How to Teach
Puzzle-based
Learning
Zbigniew Michalewicz
1
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Outline of the talk
• Getting started
•
•
•
•
Icebreakers
Effective teaching approaches
Understanding the problem
Some techniques (e.g. simplification,
performing a Gedanken: What if? So what?)
• Summary
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A book
The talk is based on a new book
(Springer, August 2014)
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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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General perspective
The ultimate goal of Puzzle-based Learning is to lay a
foundation for students to be effective problem solvers
in the real world.
At the highest level, problem solving in the real world
calls into play three categories of skills: (1) dealing
with the vagaries of uncertain and changing conditions;
(2) harnessing domain specific knowledge and
methods; (3) critical thinking and applying general
problem solving strategies.
These three skill categories are captured in the three
forms of learning – project, problem, puzzle-based:
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General perspective
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Outline of the talk
• Getting started
•
•
•
•
Icebreakers
Effective teaching approaches
Understanding the problem
Some techniques (e.g. simplification,
performing a Gedanken: What if? So what?)
• Summary
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Getting started
Two key components:
 The instructor
 Motivated students
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Getting started
Two key components:
 The instructor
 Motivated students
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Logicians
Annual International Conference on Logic:
 31 participants
 one master
 it is necessary to test, whether all participants are
logicians
 each participant gets from a master a dot of some colour
on his forehead
 the master assures one participant (who expressed a
concern) that everyone will be able to guess the colour of
his/her dot at some stage (when a bell rings…)
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Logicians
This is what happened. When the bell ring:
• 1st time: 4 people left
• 2nd time: all with red dots left
• 3rd time: no one left
• 4th time: at least one left
• shortly afterwards: a participant, who
expressed concern left together with his sister;
both had dots of different colour… At that time
there were still some people left…
How many times did the bell ring?
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Getting started
Two key components:
 The instructor
 Motivated students
– Real-world connections
– A few examples on ‘thinking’ in wrong directions…
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Connection with “real-world”
When technology is changing beneath your feet
daily, there is not much point in hiring for a
specific, soon-to-be-obsolete set of skills. You
have to try to hire for general problem-solving
capacity, however difficult that may be.
An interview puzzle is a filter to prevent bad
hires.
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Connection with “real-world”
Both the solver of a puzzle and a technical
innovator must be able to identify essential
elements in a situation that is initially illdefined. It is rarely clear what type of reasoning
is required or what the precise limits of the
problem are.
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A warm up
Below is a schematic of a printed circuit board (PCB).
The problem is to either connect each number to its
prime (1 to 1’; 2 to 2’ etc) by a continuous line or to
provide an argument that it is impossible.
1
3
1’
2
4
3’
4’
2’
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A warm up
Below is a schematic of a printed circuit board (PCB).
The problem is to either connect each number to its
prime (1 to 1’; 2 to 2’ etc) by a continuous line or to
provide an argument that it is impossible.
How about if we renumber as below?
3
1
3’
4
2
1’
2’
4’
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A warm up
The key is to first connect …
1
3
1’
2
4
3’
4’
2’
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A warm up
… and the rest is easy….
1
3
1’
2
4
3’
4’
2’
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Unusual puzzle…
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Unusual puzzle…
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Unusual puzzle…
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Unusual puzzle…
After cutting off the
right-bottom sub-square,
divide the remaining
shape into 4 identical
pieces…
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Unusual puzzle…
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Unusual puzzle…
Divide the
bottom-right
sub-square into
5 identical
pieces…
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Unusual puzzle…
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Pattern recognition
Our ability to recognize patterns is of utmost
importance. If we can identify a pattern, then we can
build a model to find a solution (e.g. to find the next
occurrence of a symbol, number, action, or event).
Marilyn Burns, in her book I Hate Mathematics,
wrote: “The password of mathematics is pattern.”
Indeed, in many branches of mathematics we search
for patterns which allow some generalizations. But we
search for patterns everywhere; we even recognize
patterns in words:
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Pattern recognition
Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it
deosn’t mttaer in waht oredr the ltteers in a wrod are,
the olny iprmoetnt tihng is taht the frist and lsat ltteer
be at the rghit pclae. The rset can be a toatl mses and
you can sitll raed it wouthit porbelm. Tihs is bcuseae
the huamn mnid deos not raed ervey lteter by istlef,
but the wrod as a wlohe.
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Pattern recognition
Try to memorize the following coding of numbers 1 – 9:
1
2
3
4
5
6
7
8
9
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Pattern recognition
1
2
3
4
5
6
7
8
9
What is the coded version of
the number 3875 ?
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Pattern recognition
The “pattern”:
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
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Pattern recognition
1
2
3
4
5
6
7
8
9
What is the coded version of
the number 3875 ?
