have failed to reach. Since people that have a problem solving mindset are conditioned to not give up on a problem they have a better sense of confidence and self-esteem when faced with adversity. This is because they know that there has to be a solution but they simply are not sure what approach will lead them to said solution and they do not become defeated when they cannot find the solution.
Benefits of Learning Problem Solving Skills
The first proven benefit of teaching students problem solving skills is that their achievement, confidence, and skills in mathematics and other curriculums increases. The main reason for this is that problem solving provides students with with ability to look at a situation from different points of view using critical and analytical thinking. By being a more critical thinker students can better foresee outcomes of a situation which allows them to decide what pathway to the desired solution would be most efficient. Another characteristic that is effected by the instruction of problem solving skills is a person's metacognitive skills. Metacognition is most often described as thinking about thinking and because problem solving is a decision making process metacognition plays a large role in the process. Metacognition is so important in the decision making and problem solving process because it allows the "problem solver" to be able to think about a plan of action and then determine if it will be effective or not by analyzing the outcome that will follow, or if the path taken does not lead to the desired solution the "problem solver" can reflect on their decision making process to find where he or she went wrong. Another benefit that students gain from learning problem-solving skills is that they learn how to collaborate and work cooperatively with their peers which will benefit them not only during school but also in sports that they may play, at home, and at current and future jobs. The ability to effectively work in a group or on a team is often a quality that employers look for because a team that works well together will produce better results than a team that does not work well together.
Other Places Problem Solving Skills are Applied
Some examples of non-classroom experiences that rely on problem solving and critical thinking skills:
- Reading a Map
- Reading Weather Reports
- Understanding Economics and Personal Finance
- Ensuring you are Getting the Best Buy
Please Share There More Examples of Where or When Problem Solving Strategies are Used Here
What is Problem Solving?
On this page we discuss "What is Problem Solving?" under three headings: introduction, four stages of problem solving, and the scientific approach.
Naturally enough, Problem Solving is about solving problems. And we’ll restrict ourselves to thinking about mathematical problems here even though Problem Solving in school has a wider goal. When you think about it, the whole aim of education is to equip children to solve problems. In the Mathematics Curriculum therefore, Problem Solving contributes to the generic skill of problem solving in the New Zealand Curriculum Framework.
But Problem Solving also contributes to mathematics itself. It is part of one whole area of the subject that, until fairly recently, has largely passed unnoticed in schools around the world. Mathematics consists of skills and processes. The skills are things that we are all familiar with. These include the basic arithmetical processes and the algorithms that go with them. They include algebra in all its levels as well as sophisticated areas such as the calculus. This is the side of the subject that is largely represented in the Strands of Number, Algebra, Statistics, Geometry and Measurement.
On the other hand, the processes of mathematics are the ways of using the skills creatively in new situations. Problem Solving is a mathematical process. As such it is to be found in the Strand of Mathematical Processes along with Logic and Reasoning, and Communication. This is the side of mathematics that enables us to use the skills in a wide variety of situations.
Before we get too far into the discussion of Problem Solving, it is worth pointing out that we find it useful to distinguish between the three words "method", "answer" and "solution". By "method" we mean the means used to get an answer. This will generally involve one or more Problem Solving Strategies. On the other hand, we use "answer" to mean a number, quantity or some other entity that the problem is asking for. Finally, a "solution" is the whole process of solving a problem, including the method of obtaining an answer and the answer itself.
But how do we do Problem Solving? There appear to be four basic steps. Pólya enunciated these in 1945 but all of them were known and used well before then. And we mean well before then. The Ancient Greek mathematicians like Euclid and Pythagoras certainly knew how it was done.
Pólya’s four stages of problem solving are listed below.
Four Stages of Problem Solving
1. Understand and explore the problem
2. Find a strategy
3. Use the strategy to solve the problem
4. Look back and reflect on the solution.
Although we have listed the Four Stages of Problem Solving in order, for difficult problems it may not be possible to simply move through them consecutively to produce an answer. It is frequently the case that children move backwards and forwards between and across the steps. In fact the diagram below is much more like what happens in practice
There is no chance of being able to solve a problem unless you are can first understand it. This process requires not only knowing what you have to find but also the key pieces of information that somehow need to be put together to obtain the answer.
Children (and adults too for that matter) will often not be able to absorb all the important information of a problem in one go. It will almost always be necessary to read a problem several times, both at the start and during working on it. During the solution process, children may find that they have to look back at the original question from time to time to make sure that they are on the right track. With younger children it is worth repeating the problem and then asking them to put the question in their own words. Older children might use a highlighter pen to mark and emphasise the most useful parts of the problem.
Pólya’s second stage of finding a strategy tends to suggest that it is a fairly simple matter to think of an appropriate strategy. However, there are certainly problems where children may find it necessary to play around with the information before they are able to think of a strategy that might produce a solution. This exploratory phase will also help them to understand the problem better and may make them aware of some piece of information that they had neglected after the first reading.
