Variation | KS1

Developing mathematical thinking through intelligent practice

Jo Harbour jo@cambridgemathshub.org

Luke Rolls lrolls@universityprimaryschool.org.uk

What makes for good practice?

1. Random shooting from around the court?

2. Making connections to previous shot and changing positions

deliberately?

Let’s try.

Defining variation?

Conceptual variation - different representations of the same idea strengthens our understanding of what ‘it’ is.

What would a child understand about the ‘sixness’ of 6 through only being exposed to it as a Numicon 6 shape?

Procedural variation - choosing to vary one aspect to expose a mathematical structure or connection.

Is there a way to structure the e.g. learning of number bonds of 6 in a way that encourages children to think mathematically, see patterns, make

connections?

The central idea of teaching with variation is to highlight the essential features of the concepts through varying the non-essential features.Gu, Huang & Marton, 2004

Variation

3 + __ = 61 + 5 = ____ + 0 = 63 + 3 = __5 + __ = 62 + 4 = __0 + 6 = __

6 = 1 + __6 = __ + 26 = __ + 36 = 4 + __6 = __ + 16 = __ + 0__ = 0 + 6

What do you notice?

What’s the same? What’s different?

What structures are exposed?

Variety

Variation

__ + 0 = 60 + 6 = __1 + __ = 65 + __ = 6__ + 2 = 6__ + 4 = 63 + __ = 6

6 = 1 + __6 = __ + 26 = __ + 36 = 4 + __6 = __ + 16 = __ + 0__ = 0 + 6

Both procedural variation. Why a

different choice of order?

Variation

Think hard… _ + _ + 3 = 6

- Commutativity - Compensating dynamic of partitions- Systematic ordering

Using procedural variation to help learn number facts

6 + 7 = __7 + 7 = __3 + 4 = ____ + 5 = 107 + __ = 156 + 5 = __3 + __ = 6

3 + __ = 6

3 + 4 = ____ + 5 = 106 + 5 = __7 + 7 = __

7 + __ = 15

6 + 7 = __

Phase 1

Modelling and/or counting to get the answer

Phase 2

Deriving answers using reasoning strategies based on known facts

Phase 3

Efficient production of answers

Three phases of basic fact fluency

Baroody (2006)

Exemplifying procedural variation

Using variation to teach comparison: equivalence, greater than, less thanChildren in UK often don’t understand the = sign…

5 = 2 + __2 + 3 = 5

2 + 3 = 4 + 12 + 5 = 4 + 34 + 5 = 6 + 3

A

B

C

D

E

Exemplifying conceptual variation Teaching = so it makes sense

Balancing scale Number Balance Cuisenaire Rods

Tens Frames Measurement: Height

Measurement: Money

Symbolic Subitizing

Exemplifying conceptual variation Teaching = so it makes sense

Children’s recordings

With thanks to David Thomas

With thanks to David Thomas

With thanks to David Thomas

With thanks to David Thomas

Over to you:

10 + 5 = 15Design a series of questions… what

structure will you try to expose?

Over to you:

10 + 5 = 15Feeding back

https://www.oxfordowl.co.uk/welcome-back/for-school-back/default/series-landing-pages/pd-books/making-numbers

Questions

On tables,