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Puzzle-based Learning

Methodology

Zbigniew Michalewicz

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Outline of the talk

• The history & motivation

• Some examples

• Current status

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Outline of the talk

• The history & motivation

• Some examples

• Current status

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Two men meet on the street (they have not seen each other for many years):

A: All three of my sons celebrate their birthday today. Canyou tell me how old each one is?

B: Yes, but you have to tell me something about them…A: The product of their ages is 36.B: I need more info…A: The sum of their ages is equal to the number of windows

in the building next to us…B: I need more info…A: My oldest son has blue eyes.B: That is sufficient!

A puzzle from 1964…

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Years 1982 – 1998

I have been introducing puzzles in many lectures, on many topics related to Computer Science, at many universities in United States, New Zealand, Australia, China, Japan, South Korea, Chile, Argentina, Mexico, Poland, Germany, France, Denmark, Sweden, Norway, Finland, Italy, Spain, Portugal, Austria…

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Aarhus University, 1998/99

Work on How to Solve It: Modern Heuristics

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Some comments

• “This book teaches you how to think of a solution for the problem you face…”

• “…anyone interested in […] human thinking should read and understand this book.”

• “I used this book in a Master's class on Heuristics (Systems Engineering, University of Virginia) and received the most positive textbook reviews I have seen in my fifteen years of teaching.”

• “Most importantly, it does so in a way that no other book I've seen does – it makes it fun and it makes you think!”

Some comments (Amazon.com) on How to Solve It: Modern Heuristics:

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Motivation

What’s missing in most curricula – from

elementary schools through to universities –

is the development of thinking skills, problem

solving skills, creative thinking, thinking “out-

of-the-box,” etc.

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Motivation

Alex Fisher, Critical Thinking (Cambridge, 2001):

“… though many teachers would claim to teach their students ‘how to think’, most would say that they do this indirectly or implicitly in the course of teaching the content which belongs to their special subject. Increasingly, educators have come to doubt the effectiveness of teaching ‘thinking skills’ in this way, because most students simply do not pick up the thinking skills in question.”

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Motivation

When we face a problem, we need a systematic (and precise) approach – many problem-solving methodologies depart from precision too far.

Basic mathematical skills are very important in problem-solving methodologies…

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The rejection of mathematics and any precision of thought in Six Thinking Hats (1999) by Edward de Bono, expresses itself in author’s statement like:

“In a simple experiment with three hundred senior public servants, the introduction of the Six Hats method increased thinking productivity by 493 percent.”

Additional motivation

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The rejection of mathematics and any precision of thought in Six Thinking Hats (1999) by Edward de Bono, expresses itself in author’s statement like:

“In a simple experiment with three hundred senior public servants, the introduction of the Six Hats method increased thinking productivity by 493 percent.”

Additional motivation

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Puzzle-based Learning

• Puzzles represent “unstructured” problems; they teach how to think “out of the box.”

• Puzzles are not attached to any chapter of any text.

• Puzzles illustrate many general and powerful problem solving techniques.

• Puzzles illustrate importance of science.

• Puzzles are fun and easy to remember!

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Puzzle-based Learning

• Puzzles represent “unstructured” problems; they teach how to think “out of the box.”

• Puzzles are not attached to any chapter of any text.

• Puzzles illustrate many general and powerful problem solving techniques.

• Puzzles illustrate importance of science.

• Puzzles are fun and easy to remember!

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A rectangular chocolate bar consists of m × n small rectangles and you wish to break it into its constituent parts. At each step, you can only pick up one piece and break it along any of its vertical or horizontal lines. How should you break the chocolate bar using the minimum number of steps (breaks)?

Chocolate bar

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Saturdays and Sundays

Which day of the week, Saturday or Sunday,appears more often as January 1st?

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A well-known puzzle

A farmer has to take a wolf, a goat, and some cabbage across a river.

His rowboat has enough room for the man plus either the wolf or the goat or the cabbage.

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A well-known puzzle

• If the farmer takes the cabbage with him, the wolf will eat the goat. • If the farmer takes the wolf, the goat will eat the cabbage.• Only when the farmer is present are the goat and the cabbage safe from their enemies.

