think of a number (single digits are easiest) multiply it by 10 add 6 divide by 2 …
DESCRIPTION
Patterns are evidence of underlying mathematical structure Patterns of same-ness; patterns of differenceTRANSCRIPT
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
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Patterns in fractions
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X X
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Patterns in Fractions
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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 [email protected]