Having explored the problem and decided on a plan of attack, the third problem-solving step, solve the problem, can be taken. Hopefully now the problem will be solved and an answer obtained. During this phase it is important for the children to keep a track of what they are doing. This is useful to show others what they have done and it is also helpful in finding errors should the right answer not be found.
At this point many children, especially mathematically able ones, will stop. But it is worth getting them into the habit of looking back over what they have done. There are several good reasons for this. First of all it is good practice for them to check their working and make sure that they have not made any errors. Second, it is vital to make sure that the answer they obtained is in fact the answer to the problem and not to the problem that they thought was being asked. Third, in looking back and thinking a little more about the problem, children are often able to see another way of solving the problem. This new solution may be a nicer solution than the original and may give more insight into what is really going on. Finally, the better students especially, may be able to generalise or extend the problem.
Generalising a problem means creating a problem that has the original problem as a special case. So a problem about three pigs may be changed into one which has any number of pigs.
In Problem 4 of What is a Problem?, there is a problem on towers. The last part of that problem asks how many towers can be built for any particular height. The answer to this problem will contain the answer to the previous three questions. There we were asked for the number of towers of height one, two and three. If we have some sort of formula, or expression, for any height, then we can substitute into that formula to get the answer for height three, for instance. So the "any" height formula is a generalisation of the height three case. It contains the height three case as a special example.
Extending a problem is a related idea. Here though, we are looking at a new problem that is somehow related to the first one. For instance, a problem that involves addition might be looked at to see if it makes any sense with multiplication. A rather nice problem is to take any whole number and divide it by two if it’s even and multiply it by three and add one if it’s odd. Keep repeating this manipulation. Is the answer you get eventually 1? We’ll do an example. Let’s start with 34. Then we get
34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
We certainly got to 1 then. Now it turns out that no one in the world knows if you will always get to 1 this way, no matter where you start. That’s something for you to worry about. But where does the extension come in? Well we can extend this problem, make another problem that’s a bit like it, by just changing the 3 to 5. So this time instead of dividing by 2 if the number is even and multiplying it by three and adding one if it’s odd, try dividing by 2 if the number is even and multiplying it by 5 and adding one if it’s odd. This new problem doesn’t contain the first one as a special case, so it’s not a generalisation. It is an extension though – it’s a problem that is closely related to the original. You might like to see if this new problem always ends up at 1. Or is that easy?
It is by this method of generalisation and extension that mathematics makes great strides forward. Up until Pythagoras’ time, many right-angled triangles were known. For instance, it was known that a triangle with sides 3, 4 and 5 was a right-angled triangle. Similarly people knew that triangles with sides 5, 12 and 13, and 7, 24 and 25 were right angled. Pythagoras’ generalisation was to show that EVERY triangle with sides a, b, c was a right-angled triangle if and only if a 2 + b 2 = c 2 .
This brings us to an aspect of problem solving that we haven’t mentioned so far. That is justification (or proof). Your students may often be able to guess what the answer to a problem is but their solution is not complete until they can justify their answer.
Now in some problems it is hard to find a justification. Indeed you may believe that it is not something that any of the class can do. So you may be happy that the children can guess the answer. However, bear in mind that this justification is what sets mathematics apart from every other discipline. Consequently the justification step is an important one that shouldn’t be missed too often.
Another way of looking at the Problem Solving process is what might be called the scientific approach. We show this in the diagram below.
Here the problem is given and initially the idea is to experiment with it or explore it in order to get some feeling as to how to proceed. After a while it is hoped that the solver is able to make a conjecture or guess what the answer might be. If the conjecture is true it might be possible to prove or justify it. In that case the looking back process sets in and an effort is made to generalise or extend the problem. In this case you have essentially chosen a new problem and so the whole process starts over again.
Sometimes, however, the conjecture is wrong and so a counter-example is found. This is an example that contradicts the conjecture. In that case another conjecture is sought and you have to look for a proof or another counterexample.
Some problems are too hard so it is necessary to give up. Now you may give up so that you can take a rest, in which case it is a ‘for now’ giving up. Actually this is a good problem solving strategy. Often when you give up for a while your subconscious takes over and comes up with a good idea that you can follow. On the other hand, some problems are so hard that you eventually have to give up ‘for ever’. There have been many difficult problems throughout history that mathematicians have had to give up on.
That then is a rough overview of what Problem Solving is all about. For simple problems the four stage Pólya method and the scientific method can be followed through without any difficulty. But when the problem is hard it often takes a lot of to-ing and fro-ing before the problem is finally solved – if it ever is!
Watch the video: 3 σπαζοκεφαλιές που θα σας τρελάνουν!!! (December 2021).