How should the farmer carry the wolf, goat, and cabbage across the river?

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Supporting evidence

• English scholar Alcuin (born around 732) published his main work, Problems to Sharpen the Young (in that book, one of the puzzles was the famous river-crossing puzzle – used 1,200 years later in all Artificial Intelligence courses all over the world!).

• The first known puzzles are from Neolithic Age (3,000 – 2,500 BC).

• Statements by famous mathematicians (Polya, Gardner, Steinhauss, Smullyan, etc.).

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Real-world “puzzles”

“Why use logic puzzles, riddles, and impossible questions? The goal of Microsoft’s interviews is to assess a general problem-solving ability rather than a specific competency. At Microsoft, and now at many other companies, it is believed that there are parallels between the reasoning used to solve puzzles and the thought processes involved in solving the real problems of innovation and a changing marketplace.”

William Poundstone, How would you move Mount Fuji? – How the World’s Smartest Companies Select the Most Creative Thinkers, 2004

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The main learning objective

Problem Solution

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Puzzle-based Learning

Puzzle-based Learning is an approach to develop thinking skills, mental stamina and perseverance at solving problems. The approach is based on unstructured, generally context-free (i.e. does not require domain knowledge) and usually entertaining problems, better known as puzzles…

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Outline of the talk

• The history & motivation

• Some examples

• Current status

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A B

Understanding the problem

• You drive a car at a constant speed of 40 km/h from A to B, and on arrival at B you return immediately to A, but at a higher speed of 60 km/h.

• What was your average speed for the whole trip?

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A B

Understanding the problem

• You drive a car at a constant speed of 40 km/h from A to B, and on arrival at B you return immediately to A, but at a higher speed of 60 km/h.

• What was your average speed for the whole trip?

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The key question is:

What is the definition of “average” speed?

The answer is:

Total distance / Total time

With this understanding:

Understanding the problem

48 km/h

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Understanding the problem

Suppose that you go from A to B at a constant

speed of 40 km/h. What should your constant

speed be for the return trip from B to A if you

want to obtain the average speed for the whole

trip of 80 km/h?

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Rule #1

Rule #1: Be sure you understand the problem, and all the basic terms and expressions used to define it.

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Intuition

Find the solution for the following: Two men play a game of Russian roulette with a gun. The gun has six chambers, where two bullets were placed in two adjacent chambers.

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Intuition

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After a random spin of the barrel, the first man puts the gun to his head and pulls the trigger. Click. Now it is the turn of the second man. What should he do to increase his chances of staying alive: to spin the barrel first or just pull the trigger?

Intuition

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Rule #2

Rule #2: Do not rely on your intuition too much; solid calculations are far more reliable.

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• You won $1,000.

• Consider 2 scenarios:

(A) You get an additional $500

(B) On flip of a coin, you get an additional $1,000 or no additional money.

Some Psychology

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• You won $2,000.

• Consider 2 scenarios:

(A) You immediately lose $500.(B) On flip of a coin, you lose $1,000

or you do not lose anything.

Some Psychology

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• You won $1,000.• Consider 2 scenarios:

(A) Get additional $500

(B) On flip of a coin,

get additional $1,000

or no additional money.

• You won $2,000.

• Consider 2 scenarios:

(A) Lose $500

(B) On flip of a coin,

lose $1,000 or you do not lose anything.

(A)You leave with $1,500

(B)You leave with either $1,000 or $2,000 (50-50 chances)

Some Psychology

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Rule #3

Rule #3: Solid calculations and reasoning are more meaningful when you build a model of the problem by defining its variables, constraints, and objectives.

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Two men meet on the street (they have not seen each other for many years):

A: All three of my sons celebrate their birthday today. Canyou tell me how old each one is?

B: Yes, but you have to tell me something about them…A: The product of their ages is 36.B: I need more info…A: The sum of their ages is equal to the number of windows

in the building next to us…B: I need more info…A: My oldest son has blue eyes.B: That is sufficient!

