Organising the Teaching of Problem Solving
Our purpose in this section is to suggest ideas for lesson structure and organisation. We break this section up into three using the headings:
Lesson Structure Scaffolding The Year Plan
In this web-site you will find lesson plans for problems sorted by both Curriculum Strand and Problem Solving Strategy. We have focussed mainly on providing single lessons with one-off problems for you to incorporate into your units of mathematics although it is of course possible to link these lessons or extend them to create problem solving units. There are some lessons, however, that are more of an investigtive nature and these will involve more than one teaching period.
One-off Problems
There are a number
of
ways
of
organising
one-off
problem
solving
lessons.
One
way
that
we
have
found
to
be
quite
successful
is
the
following
three-stage
format.
Three-Stage Lesson Format
- Introduction;
- Group Work;
- Reporting Back.
1.
Introduction.
This
is
a
whole
class
phase
where
the
problem
is
presented.
The
introductory
stage
of
the
lesson
may
last
from
a
few
seconds
to
as
much
as
15
minutes,
depending
on
what
you
are
aiming
to
get
from
it.
In the initial stages of problem solving, or when a new strategy is being introduced, the introduction to the lesson may take some time. During this period, with the help of the children, you may model the problem solving process or a particular strategy, before sending the children off to practice the point for themselves.
The simplest introduction, for older children who have had some problem solving experience, might be to just say "Can you do the problem on the board?" But this will not always suit your purpose. As in any introduction it is important to interest the children so that they are motivated to participate.
2.
Group
Work.
After
the
introduction
stage,
the
children
go
off
in
their
groups
to
tackle
the
problem.
While
they
are
working,
you
move
around
and
provide
suitable
help.
(We’ll
say
more
about
this
help
under
Scaffolding
below.)
There are a number of things that can be said about group work, but the main two questions to consider are (i) how big should groups be and (ii) how should you assign children to groups?
We have watched quite a lot of groups of different sizes at work and we think that for problem solving, groups of two seems to work best. This appears as much as anything to be a problem of the actual physical arrangement rather than a social problem. Larger groups certainly work well for activities involving equipment. Problem solving groups bigger than four, seem to break up into sub-groups unless the children are using an Act it Out strategy. Other strategies don’t seem to be able to hold everyone’s attention and co-operation.
One thing that defeats groups of three and four is the geometry of their workspace. If the group is trying to work around a table, there are usually two children on either side. When they are working with pencil and paper, pairs on each side of the table tend to split off and work together. It is just too inconvenient for them to look at work on the other side of the table that for them is upside down. Groups of three often isolate one child. This is generally not for social reasons but because it is difficult for all three to share a piece of paper and work together.
So for problem solving purposes, two appears to be the optimum number of children to engage in co-operation.
On the matter of co-operation, New Entrants children especially need some assistance before they can really co-operate with one of their peers. It is not something that comes automatically. One way to facilitate this is to have the pair work together on a single sheet of paper.
Then there is the question of how to group the children. From talking to children we know that they prefer to work with a friend. They say they are more likely to share ideas with someone they know reasonably well. You can always try this method of pairing. If friendship groups are working well, then continue with them. Some teachers prefer to put children together more or less randomly. This can be done, for instance, by picking sweets out of a box. Children with the same kind of sweet form a group (and eat the sweets).
Whatever way groups are chosen it is probably a good idea to change them from time to time. In that way children get to work with a range of others and groups that are not working well don’t have to stay together for a long time.
It is not clear whether able mathematical students should only work with able students. Certainly such pairings will often be beneficial for both the children concerned. Some teachers prefer to use mixed ability groups but these have to be monitored. The more able child might end up doing all the work and not provide learning experiences for the other member of the pair. If the more able child begins to feel put upon, that child may stop co-operating completely and the group may well break up as a result. So you have to watch the group dynamics.
But, however the groups are constituted, you will need to monitor their progress. This can be partially done by having both members of the group report back to the class in the final stage of the lesson and partially by looking in on each group in the second stage of the lesson.