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Pattern recognition
• the first symbol is “different” as the only one with red border
• the second symbol is “different” as the only one without green shade
• the third symbol is “different” as the only one which is not square
• the fourth symbol is “different” as the only one with red dot
• the fifth symbol is most different as the only one which is not unique
with respect to some property…
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An observation
For the following 23 design activities, which do you
view as the six most important?
Abstracting
Generating alternatives
Making trade-offs
Synthesizing
Brainstorming
Goal setting
Modeling
Testing
Building
Identifying constraints
Planning
Understanding the problem
Communicating
Imagining
Prototyping
Using creativity
Decomposing
Iterating
Seeking information
Visualizing
Evaluating
Making decisions
Sketching
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An observation
For the following 23 design activities, which do you
view as the six most important?
Abstracting
Generating alternatives
Making trade-offs
Synthesizing
Brainstorming
Goal setting
Modeling
Testing
Building
Identifying constraints
Planning
Understanding the problem
Communicating
Imagining
Prototyping
Using creativity
Decomposing
Iterating
Seeking information
Visualizing
Evaluating
Making decisions
Sketching
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An observation
For the following 23 design activities, which do you
view as the six most important?
Abstracting
Generating alternatives
Making trade-offs
Synthesizing
Brainstorming
Goal setting
Modeling
Testing
Building
Identifying constraints
Planning
Understanding the problem
Communicating
Imagining
Prototyping
Using creativity
Decomposing
Iterating
Seeking information
Visualizing
Evaluating
Making decisions
Sketching
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Outline of the talk
• Getting started
•
•
•
•
Icebreakers
Effective teaching approaches
Understanding the problem
Some techniques (e.g. simplification,
performing a Gedanken: What if? So what?)
• Summary
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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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A quiz
Here is a quiz that can be used on the first day of
class. Consider handling it out before any
instructions. As soon as the students are in their
seats, your first words to them can be: “Here is the
first quiz; you have 5 minutes…”
Here it is:
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A quiz
Instructions: Check one of the two boxes below.
BOX A
BOX B
Grading: If all the students in the class check box A, all the
students will receive a score of 20/20 on this quiz. If at least one
student checks box B, all the students who checked box A will
receive a score of 10/20 and all the students who checked box B
will receive a score of 15/20.
BONUS: Guess the number of people in the class that will check
box number B. [No points for this; just for fun.]
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A game
There are n dots on a plane (flat surface).
There are two players, A and B, who move
alternatively; A moves first. The rules of the
game are the same for both players: at each
move they can connect two points, but they
cannot connect points which were already
directly connected to each other or connect a
point with itself.
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A game
The winner is the one who makes the dots
connected (i.e. there is a path between any
two dots, however, not every two dots have
to be connected directly). What is the
winning strategy for player A, if such
exists?
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Outline of the talk
• Getting started
•
•
•
•
Icebreakers
Effective teaching approaches
Understanding the problem
Some techniques (e.g. simplification,
performing a Gedanken: What if? So what?)
• Summary
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Effective teaching approaches
Key questions:
1. What knowledge and skills should students learn?
2. How can I facilitate their learning?
3. How do I determine how well they have learned
via formative and summative feedback?
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Effective teaching approaches
• show the student how you solve a problem (think aloud
exercise)
• get the student to articulate his/her thought process
• present a variety of puzzle types
• give frequent homework assignments
• try “flipping the classroom” approach
• match the tone and your language to the puzzle
• select carefully puzzles for use in the classroom (purpose,
groups, hints, class discussion)
• consider online activities
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Assignments
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
What is this assignment supposed to achieve in terms of learning outcomes?
What level of knowledge do your students have in this area?
What level of knowledge do you wish your students to have after this
assignment?
How much time do you have for this?
Have your students seen a similar or related puzzle before?
Have you demonstrated this in class?
Do the students have a reason to do this assignment?
Do the students understand the value of this assignment?
Do you understand how to do this assignment?
Have you tested it?
Does the assignment require additional equipment or resources?
Do you need to provide a guide?
Do you need to provide a rubric?
Do you have a list of hints?
What is the impact if a student cannot solve any of the problems?
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Effective teaching approaches
• assessment
–
–
–
–
grading (A, B vs. 8, 7)
uniform rubric
rewarding effort and outcome
feedback is essential
• increasing and maintaining confidence
• peer teaching
• final exam questions (note the difference between in-class
puzzles, homework, exams, great-challenges)
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Outline of the talk
• Getting started
•
•
•
•
Icebreakers
Effective teaching approaches
Understanding the problem
Some techniques (e.g. simplification,
performing a Gedanken: What if? So what?)
• Summary
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Understanding the problem
A farmer has:
20 pigs,
40 cows, and
60 horses.
How many horses does he have, if he
calls the cows horses?
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Understanding the problem
A farmer has:
20 pigs,
40 cows, and
60 horses.
How many horses does he have, if he
calls the cows horses?