Constraints

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36 1 137 2 138 3 1 9 4 1 9 2 2 6 6 1 6 3 2 4 3 3

Constraints

• The product of their ages is 36:

x y z

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36 + 1 + 1 = 3837 + 2 + 1 = 2138 + 3 + 1 = 16 9 + 4 + 1 = 14 9 + 2 + 2 = 13 6 + 6 + 1 = 13 6 + 3 + 2 = 11 4 + 3 + 3 = 10

Constraints

• The sum of their ages is equal to the number of windows in the building next to us…

x y z

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We are often presented with statistical material from which it is tempting to formulate some hypothesis. One famous example was reported by Science Times on August 22, 1989, which described the ability of cats to survive large falls. The distance of the fall for 129 of the 132 cats was recorded, which ranged from 2 to 32 stories! All the data were pulled from the Animal Medical Center from June to November 1984. The most amazing part of the article was the following:

Statistical information

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“Even more surprising, the longer the fall, the greater the chance of survival. Only one of 22 cats that plunged above 7 stories died and there was only one fracture among the 13 that fell more than 9 stories. The cat that fell 32 stories on concrete, Sabrina, suffered [only] a mild lung puncture and a chipped tooth…”

Can you explain this phenomenon?

Falling cats

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Outline of the talk

• The history & motivation

• Some examples

• Current status

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Many recent books/articles

Wing, J.M., Computational Thinking, Communications of the ACM, March 2006.

William Poundstone, How would you move Mount Fuji? – How the World’s Smartest Companies Select the Most Creative Thinkers, 2004.

Danesi, M., The Puzzle Instinct: The Meaning of Puzzles in Human Life, 2002.

Restak, R. & S. Kim, The Playful Brain – The Surprising Science of How Puzzles Improve Your Mind, 2010.

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Many industries, organisations, and government agencies have introduced PBL workshops for training of their employees in problem-solving skills; these include:

• Holden• ETSA Utilities• SA Department of Education• Philip Morris International• Pernod Ricard Pacific• SA Department of Infrastructure and Transportation• Raytheon Australia• Australian Wine Research Institute• Emirates CMS power company• Victim Support Service

PBL Workshops

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More & more universities and high schools are in the process of introducing such courses/units; these include:

• University of Adelaide, Monash University, Flinders University, University of NSW (Australia)

• Carnegie Mellon University, University of Colorado, University of North Carolina, Garrett College, Clemson University, Baldwin Wallace College (USA)

• Wasatch Academy, a 9-12 independent, international college preparatory school in central Utah (USA)

• Australian Science & Maths School (Australia)

• Polish-Japanese School of Information Technology (Poland)

PBL Courses

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More & more universities and high schools are in the process of introducing such courses/units; these include:

• Tel Aviv University, Academic College of Tel-Aviv-Yaffo (Israel)

• Hosei University, Hakuoh University (Japan)

• College of Computer Engineering, Jiangsu Teachers University of Technology (China)

• Korea National University of Education (Korea)

• University of Québec, Annapolis Valley Campus of the Nova Scotia Community College (Canada)

• Universidad Técnica Federico Santa María (Chile)

• University of Kardynal Stefan Wyszynski (Poland)

PBL Courses

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2011 pilot of Puzzle-based Learning for the Australian Computer Society:

• All material was divided into 13 modules: Introduction, Understanding the problem, Intuition, Modelling the problem, Principles of problem-solving, Constraints, Optimisation, Probability, Statistics, Simulation, Pattern Recognition, Games and Strategies, and Summary.

• Each module was divided into 2, 3, 4, or 5 segments.

• There were 42 segments altogether; each segment was recorded (length – approx. 15 minutes)

Australian Computer Society

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2011 pilot of Puzzle-based Learning for the Australian Computer Society:

• There were also homework, discussion groups, additional exercises.

• Enrolled students had a full week per module; most of the work was done on weekends.

• Conclusion: PBL – an ideal material for eLearning adventure...

Australian Computer Society

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Additional information

For additional information, please, check:

www.PuzzleBasedLearning.edu.au

• news section• 8-minutes video• software pieces• and more…

... and all material is available for you on request!

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Lottery

Take your business card (or piece of paper with your name) and write one natural number (i.e. 1, 2, 3, …).

The winner is …… the person who writes the smallest number……which is unique…

Good luck!

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Thank you!