One of the problems with groups of size two, though, is that it may take you some time to get to around to all of them. You should realise that you will be unable to help all the children in your class all the time but you should be able to monitor everybody and make sure that they are all on task. You don’t necessarily have to spend a great deal of time with every group in every lesson. Rather, you should ensure that each group is given more than cursory attention at least once every other lesson. Later, when the children have learnt to work well in pairs, you might consider putting them into larger groups. Our experience is that groups of size four tend to split into two groups of two but they do keep contact as regards their progress on a problem. With larger groups it is easier for you to visit each group at least once every lesson and respond to individual children’s needs.
3.
Reporting
Back.
Generally
we
expect
some
children,
from
some
of
the
groups,
to
report
back
to
the
whole
class
at
the
end
of
each
problem.
We
deliberately
said
that
the
reporting
back
was
to
the
class
and
not
the
teacher.
Although
you
may
use
this
part
of
the
lesson
to
assess
children’s
progress
in
a
number
of
areas,
reporting
back
can
be
a
very
positive
learning
experience
for
all
the
class,
not
just
those
doing
the
reporting.
The Main Purposes of Reporting Back
To expose students to a range of thinking. In problem solving, there may be more than one way to solve the problem. Children will learn to use new strategies that they see that other children have used.
To foster mathematical communication skills. This in itself is an important process. However, it does have the valuable by-product of increasing understanding. In verbalising their own solutions, children will understand them better themselves.
To increase students’ confidence. In a friendly classroom atmosphere we have seen many students’ confidence increase, both in themselves and in their mathematical ability, by reporting back to the class. Some children will require more help than others, of course.
To clarify common misunderstandings. Verbalising seems to help many children see their errors more quickly than writing their answers out on paper. By discussing points of difference, valuable learning can be achieved.
To provide a foundation for the extension of a problem. Once the children have made sure that their solutions are correct and that they understand the solution, you can move them forward to generalise and extend the problem (see What is Mathematics?) This is an important aspect of mathematics.
To highlight the mathematics inherent in the problem. You should take the opportunity to point out any new strategies that a group has found and to show how this fits in to the other strategies that have been used. The ideas produced can also be related to the problems that have been attempted and to other aspects of the curriculum. Any generalisations and extensions can be developed to show that mathematics is about big ideas that cover many problems.
We have seen many valuable reporting back sessions where the children have been engrossed by what their peers have said and done. This can be because children sometimes understand what their peers say better than they understand what their teachers say. Maybe this is because a peer has had to struggle through a problem from scratch and shows the solution in smaller steps than a teacher might. It is also possible that children will express the solution as more of an idea in progress than a finished, polished solution. Sometimes it is easier to transmit ideas than the final product.
But as we said above, during this stage of the lesson you may take the time to monitor the children’s ability. This will give you the opportunity to assess the progress they are making and accordingly vary the experiences that you are providing them with. Particular aspects of this are listed below.
Points To Monitor
- presentation of a reasoned argument;
- making of conjectures (guesses);
- making deductions;
- proving and reporting other children’s statements;
- demonstration of flexible thinking;
- the thought processes of the children;
- the explanations children give.
It should be emphasised at this point, that a lesson may not necessarily proceed nicely through the sequence 1. Introduction, 2. Group Work, 3. Reporting Back, above, in that order. For instance, there may be good reasons for repeating the stages 1, 2, 1, 2, and sometimes there are advantages in leaving stage 3 until the following lesson. One teacher we know goes through the teps 1, 2, 3, several times in a lesson. In this way, he is ble to judge the overall progress of the class and ensure that no children get too far behind. However your lesson goes, we feel that these three basic stages will provide a useful framework for problem solving lessons.
Scaffolding Good Questions to Ask Back to Top
One of the prime goals of problem solving in school is to prepare children to tackle problems in later life. When dealing with these problems, they will not have a teacher by their side to prompt them towards a solution. Part of the reason for scaffolding, good leading questions, is to help prepare children for later problem solving. Of course, scaffolding is also important to help children solve the problem they are working on currently. Nevertheless, you should be aware that good scaffolding questions should be incorporated into children's thinking and become part of their approach to problem solving.
We now spend some time considering the types of questions that you might like to ask during problem solving lessons. We break these scaffolding questions into three types. These types correspond roughly to both Polya’s four stages of problem solving (What is Problem Solving?) and to the three stages of the problem solving lesson we have just been talking about.
A. Getting Started
- 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
- 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
- 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 class?
