the average company doesn’t invest enough in skills. that’s why they’re average!. (bob putnam,...
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The average company doesn’t invest enough in skills.
That’s why they’re average!.
(Bob Putnam, Chairman Ford UK)
Improving learning in mathematics
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Every School a Good Every School a Good SchoolSchool
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Better Mathematics
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Improving learning in mathematics
Programme Aim
To enhance the learning experiences of all pupils by promoting quality teaching of
mathematics
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Improving learning in mathematics
By the end of the two-day programme participants will be better able to:
• Challenge pupils understanding through skilful questioning• Use an appropriate variety of teaching activities and learning
strategies• Encourage pupils to think and talk about how they learn mathematics and what they have learnt• Contribute to departmental planning and the dissemination of
good practice within and across schools.
Learning Intentions
Every School a Good School
A strategy for raising achievement in literacy and numeracy.
Better Maths
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Session 1
Improving learning in mathematics
Beliefs about learning and teaching
Learning Intentions
This session is intended to help us to reflect on our current assumptions, beliefs and teaching practices
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Beliefs about learning and teachingWorkshop 1a
Improving learning in mathematics
• With your partner discuss the 6 statements in your envelope.
• At your table share your thoughts on all of the statements and as a group decide on which ones reflect good practice in a mathematics classroom.
• Use post-its to record the statements you disagree with on the bar chart.
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Improving learning in mathematics
Beliefs about learning and teaching Workshop 1b
• Discuss those statements you believe reflect good practice in a mathematics classroom.
• Choose one statement which you think :– is well addressed in your classroom– you would need work towards in the future.
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To help learners to adopt more active approaches towards learning
Improving learning in mathematics
Engage learners in discussing and explaining ideas, Challenging and teaching one another, Creating and solving each other's questions and working collaboratively to share methods and results.
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Improving learning in mathematics
To develop more 'connected' and 'challenging' teaching methods.
Traditional, 'transmission'
approaches involve simplifying ideas and
methods by explaining them to learners one
step at a time.
In contrast, this model emphasises the
interconnected nature of mathematics, and it is
'challenging' in that it seeks to confront common conceptual
difficulties head on.
2 + 2 = 5
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Improving learning in mathematics
The types of activity
Classifying mathematical objects
Creating problems
Evaluating mathemati
cal statements
Interpreting multiple
representations
Analysing reasoning
and solutions
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Improving learning in mathematics
Personal reflection
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Improving learning in mathematics
Please be back in 30 minutes
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Improving learning in mathematics
Effective Questioning
The answer to my question is 48!
What is the question?
Session 2
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Improving learning in mathematics
Bowland Charitable Trust
• What different types of questions are there?
• What different purposes do your questions serve?
• Which type of question do you use most frequently?
Record your comments on the worksheet provided
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Why Do We Ask Questions?
• To manage and organise
pupils’ behaviour
• To find out what pupils
know
• To stimulate interest in a
new topic
• To focus on an issue or
topic
• To structure a task for maximum learning
• To identify, diagnose difficulties or blocks to learning
• To stimulate pupils to ask questions
• To give pupils opportunity to assimilate, reflect and learn through discussion
Improving learning in mathematics
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What Is Effective Questioning?
• Questions are planned and related to session objectives.
• Questions are mainly open.• Teacher allows ‘wait time’.• Both right and wrong answers are followed up.• Questions are carefully graded in difficulty.• Teacher encourages learners to explain and
justify answers.• Teacher allows collaboration before
answering.• All participate e.g. using mini-whiteboards.• Learners ask questions too.
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Promoting Pupil Questioning
• Model questioning for pupils.
• Provide opportunities for pupils to practise their skills.
• Plan time for pupils’ questions and for dealing with them effectively.
Improving learning in mathematics
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Different types of questions
• Creating examples and special cases
• Evaluating and correcting
• Comparing and organising
• Modifying and changing.
• Generalising and conjecturing
• Explaining and justifying
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Creating examples and special cases
Show me an example of:• a number between and ;• a hexagon with two reflex angles; • a shape with an area of 12 square units and a
perimeter of 16 units;• a quadratic equation with a minimum at (2,1);• a set of 5 numbers with a range of 6
…and a mean of 10…and a median of 9
3
17
2
Improving learning in mathematics
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Evaluating and correcting
What is wrong with these statements?
How can you correct them?• When you multiply by 10, you add a nought.• + = • Squaring makes bigger.• If you double the radius you double the area.• An increase of x% followed by a decrease of x%
leaves the amount unchanged.• Every equation has a solution.
10
2
10
3
20
5
Improving learning in mathematics
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Comparing and organising
What is the same and what is different about these objects?
Square, trapezium, parallelogram. An expression and an equation. (a + b)2 and a2 + b2 Y = 3x and y = 3x +1 as examples of straight
lines. 2x + 3 = 4x + 6;
2x + 3 = 2x + 4; 2x + 3 = x + 4
Improving learning in mathematics
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• 1, 2, 3, 4, 5, 6, 7, 8, 9,10
• , , , , ,
•
a
y = x2 - 6x + 8; y = x2 - 6x + 10;
y = x2 - 6x + 9; y = x2 - 5x + 6
How can you divide each of these sets of objects into 2 sets?
