Contextualised group work – teaching a broader mathematics

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curriculum to first year science students:

Case study – Problem solving

Jo-ann Larkins School of Applied Sciences and Engineering Federation University Australia

Today’s presentation

• Participate in an example of contextualised group work activity – theme of problem solving • Brainstorm around how you can incorporate contextualised group work techniques in your classroom • Examples of the types of teaching strategies / activities you can use to engage groups of students

•

Polya : “The Father of Problem Solving”

George Pólya

was a Hungarian mathematician.

• He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in

heuristics

and

mathematics education.

Heuristics

•

Heuristic

(Greek:

"find"

or

"discover"

) refers to experience-based techniques for problem solving, learning, and discovery that gives a solution which is not guaranteed to be optimal.

• Where an exhaustive search is impractical, heuristic methods are used to speed up the process of finding a satisfactory solution via mental shortcuts. • Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, stereotyping, or common sense.

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Polya's Problem Solving Techniques

• In 1945 George Polya published the book

How To Solve It

which quickly became his most prized publication. • It sold over one million copies and has been translated into 17 languages. • In this book he identifies four basic principles of problem solving.

1. Understand the problem

• Do you understand all the words used in stating the problem?

• What are you asked to find or show?

• Can you restate the problem in your own words?

• Can you think of a picture or diagram that might help you understand the problem?

• Is there enough information to enable you to find a solution?

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2. Devise a plan

• There are many reasonable ways to solve problems. • The skill lies in choosing an appropriate strategy.

• This best learned by solving many problems. You will find choosing a strategy increasingly easy.

• (You are going to create a list of strategies in this session).

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3. Carry out the plan

• This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. • Persist with the plan that you have chosen. • If it continues not to work discard it and choose another. Don't be misled, this is how mathematics is done, even by professionals.

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4. Look back

• Much can be gained by taking the time to reflect and look back at what you have done, what worked, and what didn't.

• Doing this will enable you to predict what strategy to use to solve future problems.

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http://www.3vadmin.com/problem-solving-skills / 10

What strategies do you use?

• In groups come up with a list of the strategies you use to solve problems.

• You should be able to come up with at least 10 different ones.

• Think of the sort of problems you’ve had to solve in class as well as real world problems. 11

What strategies do you use?

• I want to collect your group’s output at the end of this session so write it on the blank paper provided.

• You can be creative and create a mind-map or flowchart type output rather than a list.

• You have 10 minutes. 12

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Problem Solving Strategies

• Guess and check • Look for a pattern • Make an orderly list • Draw a picture • Eliminate possibilities • Solve a simpler problem • Use symmetry • Use a model • Consider special cases • Work backwards • Use direct reasoning • Use a formula • Solve an equation • Be ingenious 14

Using contextualised group work in Teaching

We’ve just participated in an activity inviting students to consider abstract thinking skills necessary for mathematics using a group based approach.

Two questions: • Where does your unit have content that lends itself to a group approach?

• Other than discussion, how can you use groups to improve understanding / engage students?

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Teaching strategies

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What I’ve tried

• Relay maths tasks – fastest correct answers • Teach the teacher – teach your peers – develop an example and demonstrate in groups – student’s choice of area within current topic • Using props for context – containers for volume, Molecular models for 3D geometry • Creating checklists / templates / how to do strategies through student centred discussion (e.g. what’s essential in graphs and tables for lab reports from examining real life student work) 17

What I’ve tried

• Role play: You are a …. Explain this concept / graph / diagram – very effective in literacy around data and graphing.

• Case studies – problem solving and applying the skills in a discipline based context.

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Final student comment

What was that question about problem solving on the exam? Worst question ever….

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