Guidance for Teachers on Learning and Teaching Approaches in Mathematics

What are the key features of a mathematics lesson What are the key features of a mathematics lesson? Standards and Quality in Primary Schools: Mathematics 1998-2001 HMIE

The Mathematics Lesson Guidance for Teachers The Mathematics Lesson Interactive Oral Mental Mathematics Introducing the Lesson Direct Interactive Teaching The Plenary Session

Interactive Oral Mental Mathematics

Interactive Oral Mental Mathematics “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

Interactive Oral Mental Mathematics “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

Interactive Oral Mental Mathematics “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

Interactive Oral Mental Mathematics “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

Interactive Oral Mental Mathematics “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

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

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 The 5-14 Guidelines go on to state that: “ It is important, then,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 discussion of the variety of different ways of carrying out particular calculations.” Rather than using this purely to highlight the different strategies being used and advising pupils to select the most effective method, there is an advantage in being aware which strategy individual pupils are using, so that the teacher can maximise support to the child. Distribute strategies A-F. Discuss.

Interactive Oral Mental Mathematics 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

Interactive Oral Mental Mathematics Questioning 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

Interactive Oral Mental Mathematics Questioning 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

Interactive Oral Mental Mathematics 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, ...

Interactive Oral Mental Mathematics Strategies There are a variety of ways in which the session can be practised: Unison Response Show Me Cover-ups Using the Pupils

Unison Response Counting Stick 0 10 count forwards/back/random 0  20 count forwards/back/random 0  100 count forwards/back/random 0  1000 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

Unison Response Rhythm 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

Unison Response Pendulum 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

Show Me Flip 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

Show Me Digit 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

Show Me Digit 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

Show Me Digit 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?

Show Me Digit 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

Show Me Fan 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….

Show Me Number Generator 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

Show Me 2D 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

Cover Ups Grids 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

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?

Improving Achievement In Mathematics in Primary and Secondary Schools HMIE 2005 “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 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.”

Using the Pupils The Answer Is… x + 7 – 

Using the Pupils The Answer Is… 7 x 1 3·5 x 2 1·75 x 4 3 + 4 5 + 2 1·2 + 5·8 -1 + 8 7 8 – 1 10 – 3 29 – 22 35  5 63  9 70  10 77  11

Using the Pupils What Else Do You Know? 12 x 5 = 60

Using the Pupils What Else Do You Know? 60 12 = 5 60  5 = 12 6 x 5 = 30 5 x 12 = 60 12 x 5 = 60 12 x 10 = 120 1·2 x 5 = 6 11 x 5 = 55

Using the Pupils What Else Do You Know? 60 12 = 5 3 x 5 = 15 60  5 = 12 6 x 5 = 30 12 x 25 = 30 5 x 12 = 60 12 x 5 = 60 12 x 10 = 120 24 x 5 = 120 1·2 x 5 = 6 12 x 05 = 06 11 x 5 = 55 10 x 5 = 50

Using the Pupils What Else Do You Know? 20% of 60 = 12 1 12 of 60 = 5 1 5 of 60 = 12 60 12 = 5 3 x 5 = 15 60  5 = 12 6 x 5 = 30 12 x 25 = 30 5 x 12 = 60 12 x 5 = 60 12 x 10 = 120 24 x 5 = 120 1·2 x 5 = 6 12 x 05 = 06 11 x 5 = 55 10 x 5 = 50

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

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

Using the Pupils What Could the Next Number Be? 3·14

Using the Pupils What Could the Next Number Be? 3·14 4·14 13·14 3·15 3·141

Using the Pupils What Could the Next Number Be? 3·14 4·14 5·14 13·14 23·13 3·15 3·16 3·141 3·142

Using the Pupils What Could the Next Number Be? 3·14 4·14 5·14 6·14 13·14 23·13 33·13 3·15 3·16 3·17 3·141 3·142 3·143

Online Resources

Introducing the Lesson

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

Direct Interactive Teaching

Direct Interactive Teaching “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

Direct Interactive Teaching “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

Direct Interactive Teaching “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

Top Tips in Questioning 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 Ask pupils for questions

The Plenary Session

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

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

How Do We Keep Skills Sharp? Appropriate resources Mental mathematics Workcards and Worksheets Homework Problem of the Week Websites Assessment

Workcards and Worksheets Numbertime The Maths File Game Show Teaching Time Revisewise Maths Numeracy Software Schoolhouse Technologies Skillswise

Problem of the Week Nrich NZMaths Problem Solving Word Problems for Kids Mrs Renz’s Math Web Sites TES Brainteasers Count On

Websites for Interactive Whiteboards CurricuLinks – Interactive Whiteboards CurricuLinks – 5-14 Mathematics

Homework

Homework

Assessment National Assessments Tracking Database Standardised Tests Mathematics Check-ups and Pupil Assessment Files Formative Assessment Questioning

Guidance for Teachers on Learning and Teaching Approaches in Mathematics