Answer: The farmer has 60 horses…
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Critical Thinking
“Arguments for banning guns are mostly myths,
and what we need now is not more laws, but more
law enforcement. One myth is that most murderers
are ordinary, law-abiding citizens who kill a
relative or acquaintance in a moment of anger
only because a gun was available. In fact, every
study of homicide shows the overwhelming
majority of murderers are career criminals, people
with lifelong histories of violence. The typical
murderer has a prior criminal history averaging at
least six years, with four major felony arrests.”
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Critical Thinking
“Arguments for banning guns are mostly myths,
and what we need now is not more laws, but more
law enforcement. One myth is that most murderers
are ordinary, law-abiding citizens who kill a
relative or acquaintance in a moment of anger
only because a gun was available. In fact, every
study of homicide shows the overwhelming
majority of murderers are career criminals, people
with lifelong histories of violence. The typical
murderer has a prior criminal history averaging at
least six years, with four major felony arrests.”
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Intuition
Judy is thirty-three, unmarried, and quite
assertive. A magna cum laude graduate, she
majored in political science in college and was
deeply involved in campus social affairs,
especially in anti-discrimination and anti-nuclear
issues. Which statement is more probable:
– Judy works as a bank teller.
– Judy works as a bank teller and is active in the
feminist movement?
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Solving a problem
Solving a complicated problem usually
involves two distinct steps, the first is a
preparation step and the second is the solving
step. The initial preparation step involves
understanding the problem, framing the
problem, drawing a diagram, and building a
model. The second step is solving the actual
problem or solving a model of the problem.
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Solving a problem
Solving a complicated problem usually
involves two distinct steps, the first is a
preparation step and the second is the solving
step. The initial preparation step involves
understanding the problem, framing the
problem, drawing a diagram, and building a
model. The second step is solving the actual
problem or solving a model of the problem.
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Mr. and Mrs. Smith puzzle
Mr. and Mrs. Smith invited 4 married couples for dinner. When
everyone arrived, they greeted each other by a handshake.
These were the rules:
– no one shakes his/her own hand
– no handshake between married couple
– no pair does it twice
– there are pairs which did not shake their hands
Mr. Smith asked all people in the room, how many times
did they shake their hands, and all answers he got were
different.
How many hands did Mrs. Smith shake?
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Let’s try to visualize:
Mr. Smith
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…and all answers were different
4
5
3
6
2
7
1
8
0
Mr. Smith
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Let’s try to visualize (again)
4
5
3
6
2
7
1
8
0
Mr. Smith
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Let’s try to visualize (again)
4
5
3
6
2
7
1
8
0
Mr. Smith
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Let’s try to visualize (further)
4
5
3
6
2
7
1
8
0
Mr. Smith
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...and further...
4
5
3
6
2
7
1
8
0
Mr. Smith
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...and further...
4
5
3
6
2
7
1
8
0
Mr. Smith
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Now we know everything...
Mrs. Smith
4
5
3
6
2
7
1
8
0
Mr. Smith
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Outline of the talk
• Getting started
•
•
•
•
Icebreakers
Effective teaching approaches
Understanding the problem
Some techniques (e.g. simplification,
performing a Gedanken: What if? So what?)
• Summary
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A crab
A crafty crab has constructed seven blind holes in
the configuration shown. Every day at noon he
changes from one hole to an adjacent hole and
every night at midnight a raccoon comes and looks
in a single hole in the hopes of finding the crab.
After five consecutive days of failure, the raccoon
takes a day off in an attempt to discover a holechecking protocol that will guarantee he will catch
the crab. In other words, the raccoon is going to do
some System 2 thinking to discover a sequence of
hole-checking that will guarantee he catches the
crab even if the crab knows the hole-checking
protocol. What sequence of hole-checking will
inevitably trap the crab?
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Juice & water
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Juice & water
After one exchange, what amount is greater? Amount of water in
the juice or amount of juice in the water?
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Juice & water: what if…
What if we separate (conceptually) juice and water?
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Outline of the talk
• Getting started
•
•
•
•
Icebreakers
Effective teaching approaches
Understanding the problem
Some techniques (e.g. simplification,
performing a Gedanken: What if? So what?)
• Summary
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Summary
Puzzle-based learning illustrates some important aspects
of problem solving: Be patient and persistent. Remember
that every problem solver gets stumped now and then.
We must have patience to investigate and understand the
problem. Perhaps we might guess at the solution but,
even if we are wrong, this guess might lead to an
interesting discovery. However, we must understand the
problem before going on to do anything else. Sometimes
the answers seem so clear that they just have to be right,
but they are not and, of course, we should not then rush
to the solution.
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Summary
From our experience, the course evolves and matures
each time it is taught. Lesson plans are developed,
adjusted and even dropped altogether. Materials are
developed, puzzles are purchased, cabinets get filled
and the program expands. If your experience mirrors
ours you may observed the following (a) Puzzle-based
Learning is one of the most fun courses to teach and
(b) Puzzle-based Learning is one of the most fun and
memorable courses for students to take.
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Thank you!
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