Improving learning in mathematics

The average company doesn’t invest enough in skills.

That’s why they’re average!.

(Bob Putnam, Chairman Ford UK)

Improving learning in mathematics

Every School a Good Every School a Good SchoolSchool

Better Mathematics


Programme Aim

To enhance the learning experiences of all pupils by promoting quality teaching of

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

Session 1


Beliefs about learning and teaching

Learning Intentions

This session is intended to help us to reflect on our current assumptions, beliefs and teaching practices

Beliefs about learning and teachingWorkshop 1a


• 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.


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.

To help learners to adopt more active approaches towards learning


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.


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


The types of activity

Classifying mathematical objects

Creating problems

Evaluating mathemati

cal statements

Interpreting multiple

representations

Analysing reasoning

and solutions


Personal reflection


Please be back in 30 minutes


Effective Questioning

The answer to my question is 48!

What is the question?

Session 2


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

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


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.


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.


Different types of questions

• Creating examples and special cases

• Evaluating and correcting

• Comparing and organising

• Modifying and changing.

• Generalising and conjecturing

• Explaining and justifying


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


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


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


• 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


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?


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.


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.


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


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.



Bowland TrustBetter Maths

NI Curriculum AfLYoutube

Resources


Personal reflection

Revisit your thoughts on questioning ref: Bowland Trust


Lunch


Session 3


Learning from Mistakes and Misconceptions


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


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


Personal reflection


Sharing good practice electronically

Session 4










• 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


Opening the boxes

Session 5


Opening the box

5 Packs

Active

Learner

Centred

Approaches

To the

Teaching

&

Learning of

Mathematics


Improving learning in Mathematics

mathematics

Standards Unit

challenges and strategies

challenges & strategies


mathematics

Standards Unit

resource file for teaching 1



mathematics

Standards Unit




mathematics

Standards Unit

the multimedia resource

the multimedia resourcea professional development

guide


mathematics

Standards Unit

a professional development guide


mathematics

Standards Unit

introduction


mathematics

Standards Unit

activity template software

CD - ROM


mathematics

Standards Unit

an overview


mathematics

Standards Unit

an overview

DVD


mathematics

Standards Unit

exploring the approaches

DVD - ROM


mathematics

Standards Unit

exploring the approaches

CD - ROMs


mathematics

Standards Unit

the multimedia resource



DVD – ROM- Exploring the approaches top menu


DVD – ROM- Exploring the approaches - Materials

Improving learning in mathematicsDVD – ROM- Exploring the approaches – Mostly

Number


Materials – N1 - Ordering fractions and decimals


Resource files for teaching


mathematics

Standards Unit



mathematics

Standards Unit


Documents

Improving learning in mathematics