Embed Size (px)
Citation preview
Standards & Learning Effectiveness Service
Jon R. BrownStandards & Learning Consultant (0 - 19)
Calculations:policy and practice
Session aims:•Highlight some key elements that support
progression through calculations•What is an ‘effective’ calculation policy?•How does clarity and consistency
support AfL and vice versa?
Calculations
Key elements for progression: 1
A and C
Key elements for progression: 1The language of place value
including ‘zero as a place holder’
46537870013013
+
843
465378
700130
3
+
843
465378+
113 4
18
What’s missing from this picture?
46537870013013
+
843
465378
700130
3
+
843
465378+
113 4
18
Non-negotiable
htu htuhtu
The language and notation of Place Value
Understanding carrying“…thirteen, that’s
one ten and 3 units….”
The story of decimals:
1000 100 110 1/101/1000
1/100
2 7 2
2 7 2£
How do pupils ‘see’ the 7?What are their mental images?
Key elements for progression: 2
12 ÷ 3“How many threes are in twelve”
Key elements for progression: 2Calculation structures
Aggregation/Combining
Augmentation/Counting on + ++ + +
Dino Maths
Calculation structures with Year 1
Small step with Year 1 to introduce another key image.Adding or subtracting?
Key elements for progression: 3Practical equipment, manipulatives,
models and images.
+
Division: Grouping on a number line. Do we add or subtract?
Adding groups Subtracting groups
Model or image relates to number line approach- the underlying structure.
Bigger issues….Compact formal division methods generally rely on subtractive approaches.When will pupil be expected to make the transition between adding and subtracting?
Beware!
Key elements for progression: 4Side by side: linking methods and images
Key elements for progression:•Teacher and (where appropriate) pupil awareness of underlying structures.•Use of the language of place value including ‘zero as a place holder’.•Use of practical equipment, manipulatives, models and images.
15 4
UT
1 6-
4
8 6TH U
5 7+
86 + 57: Adding the units (least significant figure) first
8 0 6+
T U U
5 0 7+
8 6TH U
5 7+
8 0 6+
T U U
5 0 7+
8 6TH U
5 7+
31
8 0 6+
T U U
5 0 7+
13
8 6TH U
5 7+
31
8 0 6+
T U U
5 0 7+
13
8 6TH U
5 7+
31
8 0 6+
T U U
5 0 7+
131 3 0
8 6TH U
5 7+
31
8 0 6+
T U U
5 0 7+
13 411 3 0
8 6TH U
5 7+
31
8 0 6+
T U U
5 0 7+
13 411 3 0 +
8 6TH U
5 7+
31
8 0 6+
T U U
5 0 7+
13 411 3 0 + =
8 6TH U
5 7+
31
8 0 6+
T U U
5 0 7+
13 411 3 0 + = 143
14 x 6T U U
Grid method side-by-side with aformal vertical column method
14 x 6T U U
6
10 4x1 4T U
6x
T U U
14 x 6T U U
6
10 4x1 4T U
6x
T U U
14 x 6T U U
6
10 4x1 4T U
6x
T U U
2 424
14 x 6T U U
6
10 4x1 4T U
6x
2 424
T U U
14 x 6T U U
6
10 4x1 4T U
6x
2 424
T U U
14 x 6T U U
6
10 4x1 4T U
6x
2 424
T U U
6 0
60
14 x 6T U U
6
10 4x1 4T U
6x
2 46 0
2460
T U U
14 x 6T U U
6
10 4x1 4T U
6x
2 46 0 +
2460
T U U
Total
14 x 6T U U
6
10 4x1 4T U
6x
2 46 08 4
+
2460 84
T U U
Total
23 x 47T U T U
23 x 47T U T U
20
40 7
3
x4 7TH U
2 3x
T U U
23 x 47T U T U
20
40 7
3
x4 7TH U
2 3x
T U U
23 x 47T U T U
20
40 7
3
x4 7TH U
2 3x
T U U
12
21
757 53
45
3× 10- 3 0
4 5
3× 10
15
3× 10 - 3 0
1 5
3× 10
0
3× 5- 1 5
0
3× 5
The numberof 3’s in eachchunk/group.
