Wyche CE School Maths Evening Using the Number line for Calculation - PowerPoint PPT Presentation

Wyche CE School Maths Evening Using the Number line for Calculation 26 th March 2014

National Curriculum: Maths “Mathematics equips pupils with a powerful set of tools to understand and change the world. Mathematics is important in everyday life. Mathematics is creative and can stimulate moments of pleasure and wonder” Maths National Curriculum Introduction p14

Standards are Slipping. Who says? “More than half the adult population would fail to achieve standards expected of 11 year olds. The questions included "Multiply 16 x 21" - which defeated 40 per cent of the UK sample - and "Work out 15 per cent of 700" - which 46 per cent could not answer.” (Daily Telegraph 17 th January)

OECD Report “Half the adults in Britain have such poor levels of numeracy they cannot cope with the demands of everyday life in a complex society in basic numeracy” (Report by the OECD)

Maths as Abstract Tricks 24 +17 41 Correct Answer but 1 what about the understanding?

Maths as Abstract Tricks Teacher: “What is the 1 for?” 24 Child: “I don’t know” +17 Teacher: “What is it” Child: “I don’t know” 41 ”Why is it there?” Teacher 1 Child: Mrs Jones taught me to put it there in when we were in purple class

Maths as Abstract Tricks 204 +137 4 3 1 1

Maths as Abstract Tricks 204 +137 4 3 1 1 Put the 1 in the column by the window

Its not just about getting the right answer

Maths as Abstract Tricks Subtraction 7 12 82 -25 5 7

Maths as Abstract Tricks Where is the Subtraction ten to borrow? 800 -252

Richard Skemp • Instrumental Understanding – Skemp describes this as “rules without reasons ” • Relational Understanding - Skemp describes this as “knowing both what to do and why ” and the process of learning relational mathematics is “building up a conceptual structure”

Common Misconceptions Have a look at the sheets where you think the child’s understanding gone awry

True Understanding 2/3 of 3/4 of 11

Learning with Understanding 82 -25 60 -3 57 Answer is…

The Number Line Why?

Many strategies or Too many “There was a feeling after the introduction of the Numeracy Strategy that the children were overwhelmed by the range of mental strategies on offer and therefore their ability to pick the appropriate “Efficient and Effective” strategy for any particular calculation was weak”

The Renewed Framework The Renewed Framework (2006) stated “Children are entitled to be taught and to acquire secure mental methods of calculation and one efficient written method of calculation which they know they can rely on when mental methods are not appropriate.”

One efficient method of calculation • The school had already picked this up as an issue and had decided to choose • The Number Line as its one strategy

Foundations of Calculation The foundations of calculation hinge on two key principles: • “Shallow strategy of counting on” • “The Partitioning Predicament” See website for more details

Counting On

Counting on “There is substantial research evidence to suggest that counting should constitute the basis of the early years number curriculum ” (The development of numeracy concepts, Nunes 2001)

8+4 with counting

Counting or Calculation? “Children who don’t move on from counting are disadvantaged and require intervention; such ideas form the basis of early years mental maths .” (Munn, 2001)

Counting or Calculation? “Counting has a positive role early in development, but needs to be replaced quite rapidly by more sophisticated cognitive concepts.” (Compressing the Counting Process, 1997)

The Concept of Bridging Calculating 7+7

Numicon Calculating 7+7

Half Partitioning

Key Principles • The number line relies on principle of “Half Partitioning” 204 • The principle being that instead of “partitioning” all +137 the numbers to a unit value… • One of the numbers remains 4 3 1 as a whole • 204 +100 +30 +7 = 1

Dutch Research • Dutch research shows that when children rely on 204 partitioning as a focus strategy they lose all feel for +137 number as they reduce all the digits to the value of a 4 3 1 unit e.g. 2+1 for 200+100 1

Half Partitioning So unlike column addition only one number undergoes the partitioning process

Half Partitioning And with Bridging…

Half Partitioning And with Hundreds…

Half Partitioning Bridging with Hundreds…

School Website – www.wyche.worcs.sch.uk

School Website – www.wyche.worcs.sch.uk

Show CD here

The Numberline 773 + 658 669 - 588 17 x 13 34 x 25 184 ÷ 12 Grouping not Sharing

Number Line: Visual Tool “You can see it; it doesn’t just stay in our head like numbers do. I count in my head but sometimes I lose count but when I use the number line I can take jumps on it and I don’t get lost”

Number Line: Visual Tool “When you draw the number line you can see where the number ends and you know when to stop counting in 10’s, in my head I always go past it and get muddled. When I don’t have it I use my fingers and I lose count with that whereas when you use the number line you just look at it and count”

Number Line: Visual Tool “ I see the number line in my head that is what helps me and I make the jumps in my head as well as on the paper. I see each jump in my head”

Do we teach “The Proper Way?” • Yes but only when the children are fully conversant with the numerical principles underpinning the standard algorithms • They are taught them because: – They should be taught a range of strategies – They will be introduced to these at The Chase • However the children tend to leapfrog the traditional written methods as they develop efficient mental strategies of their own…

Efficient and Effective Strategies

Key Message The Numeracy strategy introduced in 1998 stated that: “An ability to calculate mentally lies at the heart of numeracy. You should emphasise mental methods… with regular opportunities for all pupils to develop the different skills involved” (p6)

Can I do it in my head? The number line builds mental dexterity so children will come off the number line and solve sums mentally 48 x 24= 48 x 20 = 960 48 x 2 = 96 48 x 4 = 96+96 100+100-8 = 192 960 + 192 = 960+200-8 = 1152

Learning with Understanding 82 -25 60 -3 57 Answer is…

A Mental Written Method “The number line acts as a good bridging point between the “Mental” and the “Formal” methods of calculation. In Year 2 it is introduced as a “Pen and Paper method” in the KS3 strategy it comes under the heading of “Mental Methods”. It is, in a very real sense, a mental method that is enhanced through the visual use of the line.” School Calculation Document p2

Mental or Written? The Numeracy strategy stated that: “An emphasis on mental calculation does not mean that written methods are not taught but the balance between mental and written methods is very important”

Calculating with Understanding What is the answer to: 247 + 99 = The first question children should ask is Can I do this in my head?

Sums which should be Mental Calculation Strategy solved using this method All the numbers with a ten e.g. 10 +7 or 50 +16 1890 Any calculation with a 1 in the units because 42+21 allows 1890 partitioning and/or where the number is 39 the addition of 1 unit does not require bridging Any number that has a 9 in the units should be easily 1890 added using the “Add 10 and subtract 1” method All the numbers which have a number bond to 10 in the 1000 units should be added using the known fact of the number bond to 10 e.g. 37+13 Where no bridging is involved then partitioning should be 4500 able to be done mentally e.g. 13+14 Any number where the units contain a double should be 900 mentally added e.g. 42+32 Any numbers which use “near doubles” could be mentally 900 calculated e.g. 15+16

Can I do this in my Head? Using this analysis 88% of calculations require a mental calculation strategy rather than a pen and paper method.

Can I do this in my head?

I was bored after the first few slides!!