2
13
2
4
3
5
4
6
5
7
6
Comparing and organising
Improving learning in mathematics
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Modifying and changing
How can you change:• this recurring decimal into a fraction?• the equation y = 3x + 4, so that it passes
through (0,-1)?• Pythagoras’ theorem so that it works for triangles
that are not right-angled?• the formula for the area of a trapezium into the
formula for the area of a triangle?
Improving learning in mathematics
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Generalising and conjecturing
What are these special cases of?• 1, 4, 9, 16, 25.... • Pythagoras’ theorem.• A circle.
When are these statements true?• A parallelogram has a line of symmetry.• The diagonals of a quadrilateral bisect each
other.• Adding two numbers gives the same answer as
multiplying them.
Improving learning in mathematics
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Explaining and justifying
Use a diagram to explain why:• a2 − b2 = (a + b)(a − b)Give a reason why:• a rectangle is a trapezium.How can we be sure that:• this pattern will continue:
1 + 3 = 22; 1 + 3 + 5 = 32…?Convince me that:• if you unfold a rectangular envelope, you will get
a rhombus.
Improving learning in mathematics
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Workshop 2Designing Appropriate Questions
• Use the worksheet provided to write 1 question in each category which relates to a topic you are teaching at the moment.
• Share your questions with the others at your table
Improving learning in mathematics
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What is a good question?
Robert FisherProf of Education at
Brunel University
A good question: is an invitation to think, or to do. It
stimulates because it is open-
ended.is productive – it looks for a response
will generate more questions.
Improving learning in mathematics
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Improving learning in mathematics
Bowland TrustBetter Maths
NI Curriculum AfLYoutube
Resources
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Improving learning in mathematics
Personal reflection
Revisit your thoughts on questioning ref: Bowland Trust
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Improving learning in mathematics
Lunch
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Improving learning in mathematics
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Session 3
Improving learning in mathematics
Learning from Mistakes and Misconceptions
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Improving learning in mathematics
Analysing Learner’s Work
Consider the samples of work and record the errors made and possible thinking which may have led to them.
Share your thinking around the table.
Workshop 3a
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Improving learning in mathematics
Interpreting Multiple Representations
• Working in groups of 3 take turns to match pairs of cards and place them on the table side by side.
• Explain why they make a pair.
• Partners should challenge thinking if necessary.
•When finished place the cards in order of size – smallest to largest.
Workshop 3b
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Improving learning in mathematics
Personal reflection
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Improving learning in mathematics
Sharing good practice electronically
Session 4
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Improving learning in mathematics
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Improving learning in mathematics
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Improving learning in mathematics
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Improving learning in mathematics
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Improving learning in mathematics
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Improving learning in mathematics
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Improving learning in mathematics
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Improving learning in mathematics
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Improving learning in mathematics
• Day 2 - in-school (sub-cover provided) for planning/preparation
• In your classroom - use one (or more) of the ideas/activities from today
• Online - visit the LNI site and post a comment (relating to your experience) on the discussion board
• Day 3 - out-centre – share with colleagues:• Monday, 9 March 2009 – NWTC• Tuesday, 10 March 2009 – TEC
3-Day Programme
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Improving learning in mathematics
Opening the boxes
Session 5
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Improving learning in mathematics
Opening the box
5 Packs
Active
Learner
Centred
Approaches
To the
Teaching
&
Learning of
Mathematics
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Improving learning in mathematics
Improving learning in Mathematics
mathematics
Standards Unit
challenges and strategies
challenges & strategies
Improving learning in Mathematics
mathematics
Standards Unit
resource file for teaching 1
resource file for teaching 1
Improving learning in Mathematics
mathematics
Standards Unit
resource file for teaching 2
resource file for teaching 2
Improving learning in Mathematics
mathematics
Standards Unit
the multimedia resource
the multimedia resourcea professional development
guide
Improving learning in Mathematics
mathematics
Standards Unit
a professional development guide
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Improving learning in Mathematics
mathematics
Standards Unit
introduction
Improving learning in Mathematics
mathematics
Standards Unit
activity template software
CD - ROM
Improving learning in Mathematics
mathematics
Standards Unit
an overview
Improving learning in Mathematics
mathematics
Standards Unit
an overview
DVD
Improving learning in Mathematics
mathematics
Standards Unit
exploring the approaches
DVD - ROM
Improving learning in Mathematics
mathematics
Standards Unit
exploring the approaches
CD - ROMs
Improving learning in Mathematics
mathematics
Standards Unit
the multimedia resource
Improving learning in mathematics
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Improving learning in mathematics
DVD – ROM- Exploring the approaches top menu
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Improving learning in mathematics
DVD – ROM- Exploring the approaches - Materials
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Improving learning in mathematicsDVD – ROM- Exploring the approaches – Mostly
Number
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Improving learning in mathematics
Materials – N1 - Ordering fractions and decimals
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Improving learning in mathematics
Resource files for teaching
Improving learning in Mathematics
mathematics
Standards Unit
resource file for teaching 2
Improving learning in Mathematics
mathematics
Standards Unit
resource file for teaching 1