more on calculations
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More on calculations. Dr Geoff Tennant [email protected]. Vygotsky ’ s Zone of Proximal Development. I can already drive a car I cannot fly a fighter jet without a huge amount of learning which I don’t currently have. - PowerPoint PPT PresentationTRANSCRIPT
Vygotsky’s Zone of Proximal Development
I can already drive a car
I cannot fly a fighter jet without a huge amount of learning which I don’t currently have.
BUT: while I haven’t driven a lorry, it is reasonable to suppose that, with some lessons, I could. Driving a lorry is next to what I can already do.
So, driving a lorry is in my zone of proximal development – I can’t do it now, but it is next to what I can already do.
More generally, the ZPD represents what can be learnt reasonably easily given existing knowledge and understanding.
Vygotsky’s ZPD: a case study (1)
The task is to start from 278 and work what you need to add on to reach 500. John has successfully completed this task, although he has done very few questions in the time available. When you ask him how he went about this task, he reluctantly rescues from the bin a sheet of paper on which he has drawn tallies whilst counting 279-280-281-…-499-500, and then counted them from the beginning (tallies shown on the next slide).
Brief discussion: what might you say to John?
Vygotsky’s ZPD: a case study (2)
Subtraction with a number line (1)
997 1001998 999 1000
1. 1001 – 4
Subtraction with a number line (2)
2245 07202300 0000 0700
15 m 1 hr7 hrs
20 m
So total time taken is 15 min + 1 hr + 7 hr + 20 min = 8 hrs 35 min
A train leaves London at 22.45 and arrives in Inverness at 07.20 the next day.
How long was the journey?
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Timestables
Need as many ways as practising these as we can.Already have:- People maths activities, standing up, ‘fizz
buzz’;- Can use follow on cards, Tarsia activities,
bingo, and how about Countdown?
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Challenge
Think of as many ways as possible of practising the timestable fact:
4 x 6 = 24Include simple divisions, word problems, etc.
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Countdown (1)
The answer is 32
You can use 4, 7, 9, 5
Use the following numbers and any of addition, subtraction, multiplication and division to get the answer.One answer: 4 x 7 + 9 – 5
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Countdown (2)
TaskMake up a Countdown activity suitable for your class. Try it out on people sitting next to you
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Another way of thinking about timestables….
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So that leaves us with…1 2 3 4 5 6 7 8 9 10 11 12
2 4 6 8 10 12 14 16 18 20 22 24
3 6 9 12 15 18 21 24 27 30 33 36
4 8 12 16 20 24 28 32 36 40 44 48
5 10 15 20 25 30 35 40 45 50 55 60
6 12 18 24 30 36 42 48 54 60 66 72
7 14 21 28 35 42 49 56 63 70 77 84
8 16 24 32 40 48 56 64 72 80 88 96
9 18 27 36 45 54 63 72 81 90 99 108
10 20 30 40 50 60 70 80 90 100 110 120
11 22 33 44 55 66 77 88 99 110 121 132
12 24 36 48 60 72 84 96 108 120 132 144
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What do we know about:
• The 10 timestable?• The 11 timestable?• The 5 timestable?• The 9 timestable?• The 2 timestable?• The 4 timestable?• The 8 timestable?
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See if you like Vedic squares:
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If you join the 2s:1 2 3 4 5 6 7 8 9
2 4 6 8 1 3 5 7 9
3 6 9 3 6 9 3 6 9
4 8 3 7 2 6 1 5 9
5 1 6 2 7 3 8 4 9
6 3 9 6 3 9 6 3 9
7 5 3 1 8 6 4 2 9
8 7 6 5 4 3 2 1 9
9 9 9 9 9 9 9 9 9
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So, in summary:
• Timestable facts still need to be learnt, but there are fewer to learn than may appear at first sight.
Before we move on:- Do you have ideas for learning timestables we’ve not yet shared?
Multiplication (1): traditional method
3 5 6 x ____1 7 9
4503
3
2 4 9 2 05
4
1
30 03 5 6
427
2
1
6
Multiplication (2): grid method
300 50 6
450
600
420
549
70
100 30000
21000
2700
5000
3500
63724
3204
24920
35600
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Use the grid method to do the following multiplications
• 456 x 192• 243 x 891• 24 x 1025
If you have time, compare your answers with what they look like with the traditional algorithm. Can you see the similarities?
Multiplication (3): gelosia
4 54
3 5 6
9
7
1
5
06
42
27
05
35
03
21
421
7
2 3
1 6
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Use the gelosia method to do the following multiplications
• 456 x 192• 243 x 891• 24 x 1025
If you have time, compare your answers with what they look like with the traditional algorithm and with the grid method. Can you see the similarities?
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Happy numbers• Start eg. with 25• 25 goes to 2x2 + 5x5 = 29• 29 goes to 2x2 + 9x9 = 85• 85 goes to 8x8 + 5x5 = 89• 89 goes to 8x8 + 9x9 = 145• 145 goes to 1x1 + 4x4 + 5x5 = 42• 42 goes to 4x4 + 2x2 = 20• 20 goes to 2x2 + 0x0 = 4• 4 goes to 4x4 = 16• 16 goes to 1x1 + 6x6 = 37• 37 goes to 3x3 + 7x7 =58• 58 goes to 5x5 + 8x8 = 89Can you see what happens now? What about if you start with other numbers?
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Premise of maths trails
• There is maths all around us;• Excellent as an activity eg. when children
change school• Gives a focus on school outings• Can be used to consolidate maths topics being
learnt.
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Some examples
Your taskDevise a maths trail in your groups based around this room / this campus / nearby. Be ready to present your trail to the rest of the group, being clear what purposes it will serve in learning new material / consolidating / bringing ideas together.