Drilling, filling, skilling, fulfilling: what practice can and cannot do for learners Professor Anne Watson Hong Kong January 2011

The case of division....

Rods, tubes and sweets How many logs of length 60cm. can I cut from a long log of length 240 cm? How many bags of 15 sweets can I make from a pile of 120 sweets? I have to cut 240 cm. of copper tubing to make 4 equal length tubes. How long is each tube? I have to share 120 sweets between 8 bags. How many per bag?

Each household is entitled to 7 bottles of drinking water. There are 144 packages each containing 16 bottles. How many households can be supplied with water? There are 89 households – how many bottles can they each have?

Three equal volume bottles of wine have to be shared equally between 5 people. How can you do this and how much will each get? Three equal sized sheets of gold leaf have to be shared equally between 5 art students, and larger sheets are more useful than small ones. How can you do this and how much will each get?

98 equal volume bottles of wine have to be shared equally between 140 people. How can you do this and how much will each get? 98 equal sized sheets of gold leaf have to be shared equally between 140 art students, and larger sheets are more useful than small ones. How can you do this and how much will each get?

Reflection on what we have done so far Drilling & filling: 140 ÷ 98 OR 98 ÷ 140? Skilling: deciding to do 98/140 Drilling, filling, skilling and fulfilling: 7/10 Fulfilling: how does the answer help you?

A piece of elastic 10 cm. long with marks at each centimetre is stretched so that it is now 50 cm. long. Where are the marks now? A piece of elastic is already stretched so that it is 100 cm. long and marks are made at 10 cm. intervals. It is then allowed to shrink to 50 cm. Where are the marks now?

Methods used Counting in groups Known multiplication facts Derived facts Repeated addition Partitioning Mental models Sharing out/ dealing Division algorithms Fractions of a quantity Fractions as method of division Common factors Stretch/shrink each unit

Complex needs We need multiplication facts BUT Written algorithms are based on repeated subtraction of multiples from chunks of the dividend (school arithmetic) YET The way we handle quantities efficiently depends on the context (flexible problem- solving)

Dividing is... comparing quantities partitioning quantities dealing out quotas stretching and shrinking ‘undoing’ multiplication describing quantitative phenomena bridging arithmetic and measurement doing procedures until zero, a remainder, or a decimal representation is achieved

Advantages of learning procedures Key procedures can become automatic A page of ticks boosts confidence Teaching can be automated (online worksheets with good quality feedback) Children can learn (with teacher) to anticipate answers and difficulties Children can learn (with teacher) how to extend methods to more complex situations

Problems with rote-learning All methods are limited in their usefulness Misuse of methods is common It is hard to recognise when to adapt and apply methods Associated with dislike of subject It is what machines can do Need to reflect on answers in order to fully understand method The advantages are often not developed Does not prepare students well for higher mathematical enquiry or real-world use

Advantages of complex knowledge Flexible, adaptable, knowledge Can apply mathematics Misconceptions are resolved through sense- making Mathematics is more interesting Students do better in unfamiliar and multi- stage test questions than if they only know methods

Problems with focus on concepts and problem-solving May not develop fluency – over-dependent on machines May not develop personal repertoire of appropriate methods and key facts

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