Problem Solving and Reasoning

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Developing a
reasoning classroom
Written by Tim Handley
(Mathematics Specialist Teacher, NCETM Professional Development
Lead and author of Problem Solving and Reasoning)
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Aims
 Understand what reasoning ‘is’ and why
it is important.
 Explore some reasoning activities.
 Explore ways in which reasoning can
become part of day-to-day teaching.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Running order
 Introduction: What do our children already do?
 Taking a slice: a non-numerical activity
 What is reasoning and why is it important?
 Reasoning in the past week
 Graffiti Maths with Chicken Nuggets: a numerical activity
Break
 Children’s mathematical superpowers
 Key strategies to make reasoning part of day-to-day life
in our classrooms
 Planning a reasoning activity
 Next steps
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
What do children in our school
already do?
 Have a look at the statements on the progression cards from
http://nrich.maths.org/6105.
 Highlight them green if children in your year group do these
regularly and independently.
 Highlight them orange if children in your year group do these
regularly but require support.
 Leave them blank if children in your year group struggle to do
these, or only do them infrequently.
 Then identify the 4 statements which you consider to be age
appropriate, but which children in your year group struggle
with the most and mark these with a star.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
What can children
do independently?
Problem Solving and Reasoning
What do they
find difficult?
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Take a slice: a non-numerical activity
Here is one way of splitting a 4 x 4 board into 2 identical
pieces:
How many other ways can you find?
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Take a slice: a non-numerical activity
Things to
think about
So, what were you doing?
What key phrases were you using?
What were you thinking?
We were all
reasoning!
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
So, what is reasoning?
Some things reasoning can be seen as:
Thinking about mathematics
Making connections
Application of facts and knowledge – either explicitly or
implicitly
Justifying
Convincing yourself and others
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Why is reasoning
important …?
What do you
think?
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
What OTHERS say ...
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Academics
“In our understanding, we have two components – the range of
mental representations and the reasoning linking them
together. Therefore we must also develop the reasoning that
we carry out between representations.”
Barnaby et al, Primary Mathematics, Teaching for understanding.
And lots, lots more…
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Examiners
Just some of the questions that require children to
reason from the 2009 KS1 + 2012 KS2 SATs tests.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Ofsted
The reports Mathematics: made to measure (Ofsted, 2012)
and Good practice in primary mathematics: evidence from
20 successful schools (Ofsted, 2011) make clear that:
Real mathematical thinking lies at the heart of learning
mathematics. It involves making sense of mathematics
through engagement, reasoning and making connections.
Sometimes this may involve significant challenge in terms
of the requirement to think. But it is only through their own
thinking that children will develop real depth of learning.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Ofsted Maths subject-specific criteria
Outstanding
 Pupils understand core concepts and make connections.
 They think for themselves.
 They show exceptional independence when solving problems in a
range of contexts.
 They think for themselves and persevere with challenges, knowing
they will succeed.
 They embrace the value of learning from mistakes and false starts.
 They reason, generalise and make sense of solutions.
Requires improvement
 Children use techniques correctly.
 They have an insecure understanding of concepts.
 They solve routine problems in known contexts.
 They follow the teacher’s example.
 They avoid making mistakes.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
The DfE: 2014 National Curriculum
Aims
It is expected that these aims are applied
throughout the programmes of study.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
How have our children been
reasoning over the past week?
Share with
your table!
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Graffiti maths
Graffiti maths is an approach to problem solving and
reasoning tasks which encourages children to think
and work ‘big’.
 See page 12 of the Introduction in the Problem Solving
and Reasoning books for an explanation of how it works.
 See the Key Strategies Maths Stories section on the
DVD for footage of Graffiti maths in action.
Now use the graffiti maths approach to try
out the Chicken nuggets problem from
Problem Solving and Reasoning Year 5.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
BREAK!
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Mathematical Superpowers
All children
have them!
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Mathematical superpowers
John Mason identified a set of 8 ‘Mathematical Powers’
that all children posses. They come in pairs:
Conjecturing and
Convincing
Imagining and
Expressing
Specialising and
Generalising
Organising and
Classifying
See page 8 of the Introduction in the Problem Solving
and Reasoning books for more details.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
The Story of a Number
An easy way to get children
reasoning, pattern spotting
and generalising.
