Embedding Puzzle-based
Learning in STEM Teaching
Colin Thomas
on behalf of the PZBL project team
Aims of Meeting
• Dissemination and discussion of our ideas on
Puzzle-based Learning
• Help with making the draft Guide more useful,
especially with respect to specific STEM
subjects
• Help you use puzzles in your teaching
• Have fun
Overview of Meeting
11:00 to 11:30 am
11:30 am to 12:30 pm
12:30 pm to 1:30 pm
1:30 pm to 2:30 pm
2:30 pm to 3:30 pm
3:30 pm
Puzzle-solving warm-up
Plenary and discussion
Lunch
Group work preparing
STEM specific variants
Reporting back
Refreshments and close
Acknowledgements
Initial funding: Centre for Learning and
Academic Development, UoB
Meeting funding: Higher Education Academy
Matthew Badger: collected puzzles
Chris Sangwin: drafted first version of Guide
Esther Ventura-Medina
Colin Thomas
.. and many puzzle setters and solvers
over several millennia!
Project Aim
To develop new learning resources to enable staff
working in STEM to incorporate puzzle-based
learning in their teaching
We believe embedding puzzles will
• develop students’ general problem-solving
skills
• develop group working and independent
learning skills
• increase students’ motivation to learn
mathematics
Puzzle-based Learning (PzBL)
• long tradition in mathematics
• Michalewicz and Michalewicz (2008) book:
Puzzle-based Learning: Introduction to
critical thinking, mathematics, and
problem solving
• courses at the University of Adelaide
• we wanted to embed PzBL in our modules
What is a Puzzle?
Initial thoughts
• task: any activity given to a student
• (mathematical) exercise: a task that can be
solved by a routine, well-established
technique
– often part of direct instruction and traditional
teaching involving drill and imitation
– usually relates closely to what has just been
taught
• problem: more than an exercise
– often posed in words, especially in Engineering
A large steel cylindrical tank is required to have a
volume of 32 m3 and to use the smallest amount
of steel in its construction. What height will it
have to be to satisfy these conditions?
– needs the student to decide how to proceed, undressing the task and isolating the essentials
• experience of problem solving can make
problems less challenging needs thinking 
Writing down equations from word problems
can pose some serious difficulties
For example:
Write an equation for the following statement:
“There are six times as many students as
professors at this university”.
Clement et al. (1981): of 150 calculus level
students, 37% answered incorrectly and
6S = P accounted for two thirds of the
errors
Fermi Problems
Require estimation and approximation
- useful skill for all STEM students
e.g. to check answers to Engineering design
problems
- orders of magnitude
What is a Puzzle?
• Michalewicz and Michalewicz (2008)
Four criteria for a puzzle:
– generality
– simplicity
– “Eureka” factor
– entertainment factor
What is a Puzzle?
A puzzle is a problem that is perplexing and
can only be solved by applying considerable
ingenuity
?
Solving the puzzle usually results in a
“Eureka” moment and the process of finding
a solution is both frustrating and entertaining
We contend that a puzzle should contain all
the information needed for its solution e.g.
Two squares intersect as
shown in the diagram. The
smaller square has side length
30 cm, the larger 40 cm, and
the top left corner of the
larger square sits at the
centre of the smaller. Find the
area of the intersection of the
two squares.
The angle is not given so we can infer it does
not matter:
Can a problem with no solution be a
valid puzzle?
Below is part of an infinite integer lattice. A
lattice triangle is a triangle where the
coordinates of all vertices are integers.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
What is the size of the smallest equilateral
lattice triangle?
Puzzles for embedding
• We sought puzzles that had both a
conventional and a lateral thinking solution
– much harder than one might think
– can provide great pedagogic value
Two segments are drawn as diagonals on the
sides of a cube so that they meet at a vertex
of the cube. What is the angle between the
segments?
This was set in a class on
vectors. Vectors were
therefore used to get the
answer (60°)!
The red triangle is
equilateral so the angle
is 60°
Messages:
Think first!
If an answer is simpler than one might
expect, why?
Variants
Perhaps we should make our puzzles subject
specific e.g.
As part of her plant layout, as shown in Figure 1, a chemical
engineer has a pipe going from point A on a cubical tank up
to point B, and then across to point C. What angle does the
pipe have to be bent to fit the tank?
As part of his latest engine block design shown in Figure 1,
a mechanical engineer has a pipe going from point A in his
cubical block up to point B, and then across to point C.
What angle does the pipe have to be bent to fit the block?
How not to do variants?
“Each day,” said the demanding boss to the metallurgist, “you
must fill some casting moulds with molten titanium, and you will
continue to do this until all the moulds are full. Moreover, each
day your work will become more strenuous. On each day after the
first, you must fill double the number of moulds that you have so
far filled. For example, if you fill 3 moulds with titanium on the
first day, you will fill 6 on the second, 18 on the third and so on.
Clear?”
“Perfectly clear,” said the metallurgist who summoned his team
and with great skill and dedication, the moulds began to fill. After
a week, a third of the available moulds were full. How long did it
take them to do the job? Prove this mathematically.
Another reason not to do variants?
Alice and Bob take two hours to dig a hole. Bob and Chris take
three hours to dig the hole, while Chris and Alice would take
four hours. How long would they take working together?
There are three construction companies: AcMe, BeMe and
CeMe. Working together, AcMe and BeMe take two days to
erect a building. BeMe and CeMe would take three days to
build a similar building whilst AcMe and CeMe would take four
days. How long would AcMe, BeMe and CeMe take working
together?
Surprising, but not a puzzle?
A railway track is exactly 1 km long. It sits on a piece of ground
that is flat. One day, under intense heat from the sun, the
track expands 1 m in length. Its ends remain fixed to the
ground, so the track bows up to form a circular arc of length
1001 m. At the centre of the arc, how high is the track above
the ground?
What is the educational value of
problems?
•
•
•
•
•
In solving problems a student needs to
take personal responsibility
make choices
develop modelling skills
develop tenacity
practice recognition of cases, reducing
problem situations to exercises
What is the educational value of
puzzles?
We contend that puzzles can provide
• additional challenges
• additional insights
• entertainment
all of which can increase student
engagement and promote independent
learning
• Puzzle-based Learning is a subset of
• Problem-based Learning is a subset of
• Enquiry-based Learning
Puzzling in practice
• Modelling Concepts and Tools
• The Moore Method
Originally the puzzles were bolted on, now
they are embedded e.g.
Alice and Bob take two hours to dig a hole. Bob
and Chris take three hours to dig the hole,
while Chris and Alice would take four hours.
How long would they take working together?
The Moore Method
Chris Sangwin
What are the differences between a problem
and a puzzle?
Should a puzzle contain all the information
needed for its solution?
Can a problem with no solution be a puzzle?
Is surprise enough to make a puzzle?
Should puzzles be embedded or taught in
stand alone modules or either?
Are the most useful puzzles those with lateral
solutions?
How important is it to have subject specific
variants?
Aim of Group Work
• To prepare as many subject-specific variants to
puzzles as possible
• To present one variant to the meeting for
discussion
• To identify any issues there might be in
embedding puzzles in STEM teaching