The numberof 3’s in eachchunk/group.
75 ÷ 3 = 25
The rapid recall of multiplication and division facts is extremely important and underpins mental and written calculations. Of equal importance is a pupil’s ability to find new facts from their existing bank (or box) of known facts and familiar processes.
Key elements for progression: 5Balancing rapid recall and
the use of known facts/processes
Instrumental understanding: “Rules without reasons”. ‘Memorising’ a procedure that works for a specific problem and learning a different procedure for each new ‘class’ of problems.
Relational understanding: “Knowing what to do and why”. Building up a ‘conceptual structure’ where packages of knowledge are interconnected.
Skemp (1976)
‘The fundamental issue for teachers is how better to develop pupils’ mathematical understanding. Too often, pupils are expected to remember methods, rules and facts without grasping the underpinning concepts, making connections with earlier learning and other topics, and making sense of the mathematics so that they can use it independently.’
(UNDERSTANDING THE SCORE: OFSTED Sept 2008, p.5)
Briefly summarise (in under a minute)the ‘bigger picture’ when describingprogression through calculations.
How would your colleagues respond?
Key elements for progression: 5Balancing rapid recall and
the use of known facts/processes
Counting Stick
An informal ‘ad hoc’ multiplication approachFind 19 ‘lots of’ 24
1 ‘lot of 24 is’ 24
2 ‘lot of 24 is’ 48
4 ‘lot of 24 is’ 96
8 ‘lot of 24 is’ 192
16 ‘lot of 24 is’ 384
Find 19 ‘lots of’ 24
1 ‘lot of 24 is’ 24
2 ‘lot of 24 is’ 48
4 ‘lot of 24 is’ 96
8 ‘lot of 24 is’ 192
16 ‘lot of 24 is’ 384
An informal ‘ad hoc’ multiplication approach
Find 19 ‘lots of’ 24
1 ‘lot of 24 is’ 24
2 ‘lot of 24 is’ 48
4 ‘lot of 24 is’ 96
8 ‘lot of 24 is’ 192
16 ‘lot of 24 is’ 384
I have 16 ‘lots of’ 24. How many more do I need?
An informal ‘ad hoc’ multiplication approach
Find 19 ‘lots of’ 24
1 ‘lot of 24 is’ 24
2 ‘lot of 24 is’ 48
4 ‘lot of 24 is’ 96
8 ‘lot of 24 is’ 192
16 ‘lot of 24 is’ 384
Final addition:Mental, Horizontal, Vertical, Ad hoc, Formal?
I have 18 ‘lots of’ 24. How many more do I need?
An informal ‘ad hoc’ multiplication approach
Find 15 ‘lots of’ 24
1 ‘lot of 24 is’ 24
2 ‘lot of 24 is’ 48
4 ‘lot of 24 is’ 96
8 ‘lot of 24 is’ 192
16 ‘lot of 24 is’ 384
Progression is shown by the efficiency of the calculation.
An informal ‘ad hoc’ multiplication approach
Find 15 ‘lots of’ 24
1 ‘lot of 24 is’ 24
2 ‘lot of 24 is’ 48
4 ‘lot of 24 is’ 96
5 ‘lot of 24 is’ 120
10 ‘lot of 24 is’ 240
An informal ‘ad hoc’ multiplication approach
Progression is shown by the efficiency of the calculation.
Find 102 ‘lots of’ 24
1 ‘lot of 24 is’ 24
10 ‘lot of 24 is’ 240
100 ‘lot of 24 is’ 2400
2 ‘lot of 24 is’ 48
An informal ‘ad hoc’ multiplication approach
Progression is shown by the efficiency of the calculation.
