think of a number (single digits are easiest) multiply it by 10 add 6 divide by 2 …

Reflecting on calculation: when drilling becomes fulfilling

Anne WatsonSingapore

2012

Think of a number ... Think of a number (single digits are

easiest) Multiply it by 10 Add 6 Divide by 2 Add 2 Divide by one more than your original

number The answer is ......

Reflection to see patterns

Patterns are evidence of underlying mathematical structure

Patterns of same-ness; patterns of difference

Different same Different starting numbers same

relationship

Same different60 ÷ 1 = 6060 ÷ 2 = 3060 ÷ 3 = 2060 ÷ 4 = 1560 ÷ 5 = 1260 ÷ 6 = 1060 ÷ 7 = ??

= 0.1 = 10%

= 0.333 ... = 33%

Expectations Teacher – generated (deliberate or

accidental)

Creating expectations deliberately

1 ÷ 9 = 0.111 …2 ÷ 9 = 0.222 …3 ÷ 9 = 0.333 …4 ÷ 9 = 0.444 …5 ÷ 9 = 0.555 …

6 ÷ 9 = 0.666 …7 ÷ 9 = 0.777 …8 ÷ 9 = 0.888 … 9 ÷ 9 = 0.999 …

Creating student-generated expectations accidentally

23 ÷ 10 = 2.3 23 ÷ 20 = 0.23

Long division 222222 ÷ 13 = 444444 ÷ 13 = 666666 ÷ 13 = 777777 ÷ 13 = 101010101010 ÷ 13 = 131313131313 ÷ 13 121212121212 ÷ 13

170943418851282598297770007770 of course???

Expectations Generated from past experience

Reflection on the effects of actions

Patterns in the effects of actions can be evidence of mathematical meaning

Patterns in equivalent fractions

~~~~~~

===

Patterns in fractions

XXX

X X

X

===

Patterns in Fractions

+++

+ +

+

===

Patterns in fractions

Patterns in fractions

Reflecting on actions: stretching and shrinking

Reflection/analysis of a whole piece of work

Reflection on a whole piece of work can reveal similarities and differences with other objects

A structured collection of questions

Think of a number› Multiply it by 12 and divide by 3› Now multiply the output by 3 and divide by 12

Think of a number Multiply it by 15 and divide by 5 Now multiply the output by 5 and divide by 15

Think of a number› Multiply it by a and divide by b› Now multiply the output by b and divide by a

Reflection on a whole object Two triangles enlarged additively and

multiplicatively

Reflection on the learning process

› What do I know now that I did not know an hour ago?

› What could I do now that I would not have thought of an hour ago?

› How did I come to know these new things?

Reflection can trigger awareness of change and growth in knowledge

Therefore learners need to: reflect on what they have produced,

to look for patterns reflect on the effects of their actions,

so they can make predictions, check their work, invent short cuts

know that reasoning from experience can lead to false conjectures so we need logical reasoning too

Therefore teachers need to: Provide sequences of tasks that reveal

mathematical patterns: different/same Encourage learners to reflect on the

effects of their actions, so they can make predictions, check their work, invent short cuts

Provide tasks that show how reasoning from experience can sometimes be misleading

Summary• well-structured tasks and prompts can

make reflection a habit• methods and concepts of the

multiplicative relationship interact• reflection can transform knowledge

about calculation• engaging the reflective mind can

enrich understanding

Thankyou anne.watson@education.ox.ac.uk