General Problem Solving Interventions
Skill- Math Problem Solving
Intervention – Strategy Application
Source or adapted from – Ministry of Education, New Zealand Maths
http://www.nzmaths.co.nz/
Instructions for administration:
There are a number of common strategies that children of primary age can use to help them solve problems. These
should be both encouraged and explored.
Common Problem Solving Strategies
1.
2.
3.
4.
5.
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)
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.
1 Guess
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.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.
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
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.
3 Draw
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 color. 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 Organized 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 organize 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 Organized 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 organized. They usually grow haphazardly as you
think of each item. A little thought might make them organized. Putting all the meat together, all the vegetables
together, and all the drinks together, could do this for you. Even more organization 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.
5 Think
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 organized 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.
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. The trick here
is to see which skills that you know can be applied to the problem in hand. 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.
Suggested Questions for Teacher To Use:
A. Getting Started
1.
2.
3.
4.
5.
6.
7.
8.
9.
Has anyone seen a problem like this before?
What are the important ideas in this problem?
Can you rephrase the problem in your own words?
What is this problem asking you to find out?
What information has been given?
What conditions apply?
Can you guess what the answer might be?
What strategies might you use to get started?
Which of these ideas are worth pursuing?
B. While Working on the Problem
1.
2.
3.
4.
5.
6.
7.
Tell me what you are doing?
Why (How) did you think of that?
Why are you doing this?
What will you do with the result of that work when you’ve got it?
Why is this idea better than that one?
You’ve been trying that idea for 5 minutes. Is it time to try something else?
Can you justify that step?
C. At the Finish
1.
2.
3.
4.
5.
6.
7.
Have you answered the question?
Have you considered all possible cases?
Have you checked your solution?
Does the answer look reasonable?
Is there another answer?
Is there another solution?
Can you explain your solution to the me?
8.
9.
Is there another way to solve the problem?
Can you generalize or extend the problem?
Suggested Progress Monitoring Tool:
Directions:
1. Display or give each student a copy of the word problem.
2. Students should use a problem solving strategy such as modeling with cubes,
drawings, or equations to represent the problem. Students should display their
work on the page or model for the teacher for younger students.
3. Consider having students work in pairs/groups and then share the variety of
ways they solved the problem.
Considerations:
Watch to see if students are able to understand the problem.
 Did each student use an appropriate strategy to solve the problem?
 Do the students arrive at the correct solution?
Collecting Data:
Student performance can be scored with a provided task rubric or a rubric created by
the teacher.
Data can be recorded on a score sheet.
Not yet: Student shows evidence of
misunderstanding, incorrect concept or
procedure
0 Unsatisfactory:
1 Marginal:
Little
Partial
Accomplishment
Accomplishment
Got It: Student essentially understands the
target concept.
2 Proficient:
Substantial
Accomplishment
3 Excellent:
Full
Accomplishment
The task is
attempted and some
mathematical effort
is made. There may
be fragments of
accomplishment but
little or no success.
Further teaching is
required.
Student could work
to full
accomplishment with
minimal feedback
from teacher. Errors
are minor. Teacher is
confident that
understanding is
adequate to
accomplish the
objective with
minimal assistance.
Strategy and
execution meet the
content, process,
and qualitative
demands of the task
or concept. Student
can communicate
ideas. May have
minor errors that do
not impact the
mathematics.
Part of the task is
accomplished, but
there is lack of
evidence of
understanding or
evidence of not
understanding.
Further teaching is
required.