Mathematical thinking and learning as problem-solving Anne Watson Roskilde February 2011

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?

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?

What is problem-solving? Understand the problem Make a plan Act out the plan Reflect on the outcomes Adapt the plan or make a new one Act out the new plan What is missing?

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? Use different multiples; and think about getting back to where you started

What is problem-solving? Understand the problem Make a plan Act out the plan Reflect on the outcomes Adapt the plan or make a new one Act out the new plan What is missing?

Look for patterns ?

What is problem-solving? Understand the problem Make a plan Act out the plan Reflect on the outcomes Adapt the plan or make a new one Act out the new plan What is missing?

What is problem-solving? Understand the problem Make a plan Act out the plan Reflect on the outcomes Adapt the plan or make a new one Act out the new plan What is missing?

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What is problem-solving? Understand the problem Make a plan Act out the plan Reflect on the outcomes Adapt the plan or make a new one Act out the new plan What is missing?

Understand the problem Identify variables – same/different Think about structure What is relevant and what is irrelevant

Make a plan What do I know? What do I need to know? Change some variables and hold others constant Be systematic or choose particular values Choose representations Choose methods from repertoire Think about implications of known facts

Act out the plan Vary variables Compare representations Manipulate notations Follow lines of argument

Reflect on the outcomes What varies and what stays the same? What are the relations, rules, covariation? What was a surprise? What was expected? What general relations, connections and patterns emerged? Does it make sense with what you already know?

Make a new plan Based on new relations, covariations, conjectures, properties, or based on new/different knowledge and methods

Act out the new plan Vary properties – same/different – what changes and what stays the same?

And so on ……

What is missing? The teacher provides the mathematical knowledge and insight – naming new objects – drawing attention to worthwhile features – providing the conventional notation – knowing what is mathematically important – taking it to a new level of mathematics – making connections with other mathematics – ways of reasoning that are specially mathematical

Thankyou