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Putting Problems First
A look at how a "problem first" approach to learning and teaching Mathematics can improve pupil engagement and help develop perseverance and essential higherorder skills. Presented at the 2014 SMC conference in Stirling.
by
TweetStuart Welsh
on 21 May 2015Transcript of Putting Problems First
Go create
A curious mind
(1, 7)
(9, 3)
(3, 1)
The problem
We need problem solvers
but
Many pupils struggle
with problem solving
We know that many learners find problem solving difficult.
However, there is an increasing demand for
independent thought
and highlevel
problemsolving ability
from workers at all levels in the modern workplace.
Employers want people who can
investigate and understand a problem
, explore problems from different angles,
process and evaluate
information, and make reasoned, evidence based conclusions.
In short, our learners must develop
higher order thinking
and problem solving skills if they are to seriously compete in the modern workplace.
There is no better subject than Mathematics in which to develop these skills. I don’t mean throw a couple of maths challenges at them every term.
We are talking about embedding rich
problem solving
and opportunities for
discovery learning
into
lessons
, accessible to
all abilities
, from 3 – 18.
A fundamental shift is needed. CfE is on the right track but there’s a lot more to be done, and we're the ones to do it!
Putting Problems First
Stuart Welsh
sjw@hsog.co.uk
What do you see?
Current Practice
Today, we are learning to... "
express one number as a percentage of another
"
Take a moment to think about what goes through our learners' minds when they see this learning intention?
Typically we demonstrate a simple process, get learners to practise this process, demonstrate a more complicated process, practise, repeat, then do one or two problems before it is time for the next topic.
Some pupils never reach the problems.
Assessment
Our assessments typically focus on
remembering
content /
reproducing
a process
We
tell
learners what to do, e.g. "Evaluate" / "Multiply out the brackets" etc.
Questions go from easier > harder with "
non routine
" questions regularly the most difficult for learners
End of topic tests focus on quantity of questioning, not quality. They
test process, not understanding
.
What's CfE saying?
Learners should have a secure
understanding
of concepts, principles and processes
Be able to apply these in different
contexts
Engage
with abstract mathematical concepts
Develop important
new kinds of thinking
Understand the
application
of mathematics
Creatively and logically
solve problems within a variety of contexts
Plus...
Learning experiences should include
Investigative
or
project
based tasks
Learners should have the opportunity to work
collaboratively and independently
on
tasks which engage.
solve problems,
explaining thinking
to others
Make links
across the curriculum
Interpret a situation requiring maths and
select a strategy
Solution
Get Creative
Turn it on its head
How to develop problem solving?
Sounds great in theory
How do we actually put it into practice?
We need to teach in a way that will
develop problem solving skills, as well as teach the processes
required by the curriculum
We need to teach in a way that will instil and nurture an intrinsic motivation for solving problems.
Problems 1st  process 2nd
Giving students a problem to solve presents them with an unfinished picture that begs to be completed and that draws them into the task.
Pupils actually
need the maths
!
An everyday lesson plan for problem solving
Replace learning intentions with problems
W.A.L.T.
Express one number as a percentage of another
vs
Here are this week's and last week's performance figures for 3 employees at a telesales call centre.
The workers all have different targets.
Compared with their targets, whose sales have improved the most over the past week, and who is getting fired?
Make three of your class the employees, the rest can be the board members deciding who gets fired. Make it real!
Decisions, decisions!
Replace learning intentions with problems
WALT:
Factorise a trinomial
vs
Apple dropped in water.
Depth modelled by d(t).
How long was it underwater?
Bring in an apple, drop it in a bucket of water! Make it real.
The process of factorising isn't the goal, solving the problem is the goal, the process becomes a requirement.
Replace learning intentions with problems
W.A.L.T. Calculate the gradient of a straight line from vertical and horizontal distances
vs
Climbs in the Tour de France are classified according to their steepness. Cat 3 > 0.05, cat 2 > 0.07, cat 1 > 0.1 etc.
You have been asked by the race organiser to put these climbs into the
correct category.
Learners select a strategy then make decisions.
Give them responsibility. Make it real!
Everything in "rich" context
We must use our
creativity
and
imagination
to generate the interest needed to make instruction more enjoyable for our students.