- Is there another way to solve the problem?
- Can you generalise or extend the problem?
All of the above questions can be used with almost any problem. They are generic questions that fit almost any situation. The difficult scaffolding questions are the ones you have to use with particular children working on particular problems. It takes practice to be able to say the right thing at the right time. There are a few guiding principles, however.
Wait time. The research seems to suggest that most of us are too anxious when it comes to receiving answers from children. We don’t leave them sufficient time to think about their answer and then reply. So try to give children the time they need to respond.
Another aspect of this relates to group questions. One of the most successful teachers we have seen stands beside a group for a while before she asks any questions at all. During this time she is able to gauge what they are doing and where they are with the problem. Hence she can obtain a good idea of what help they need, or what assessment she is going to try, before she commits herself to asking questions.
Of course, it is not always necessary to ask a question. You might simjply say "very nice idea"and walk away. All of us respond very favourably to praise so we should not be afraid to use it with groups.
The guide by the side, not the sage on the stage. The point is that you are there to give assistance and to help the student over the hurdles. We do not feel that you can develop good problem solving in the children solely by standing at the front and explaining. Problem solving in that sense is more like coaching. Remember that the person who does the thinking does the learning. What’s more one of the things they learn, and one of the things that we want them to learn, is how to think.
Open rather than closed. As a consequence of the above philosophy, the questions that you ask should be open rather than closed (as in the list above under Getting Started, While Working on the Problem and At the Finish). Questions should provoke thinking and encourage ideas. They should enable the children to make their own progress. The idea is to try to ask questions that do not have a fixed answer. Good questions can be answered in a number of ways, each of which is effective.
Small grunts. Minimal encouragers are also valuable as children will then tell you their ideas. "Right", "OK", etc., provides support. Phrases such as "I like that idea" also give positive reinforcement. At the same time, none of these responses takes away the opportunity for children to find the key to the problem themselves.
In all of this, you will be modelling behaviour that is ultimately expected of all children. One of the main objectives is to get children to internalise the questions (scaffolding) that you provide. The point is that when a child runs into difficulties and you are not around to provide assistance, the child can go through their scaffolding armoury and ask the questions that you might have asked. In this way, the child may be able to surmount the current difficulty without external assistance. Surely this is one of the goals of education, that children can think for themselves.
One word of warning, good questioning doesn’t always work. We have seen what were apparently good scaffolding attempts by teachers, fail. The children being scaffolded apparently took no notice of the scaffolding when the teacher moved on to another group. We think that there are at least three reasons for this. First, the children may have been so engrossed in their approach to the problem that they were not able to "hear" what the teacher said. In situations like this the children need to be shown the error in their thought processes before they will consider a change in direction. Even when some children are shown to be wrong, they may not immediately acknowledge their error. Rather they often try to get around the difficulty somehow before they will listen to a new idea.
Second, although the teacher’s scaffolding appeared to be excellent to the adult observer, it may not have been seen that way by the children. On a number of occasions we have seen teachers, through their use of language, go over the children’s heads. There are occasions when teachers are unable to communicate with their students. Strangely enough, children can sometimes communicate mathematical ideas to children better than teachers can.
The third reason for scaffolding not being effective is that it is outside the children’s zone of proximal development and so it "goes over their heads".
Vygotsky was a Russian psychologist who coined the phrase zone of proximal development. He observed children who were operating both with and without the assistance of adults. Generally there was a difference of achievement between the occasions when the children were on their own and the occasions when they had adult help. This difference he referred to as their zone of proximal development.
Another
way
of
thinking
of
it
is
to
imagine
the
child
as
a
point
somewhere
in
space.
Where
the
child
is
represents
their
capacity
for
action
of
various
kinds.
It
represents
their
ability
at
a
particular
time.
Around
this
child
is
some
region.
This
represents
the
things
that
the
child
can
do,
given
appropriate
assistance.
This
region
is
called
the
child’s
zone
of
proximal
development.
We
show
this
diagrammatically
below.
Implicit
in
Vygotsky’s
writings
is
the
notion
of
scaffolding.
This
is
the
assistance
required
to
move
the
child
from
their
current
state
to
a
point
in
their
zone
of
proximal
development.
