Quality Improvement Guidance for Teachers on Learning and Teaching Approaches in Mathematics.

Quality Improvement

Guidance for Teachers on

Learning and Teaching Approaches in Mathematics

Quality Improvement

What are the key features of a mathematics lesson?

Standards and Quality in Primary Schools: Mathematics 1998-2001 HMIE

Quality Improvement

Guidance for Teachers

The Mathematics Lesson

• Interactive Oral Mental Mathematics• Introducing the Lesson• Direct Interactive Teaching• The Plenary Session

Quality Improvement

Interactive Oral Mental Mathematics

Quality Improvement

“It is important to recognise that when calculating mentally we usually make use of different methods from those we have used for written calculations. We should encourage pupils to develop and practise flexible mental approaches. One way of doing this is through class or group discussion of the variety of different ways of carrying out different operations.”

Mathematics 5-14 National Guidelines SOED 1991 p52


Quality Improvement

“At all stages, many pupils tried to perform mental calculations by attempting paper and pencil methods in their heads, seldom if ever the most efficient approach. As a result they responded slowly and often wrongly. Overall there was insufficient practice in mental calculation and, even when mental work was done, almost no discussion of how answers were obtained.”

Improving Mathematics 5-14 SOEID 1997 p7


Quality Improvement

“In all classes there should be regular mental calculation sessions to provide opportunities to practise skills, discuss alternative mental methods and be shown how to use efficient approaches.”

Improving Mathematics 5-14 SOEID 1997 p8


Quality Improvement

“In less effective courses, the mental mathematics programme consisted merely of a series of regular mental tests or involved pupils working individually through mental exercises. This approach missed opportunities for teachers and pupils to work together, for pupils to share their thinking with teachers, reveal evidence about their understanding and use mathematical words and phrases in explanations and discussions.”

Standards and Quality in Primary Schools: Mathematics 1998-2001 HMIE p13


Quality Improvement

“Most schools had made improvements to the emphasis on the progressive development of pupils’ mental mathematics skills, with emphasis on oral rather than written calculation in the early stages. Some schools included short daily sessions of oral activities to keep pupils’ mental skills sharp. The best sessions involved regular discussion of the different strategies used by pupils to answer questions in mental mathematics so that they became aware of the variety of possible approaches.”

Standards and Quality in Primary Schools: Mathematics 1998-2001 HMIE p13


Quality Improvement

Mental Calculation 38 + 47

• count on in tens from 40, 50, 60, 70, add 8 to get 78 and then 7 to get 85;

• first add 38 and 40 to get 78, then add on 7 to get 85;

• add 30 and 40 to get 70. Add 8 and 7 to get 15. Add 70 and 15 to get 85;

• add 38 and 2 (from the 47) to get 40, add 40 and 40 to get 80 and the 5 left to get 85;

• add 40 and 50 to get 90, take away 2 (for the 38) to get 88 and take away 3 (for the 47) to get 85.

Mathematics 5-14 National Guidelines SOED 1991 p82

Quality Improvement

Mental Calculation 47 - 18

• 18 add 2 gives 20, then 20 gives 40, now 7, so 29 in all;

• 47 - 10 leaves 37, 37 take away 7 is 30, then take away another 1, leaves 29;

• 47 - 20 + 2 = 27 + 2 = 29;

• 47 - 18 = 47 - 17 - 1 = 30 - 1 = 29

Quality Improvement


This session can be used to:

• Consolidate previous work• Establish knowledge of an attainment target• Focus on skills needed in the main part of the

lesson• Practise mental calculations and rapid recall of

number facts in a variety of ways• Establish new facts from known facts and explain

the strategies used• Develop a particular strategy

Quality Improvement

Interactive Oral Mental MathematicsQuestioning

Questioning should effectively:

• give pupils time to think before providing an answer

• involve as many pupils as possible• allow time to explore reasons for wrong answers• take account of pupils’ attainments

Quality Improvement

Interactive Oral Mental MathematicsQuestioning

Different types of questioning can be used:

• Quick Recall• Mental Calculation• Oral Responses to Written Problems• Oral Questions in Context• Oral Questions with Multiple Answers

Quality Improvement


• What is 6 - 4? Show me two numbers with a difference of two.

• Is 16 an even number? What even numbers lie between 10 and 20?

• What are four threes? Show me two numbers with a product of 12.

• How many centimetres in a metre? Tell me two lengths that together make a metre.

• Continue this sequence: 1, 2, 4, … Find different ways to continue this sequence: 1, 2, 4, ...

Quality Improvement

Interactive Oral Mental MathematicsStrategies

There are a variety of ways in which the session can

be practised:

• Unison Response • Show Me• Cover-ups• Using the Pupils

Quality Improvement

Unison ResponseCounting Stick

• 0 10 count forwards/back/random




• 6 count on…13 count on…42 ...

• 16 count back... 23 count back...

• 0 50 5x table, 0 70 7x table

• Zero in middle

Quality Improvement

Unison ResponseRhythm Counting

• count forward in ones to ten - then back

• count forward in tens - then back

• count in fives, threes...