Watch the Maths Stories
clip on the DVD.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Money boxes
• Try the Money boxes problem from Problem
Solving and Reasoning Year 3.
• Can you work out how much money could
be in each money box?
Add in questions to draw out the reasoning…
If we know that half of the coins are £1, what else do we
know?
Which of these clues are hard/easy to solve? Why?
Which of the clues is the odd one out why?
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Key strategies
To help include reasoning as part of
our day-to-day teaching.
The Problem Solving and Reasoning
books include 14 key strategies.
Watch the clips
on the DVD.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Always, sometimes, never
• Look at the statements from the Number
Knowledge problem Resource Sheet 13.1
from Problem Solving and Reasoning Year 6.
• Discuss with your partner. Are they always,
sometimes or never true?
• Make sure you give examples.
Always, sometimes, never can also be used as part
of shared learning.
Can you come up with some ‘always, sometimes,
never’ statements for your year group?
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Other examples for always,
sometimes, never…
The picture on a pictogram represents one piece of data.
If the answer is in cm2 you are measuring area.
The diagonals of a quadrilateral bisect each other.
Two negatives make a positive.
All multiples of 3 have a digital root of 3, 6 or 9.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Another, another, another…
An even number…
Number with the product of 60
Factors of 36
A triangle
A prime number
Draw out the links and generalisations between the sets
generated.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Convince me ...
All angles in a triangle always add up to 180 degrees.
That the next whole number after 119 will be 120.
That coordinates are useful for identifying points.
An odd number plus an odd number always equals
an even number.
That the area of a rectangle divided by the length of
one side gives the length of the other side.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Hard and easy
A calculation whose answer is 7
A three digit subtraction
A word problem
A region made up of rectangles whose area and
perimeter are to be found
A calculation to be carried out on a calculator
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
If this is the answer, what’s the question?
24
Impossible
No
Square numbers
15.3
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Odd one out (and why?)
3, 6, 9 …
3 x 10 = 30, 31 x 10 = 310, 423 x 10 = 4230, 0.3 x 10 = 3,
1111 x 10 = 11 110
1/4 , 8/36,
6/
32,
10/
40
11, 101, 1001, 101
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
POGs: Peculiar, Obvious, General
A quadrilateral
A multiple of 2
A factor of 36
A fraction equivalent to ¾
A number
A set of 3 numbers with a product of 10
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
What do you notice?
2, 4, 6, 8, 10, 12, 14, 16
2, 3, 5, 7, 11,13 (Can you give me more in this set?)
1 / 2 , 2 / 4 , 3 / 6 , 4 / 8
4, 9, 16, 25, 36, …
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
What else do we know? (and why?)
5 x 8 = 40
3 + 4 =7
1 cm = 10 mm, 2 cm = 20 mm
All even numbers end in an even digit: give me an even
number over 1000.
50 is 5 lots of 10, 40 is 4 lots of 10, …
1/2 = 2/4 = 4/8 = 3/6
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
What’s the same? What’s different?
One and 1
9 and 18
Multiples of 3 and Multiples of 6
1/2, 0.5, 50%, 4:8, 3/6, 1:2
Square numbers and numbers that are not square

Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
What’s the same? What’s different ?
Choose any two of the three. In what way are they
the same as each other and different from the third?
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Zooming in
Write down a:
Multiple of 4...
... that is greater than 30...
... that is also square...
... where both digits are even.
Draw a:
Quadrilateral...
... that has at least one line of symmetry...
... and only one pair of parallel sides...
... and at least one acute angle.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Is it a good explanation of…?
Is “having three sides” a good explanation of a triangle?
Is “adding a zero” a good explanation of multiplying by 10?”
Is “a shape with 4 sides” a good explanation of a square?
Is “moving the digits 3 places to the left” a good explanation
of multiplying 1.3 by 1000?
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Planning a reasoning activity
In your year group, plan a reasoning activity that you
could use next week.
This could be a key question, it could use one of the
strategies we’ve just looked at, or be a full lesson activity.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Next steps
Encourage and support our children to reason!
Over the next 2 weeks, ensure there are at least 6
opportunities for your children to reason.
Try also to include reasoning and some of the key
strategies in other subjects.
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
Developing a reasoning classroom
Resources to help
14 key strategies with example questions for each year
18 activities and investigations per year level
Problem Solving and Reasoning
CDP DVD For Years 1 to 6
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