Aims of the NC: Fluency … through varied and frequent practice with
increasingly complex problems …, so that pupils develop conceptual understanding and the ability to recall and apply
knowledge rapidly and accurately. Mathematical reasoning: following a line of enquiry,
conjecturing, generalising, justifying and proving.Problem solving: Applying their mathematics. Breaking
down problems and persevering.
Key elements for progression:•Teacher and (where appropriate) pupil awareness of underlying structures.•Use of the language of place value including ‘zero as a place holder’.•Use of practical equipment, manipulatives, models and images.•Side by side: linking methods and images•Balancing rapid recall and use of known facts and processes
Some key changes in Year 3•children are now expected to count in multiples of 4, 8, 50 and 100 •they are expected to mentally calculate with three-digit numbers •learning the eight-times table has been included •tenths are new to Year 3 •children now need to add and subtract fractions of the same denominator •measuring perimeters of simple shapes was in Year 4 but now it is in Year 3 •children are expected to be able to tell 24-hour time - this previously appeared in Year 5 •they are also expected to be able to read the time on clocks with Roman numerals •children need to be able to identify perpendicular and parallel lines (Dynamic geometry)
NCETM
Creating an ‘effective’ calculation policy:• School Action Plan priority?• Subject leader action plan?
o Initial audit
Creating an ‘effective’ calculation policy:• School Action Plan priority?• Subject leader action plan?
o Initial audit
Creating an ‘effective’ calculation policy:• School Action Plan priority?• Subject leader action plan?
o Initial audit including staff and pupil voice
Creating an ‘effective’ calculation policy:• School Action Plan priority?• Subject leader action plan?
o Initial audit including staff and pupil voiceo Calculation guidance. To be used in conjunction withan existing policy or drawn upon to develop/enhancea policy based on pupil needs and preferred approaches.
Creating an ‘effective’ calculation policy:• School Action Plan priority?• Subject leader action plan?
o Initial audit including staff and pupil voiceo Calculation guidance. To be used in conjunction withan existing policy or drawn upon to develop/enhancea policy based on pupil needs and preferred approaches.o Calculation policy.
Part 1: What Part 2: How
Calculation GuidanceExpectations
Creating an ‘effective’ calculation policy:• School Action Plan priority?• Subject leader action plan?
o Initial audit including staff and pupil voiceo Calculation guidance. To be used in conjunction withan existing policy or drawn upon to develop/enhancea policy based on pupil needs and preferred approaches.o Calculation policy linked in to long and medium term plans.
Creating an ‘effective’ calculation policy:• School Action Plan priority?• Subject leader action plan?
o Initial audit including staff and pupil voiceo Calculation guidance. To be used in conjunction withan existing policy or drawn upon to develop/enhancea policy based on pupil needs and preferred approaches.o Calculation policy linked in to long and medium term plans.o Future proof – Informed flexible updates (Age or stage??)
Yr 4 should focus on additive approaches to grouping on the number line.Suggested activities:•Dino division•Grouping carousel•Side-by-side with number rods•Etc.
Creating an ‘effective’ calculation policy:• School Action Plan priority?• Subject leader action plan?
o Initial audit including staff and pupil voiceo Calculation guidance. To be used in conjunction withan existing policy or drawn upon to develop/enhancea policy based on pupil needs and preferred approaches.o Calculation policy linked in to long and medium term plans.o Future proof – Informed flexible updates (Age or stage??)
SLT and Maths SL working together strategically.Regular joint meetings.Review, track impact and pupil progress.Planning updates, interventions and QFT approaches.
Supporting AfL: Success criteria
Formative assessmentand feedback.
Aims of the NC: …….. and persevering.
Session aims:•Highlight some key elements that support
progression through calculations•What is an ‘effective’ calculation policy?•How does clarity and consistency
support AfL and vice versa?
Calculations
Aims of the NC: Fluency … through varied and frequent practice with
increasingly complex problems …, so that pupils develop conceptual understanding and the ability to recall and
apply knowledge rapidly and accurately.
Stepping off thelevel ladder andhaving a look around.