• count on in twos from 5...

• count on in threes from 55...

• count back in fours from 40...

• count in halves from one half

• count in quarters - and simplify

Quality Improvement

Unison ResponsePendulum Counting

• count forward in ones to ten - then back

• count forward in tens - then back

• count in fives, threes...

• count on in twos from 5...

• count on in threes from 55...

• count back in fours from 40...

• count in halves from one half

• count in quarters - and simplify

Quality Improvement

Show MeFlip Flops

• how many can you see?

• show me more/less/same/one more/less

• how many can I see?

• in pairs - make 12

• in threes - make 15

Quality Improvement

Show MeDigit Cards

• show me 6, 15, 36, 85

• show me an even, odd number

• show me a number greater than 4

• show me a number between 16 and 20

• show me number bonds +, -, x, ÷

• show me double 5, 9, 36

• show me the remainder: 18÷5

• show me 1 less than 70

Quality Improvement

Show MeDigit Cards in Pairs

• show me pairs of numbers that total 10

• show me four factors of 24

• show me two multiples of 7

• show me square numbers

• show me a four digit multiple of 3

Quality Improvement

Show MeDigit Cards - use four (1, 2, 4, 7)

• show me the largest/smallest number

• show me the number nearest to 3000

• in groups of 3 - make different numbers

• who has the largest/smallest?

• who has the nearest to 5000?

Quality Improvement

Show MeDigit Cards - use two (3, 4)

• show me the sides on a quadrilateral

• show me the difference between 129 and 132

• show me 24÷6

• show me 27÷9

• show me the lines of symmetry on an equilateral triangle

Quality Improvement

Show MeFan Cards

• show me 2/3 digits which total 8, 10..

• show me a 2 digit number which is odd

• show me a 2 digit number between 50 and 75

• show me a 3 digit number between 250 and 500

• show me a 2/3 digit number divisible by 4

• show me a 2 digit prime number

• I’m thinking of a number….

Quality Improvement

Show MeNumber Generator

• show me 457, 12 670, 238 300

• show me 1750, make it 10 more...

• show me 22 000, make it 100 less...

• show me 467 - now add 4

• show me 342 - now double it

• rounding to nearest 10, 100, 1000

• multiply/divide by 10, 100, 1000

Quality Improvement

Show Me2D Shape

• show me the shape behind the wall

• show me a hexagon

• show me a triangle with an angle greater than 90°

• show me a four sided shape with only two lines of symmetry

Quality Improvement

Cover UpsGrids

• 0-99 grid - recognition/one more/less

• 1-100 grid

• shapes - names, symmetry, angles

• fractions

• tables - cover up answers to 9x, 4x...

• money

• 24 hour time

Quality Improvement

Using the Pupils

• Loop Cards

• Human Lines - change places with number 15; number before/after 12; number more/less than 6...

• Decimal Point - Hold 3.2 One person holding 1 to join in. Where to stand to make number smaller/larger? Could she stand elsewhere?

Quality Improvement

“Schools did not place sufficient emphasis on developing pupils’ skills in written and mental calculation. Too often the approach was restricted to “ten-a-day” which did not develop pupils’ knowledge and understanding of techniques in mental calculation.”

Improving Achievement In Mathematics in Primary and Secondary Schools HMIE 2005

Quality Improvement

Features of effective learning and teaching at primary stages include:

“Teachers focused their questioning on checking pupils’ understanding, rather than simply eliciting correct answers.”

“Teachers asked pupils to explain their thinking, and responded skilfully when pupils made mistakes using these as a basis for further learning.”

Improving Achievement In Mathematics in Primary and Secondary Schools HMIE 2005

Quality Improvement

Using the PupilsThe Answer Is…

7

x

–

+

Quality Improvement

Using the PupilsThe Answer Is…

7

7 x 13·5 x 21·75 x 4

35 563 970 1077 11

8 – 110 – 329 – 22

3 + 45 + 21·2 + 5·8-1 + 8

Quality Improvement

Using the PupilsWhat Else Do You Know?