It
is
worth
pointing
out
though,
that
despite
the
emphasis
we
give
to
questioning,
scaffolding
is
not
solely
achieved
by
questioning.
Your
attitude
and
demeanour
are
very
important
in
the
scaffolding
process.
The
way
you
set
up
your
classroom
can
also
help
to
scaffold
children’s
learning.
Although we advocate trying to lure children into providing answers for themselves, there are times when you have to give up and provide the answers. This may be because you can think of no more open questions to ask. It may also be that the child is in such an emotional state that they would not give the answer even if they knew it. Again, it may be because the problem is outside the children’s zone of proximal development. No amount of encouraging and scaffolding will help here. For this reason the choice of problem is of the utmost importance. If the problem is too hard, then the best strategy may be to back off and come back to it later when the children have developed a little in maturity and mathematics.
Despite this, good scaffolding does produce good results. It is a facet of teaching that is well worth persisting with. Of course all this is well and good if you are on top of your game. It’s a very different story if you have a heavy cold or if it’s Friday afternoon. However, scaffolding does become easier with practice. The positive results you will get with good questioning make it worth persevering with.
Finally in this section, it is worth while saying something about metacognition. This is essentially "thinking about your own thinking". It is a useful process that enables a person to monitor what they are doing. There are at least two reasons for teacher scaffolding. The first is to get children over the hurdle that they are currently facing. The second is to model practice that they will find useful in future situations. When they have internalised this scaffolding it becomes part of their metacognitive processing and should make tackling problems easier in the future.
II Investigations
We noted earlier that on this web site there are now several investigations that require extended thought. All of what we have said above in I applies to investigations too. But there is a little more to investigations. These are usually undertaken in Levels 4 to 6. At this point in their development you will probably want the children to write up the results of their investigation in a report. Hence you may want to use a Four Stage Lesson Format consisting of
- Introduction;
- Group Work;
- Reporting Back;
- Writing.
Stages 1 to 3 will probably be used several times during an investigation. This may also be true of Stage 4, because it may be useful for children to write notes of their progress as they go along. These notes can then be combined to make their final report.
The actual format of the final write-up is entirely up to you and reflect what you plan to achieve from this stage of their work. Point form is fine if you just want a summary of what was done. On the other hand, a complete explanation of an investigation will enable you to assess certain communication skills that are not easily assessed any other way.
Finally here it is worth noting that the Four-Stage Lesson Format is a good one for regular lessons from Level 5 upward. It enables the introduction of a topic using a problem solving approach and it allows children the opportunity to make their own notes for future use.
As problem solving is a component of the Mathematical Processes Strand you need to consider how you are going to include it within your mathematics programme. In addition to determining how you are going to structure problem solving within your maths lesson you need to consider how you are going to show coverage of the achievement objectives within your long-term plan.
One way to ensure coverage of the problem solving achievement objectives is to incorporate problem solving lessons within all of your planned units of work. Some teachers do this by having a problem solving lesson one day every week, while others do problem solving for the final two days of a unit.
In addition you may decide to plan a unit of work with problem solving as the focus. This is especially useful if the children are beginning problem solvers. In a problem-solving unit you may introduce a number of problem solving strategies. You may also use a unit of problem solving to reinforce the children’s use of the problem solving steps.
The table that follows is one example of a term plan that illustrates the inclusion of problem solving in the maths programme. Further examples can be found in the Ministry of Education Publication Developing Mathematics Programmes.
Sample Maths Plan – Term 2 (level 2)
Week | Unit | Content Strands | Mathematical Processes Strand |
1 | Seeing Shapes | Geometry Make, name and describe shapes... |
PS – Strategy (draw a picture) |
2 | C – Using own language | ||
3 | Teddy Bear Problem Solving | Algebra, Number, Geometry | PS - Strategies |
4 | Thinking numbers | Number Make sensible estimates... Mentally perform calculations... Write and solve story problems. |
PS – Posing questions |
5 | C – using own language | ||
6 | |||
7 | Animal sizes | Measurement Practical measuring tasks using metric units for length, mass. |
PS – Strategy (Organised list) |
8 | L& R – classify objects | ||
9 | Surveying streets | Statistics Collect and display category data... Talk about features... |
C – Record results |
10 |