Placing course content in the context of a real world scenario provides the frame of reference the mind needs to retain that content.
Rich
Context?
1.2 m
2.2 m
Richer Context?
A builder needs new ladder. He has 3 lengths to choose from. Longer ladders cost more and times are tight, he doesn't want to spend any more than he has to.
H & S regulations say the foot of any ladder must be a minimum distance from the foot of the wall. This distance is 15% of the height the ladder reaches up the wall.
He needs a ladder that will reach 6 metres up the wall. Which one should he buy?
vs.
Real life problems don't come in nice packages
Relevant, reallife context
Decisions, decision
If pupils struggle with reading and interpreting, why not act out the whole situation? Video a senior pupil pretending to be a builder.
Make it real!
Shhh... Don't tell
Resist the temptation to tell.
Telling has a powerful hold on the teaching profession. Often telling is our main method of instruction, even though there are strong indicators that
telling
is the
least effective
way for students to learn.
Don't underestimate the value of giving learners time to think about, discus and investigate possible solutions to problems. They'll dig into their prior knowledge, make connections and come up with solutions you would never have thought of. Eventually though, they will say "We know what we need figure out  we just don't know how to do it."
This is the time to tell them, once there is a need.
Bring it all together
Present a problem:
Give plenty of
time
for learners to think about (and discuss) ways to solve it.
Establish a need
for a new mathematical process.
Introduce process:
Instruction is still required  but now there is need
Solve the problem together

then present another problem, one that requires the new process.
Consolidate process:
Rigour
is still necessary, only hopefully motivation to practise is increased.
Area of a circle
This would be the starting point but really lacks any engagement...
3cm
Calculate the area
This example is a slight step forward but still doesn't really engage learners
8cm
Calculate the radius
Area = 40 square centimetres
Traditionally this question presents a nonroutine problem which not every pupil will progress to...
Calculate the area of this circle
The square has perimeter 28 cm
By starting the journey here, pupils are encouraged to think more deeply about the problem.
A great opportunity to work together, discuss options and make decisions.
What's the question?
Further reading
Ted McCain  "Teaching for tomorrow"
TED
talks:
Sir Ken Robinson, Dan Meyer
Web:
www.blog.mrmeyer.com
www.101qs.com
www.brilliant.org
Full transcriptA curious mind
(1, 7)
(9, 3)
(3, 1)
The problem
We need problem solvers
but
Many pupils struggle
with problem solving
We know that many learners find problem solving difficult.
However, there is an increasing demand for
independent thought
and highlevel
problemsolving ability
from workers at all levels in the modern workplace.
Employers want people who can
investigate and understand a problem
, explore problems from different angles,
process and evaluate
information, and make reasoned, evidence based conclusions.
In short, our learners must develop
higher order thinking
and problem solving skills if they are to seriously compete in the modern workplace.
There is no better subject than Mathematics in which to develop these skills. I don’t mean throw a couple of maths challenges at them every term.
We are talking about embedding rich
problem solving
and opportunities for
discovery learning
into
lessons
, accessible to
all abilities
, from 3 – 18.
A fundamental shift is needed. CfE is on the right track but there’s a lot more to be done, and we're the ones to do it!
Putting Problems First
Stuart Welsh
sjw@hsog.co.uk
What do you see?
Current Practice
Today, we are learning to... "
express one number as a percentage of another
"
Take a moment to think about what goes through our learners' minds when they see this learning intention?
Typically we demonstrate a simple process, get learners to practise this process, demonstrate a more complicated process, practise, repeat, then do one or two problems before it is time for the next topic.
Some pupils never reach the problems.
Assessment
Our assessments typically focus on
remembering
content /
reproducing
a process
We
tell
learners what to do, e.g. "Evaluate" / "Multiply out the brackets" etc.
Questions go from easier > harder with "
non routine
" questions regularly the most difficult for learners
End of topic tests focus on quantity of questioning, not quality. They
test process, not understanding
.
What's CfE saying?
Learners should have a secure
understanding
of concepts, principles and processes
Be able to apply these in different
contexts
Engage
with abstract mathematical concepts
Develop important
new kinds of thinking
Understand the
application
of mathematics
Creatively and logically
solve problems within a variety of contexts
Plus...