12 x 5 = 60

Quality Improvement

12 x 5 = 60

60 5 = 12

5 x 12 = 60

60 12 = 5

6 x 5 = 30

1·2 x 5 = 612 x 10 = 120

11 x 5 = 55


Quality Improvement


12 x 5 = 60

60 5 = 12

5 x 12 = 60

60 12 = 5

6 x 5 = 30 12 x 25 = 30

1·2 x 5 = 6

3 x 5 = 15

12 x 05 = 06

24 x 5 = 12012 x 10 = 120

11 x 5 = 55 10 x 5 = 50

Quality Improvement


12 x 5 = 60

60 5 = 12

5 x 12 = 60

60 12 = 5

6 x 5 = 30 12 x 25 = 30

1·2 x 5 = 6

3 x 5 = 15

12 x 05 = 06

24 x 5 = 12012 x 10 = 120

of 60 = 5 of 60 = 12

11 x 5 = 55 10 x 5 = 50

15

112

20% of 60 = 12

Quality Improvement

Mental Calculation Multiplication

• 3 x 13 = (3 x 10) + (3 x 3) = 30 + 9 = 39

• 3 x 14 = 3 x 2 x 7 = 6 x 7 = 42 or

• 3 x 14 = 3 x 7 x 2 = 21 x 2 = 42

• 3 x 15 = (3 x 30) ÷ 2 = 90 ÷ 2 = 45 or

• 3 x 15 = (2 x 15) + (1 x 15) = 30 + 15 = 45

• 3 x 16 = 6 x 8 = 12 x 4 = 24 x 2 = 48 or

• 3 x 16 = (3 x 15) + 3 = 45 + 3 = 48

• 3 x 17 = (3 x 20) - (3 x 3) = 60 - 9 = 51

Quality Improvement

Mental Calculation Division

• 51 ÷ 3 3 x 10 = 30, this leaves 21 ÷ 3 = 7 and 10 + 7 = 17

• 90 ÷ 6 = (90 ÷ 3) ÷ 2 = 30 ÷ 2 = 15

• 115 ÷ 5 = double 115 ÷ 10 = 11.5 + 11.5 = 23

Quality Improvement

Using the PupilsWhat Could the Next Number Be?

3·14

Quality Improvement


3·14 4·14

13·14

3·15

3·141

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3·14 4·14

13·14

3·15

3·141

5·14

23·13

3·16

3·142

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3·14 4·14

13·14

3·15

3·141

5·14

23·13

3·16

3·142

6·14

33·13

3·17

3·143

Quality Improvement

Online Resources

Quality Improvement

Introducing the Lesson

Quality Improvement

Introducing the Lesson

“By sharing the purposes of a piece of learning and the expected outcomes with pupils, the learning within a classroom becomes more focused for the pupils as well as the teacher.”

Direct Interactive Teaching SCCC 2000

Quality Improvement

Direct Interactive Teaching

Quality Improvement


“Direct interactive teaching is characterised by high levels of questioning, explaining and worked examples where pupils together explain their understanding and solution to a problem.

Alternative ways of solving problems and reaching answers are sought and shared. Pupils play a very active part in the lesson. There is a high level of group interaction and cooperative working.”

The Structure and Balance of the Curriculum Guide for Teachers and Managers 5-14 Learning and Teaching Scotland 2000

Quality Improvement


“If used skilfully, a series of questions can be used together with pupil replies to explore and expand the area being taught rather than a lengthy teacher talk.

Effective questioning also provides a platform for effective differentiation, with questions being framed to provide different levels of challenge as appropriate.”

Direct Interactive Teaching SCCC 2000

Quality Improvement


“The least effective teachers relied too heavily on course material, sometimes reading out tasks from a textbook rather than explaining the principles of a topic. Where learning was less successful, pupils often spent too much time waiting for instructions from the teacher or working on repetitive exercises with little opportunity for discussion.”

Standards and Quality in Primary Schools: Mathematics 1998-2001 HMIE

Quality Improvement

Top Tips in Questioning

• Ask pupils for questions

• Plan• Sequence• Distribute questions• Show interest and respect all answers• Don’t give clues• Encourage contribution• Increase wait time• Check how answers were arrived at• Use wrong responses positively• Use more higher order and open questions

Quality Improvement

The Plenary Session

Quality Improvement

The Plenary Session

This part of the lesson can be used to:

• ask pupils to present their work, summarising what has been learned• mark or discuss particular examples, reviewing

the different methods of calculation used, rectifying any misconceptions• give constructive feedback to pupils• make conclusions or establish rules • highlight pupils’ progress towards attainment targets• provide tasks for homework

Quality Improvement

The Plenary Session

The teacher should:

• have a clear idea of the purpose and content of the plenary session

• leave enough time for this to be achieved • evaluate their own teaching by reflecting on what

pupils have learned and what progress has been made

Quality Improvement

How Do We Keep Skills Sharp?

Appropriate resources

Mental mathematics

Workcards and Worksheets

Homework

Problem of the Week

Websites

Assessment

Quality Improvement

Numbertime

The Maths File Game Show

Teaching Time

Revisewise Maths

Numeracy Software

Schoolhouse Technologies

Skillswise

Workcards and Worksheets

Quality Improvement

Nrich

NZMaths Problem Solving

Word Problems for Kids

Mrs Renz’s Math Web Sites

TES Brainteasers

Count On

Problem of the Week

Quality Improvement

CurricuLinks – Interactive Whiteboards

CurricuLinks – 5-14 Mathematics

Websites for Interactive Whiteboards

Quality Improvement

Homework

Quality Improvement

Assessment

• National Assessments

• Tracking Database

• Standardised Tests

• Mathematics Check-ups and Pupil Assessment Files

• Formative Assessment

• Questioning

Quality Improvement

Guidance for Teachers on

Learning and Teaching Approaches in Mathematics

Quality Improvement Guidance for Teachers on Learning and Teaching Approaches in Mathematics.

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Transcript of Quality Improvement Guidance for Teachers on Learning and Teaching Approaches in Mathematics.