Learning experiences should include
Investigative
or
project
based tasks
Learners should have the opportunity to work
collaboratively and independently
on
tasks which engage.
solve problems,
explaining thinking
to others
Make links
across the curriculum
Interpret a situation requiring maths and
select a strategy
Solution
Get Creative
Turn it on its head
How to develop problem solving?
Sounds great in theory
How do we actually put it into practice?
We need to teach in a way that will
develop problem solving skills, as well as teach the processes
required by the curriculum
We need to teach in a way that will instil and nurture an intrinsic motivation for solving problems.
Problems 1st  process 2nd
Giving students a problem to solve presents them with an unfinished picture that begs to be completed and that draws them into the task.
Pupils actually
need the maths
!
An everyday lesson plan for problem solving
Replace learning intentions with problems
W.A.L.T.
Express one number as a percentage of another
vs
Here are this week's and last week's performance figures for 3 employees at a telesales call centre.
The workers all have different targets.
Compared with their targets, whose sales have improved the most over the past week, and who is getting fired?
Make three of your class the employees, the rest can be the board members deciding who gets fired. Make it real!
Decisions, decisions!
Replace learning intentions with problems
WALT:
Factorise a trinomial
vs
Apple dropped in water.
Depth modelled by d(t).
How long was it underwater?
Bring in an apple, drop it in a bucket of water! Make it real.
The process of factorising isn't the goal, solving the problem is the goal, the process becomes a requirement.
Replace learning intentions with problems
W.A.L.T. Calculate the gradient of a straight line from vertical and horizontal distances
vs
Climbs in the Tour de France are classified according to their steepness. Cat 3 > 0.05, cat 2 > 0.07, cat 1 > 0.1 etc.
You have been asked by the race organiser to put these climbs into the
correct category.
Learners select a strategy then make decisions.
Give them responsibility. Make it real!
Everything in "rich" context
We must use our
creativity
and
imagination
to generate the interest needed to make instruction more enjoyable for our students.
Placing course content in the context of a real world scenario provides the frame of reference the mind needs to retain that content.
Rich
Context?
1.2 m
2.2 m
Richer Context?
A builder needs new ladder. He has 3 lengths to choose from. Longer ladders cost more and times are tight, he doesn't want to spend any more than he has to.
H & S regulations say the foot of any ladder must be a minimum distance from the foot of the wall. This distance is 15% of the height the ladder reaches up the wall.
He needs a ladder that will reach 6 metres up the wall. Which one should he buy?
vs.
Real life problems don't come in nice packages
Relevant, reallife context
Decisions, decision
If pupils struggle with reading and interpreting, why not act out the whole situation? Video a senior pupil pretending to be a builder.
Make it real!
Shhh... Don't tell
Resist the temptation to tell.
Telling has a powerful hold on the teaching profession. Often telling is our main method of instruction, even though there are strong indicators that
telling
is the
least effective
way for students to learn.
Don't underestimate the value of giving learners time to think about, discus and investigate possible solutions to problems. They'll dig into their prior knowledge, make connections and come up with solutions you would never have thought of. Eventually though, they will say "We know what we need figure out  we just don't know how to do it."
This is the time to tell them, once there is a need.
Bring it all together
Present a problem:
Give plenty of
time
for learners to think about (and discuss) ways to solve it.
Establish a need
for a new mathematical process.
Introduce process:
Instruction is still required  but now there is need
Solve the problem together

then present another problem, one that requires the new process.
Consolidate process:
Rigour
is still necessary, only hopefully motivation to practise is increased.
Area of a circle
This would be the starting point but really lacks any engagement...
3cm
Calculate the area
This example is a slight step forward but still doesn't really engage learners
8cm
Calculate the radius
Area = 40 square centimetres
Traditionally this question presents a nonroutine problem which not every pupil will progress to...
Calculate the area of this circle
The square has perimeter 28 cm
By starting the journey here, pupils are encouraged to think more deeply about the problem.
A great opportunity to work together, discuss options and make decisions.
What's the question?
Further reading
Ted McCain  "Teaching for tomorrow"
TED
talks:
Sir Ken Robinson, Dan Meyer
Web:
www.blog.mrmeyer.com
www.101qs.com
www.brilliant.org