Developed by ERLC/ARPDC as a result of a
grant from Alberta Education to support
implementation
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john.scammell@ebsb.ca
Twitter: @thescamdog
Blog: www.thescamdog.wordpress.com
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This session is aimed at High School Math
teachers and is intended to help support the
implementation of the revised program of
studies.
Participants will see examples of how
students can learn THROUGH problem
solving.
ERLC – Learning Through Problem Solving
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In the polling response section, please enter
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A if you are a High School Teacher
B if you are a Junior High School Teacher
C if you are an Elementary School Teacher
D if you are a consultant or administrator
E if you are some combination of the above
ERLC – Learning Through Problem Solving
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In the polling response section, please enter
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A if you are from Edmonton
B if you are from the greater Edmonton area
C if you are from elsewhere in Alberta
D if you are outside of Alberta
ERLC – Learning Through Problem Solving
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There are seven mathematical processes in
the front matter of the revised program of
studies. These processes are to permeate
teaching and learning.
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Communication (C)
Connections (CN)
Mental Math and Estimation (ME)
Problem Solving (PS)
Reasoning (R)
Technology (T)
Visualization (V)
ERLC – Learning Through Problem Solving
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“The mathematical processes are the content,
and the outcomes are the context.”
Dr. Peter Liljedahl, Simon Fraser University – Secondary Math
Education
ERLC – Learning Through Problem Solving
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The description of the Problem Solving
process contains the phrase:
“Learning through problem solving should be
the focus of mathematics at all grade levels.”
ERLC – Learning Through Problem Solving
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We learned FOR problem solving.
ERLC – Learning Through Problem Solving
The top of Ken’s new desk is
3  1 m wide and 5 m
long. Find the area of the
top of his desk.
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ERLC – Learning Through Problem Solving
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We learned ABOUT problem solving.
ERLC – Learning Through Problem Solving
1. Understand the Problem
2. Develop and
3. Carry Out a Plan
4. Look Back
ERLC – Learning Through Problem Solving
Learning through problem solving should be
the focus of mathematics at all grade levels.
- Alberta Program of Studies
ERLC – Learning Through Problem Solving
An example of learning
THROUGH
problem solving.
http://www.amaranthpublishing.com/MozartDiceGame.htm
ERLC – Learning Through Problem Solving
Mozart's Dice Game
Please roll your dice and enter
the results.
Mozart's Dice Game
http://sunsite.univie.ac.at/Mozart/dice/collaborate.cgi?tables=yes
ERLC – Learning Through Problem Solving
Mozart's Dice Game
If this is an activity used in
class, the next thing I would
tell the students is that I am
incredibly confident that no
one else has ever heard the
minuet they just played.
What questions might they
have? Can we distill it down
to one single question?
ERLC – Learning Through Problem Solving
Mozart's Dice Game
Possible questions:
Has the minuet we just composed ever been heard
before?
How many minuets could be created using the
parameters of the problem?
How long would it take you to listen to them all,
assuming that each takes about 30 seconds?
ERLC – Learning Through Problem Solving
Mozart's Dice Game
Debrief. Discuss.
What mathematical concept is addressed?
Could students arrive at this concept on their own?
Where will they struggle?
How will you scaffold for those who struggle?
ERLC – Learning Through Problem Solving
What makes for a good problem
for learning through problem
solving experience?
ERLC – Learning Through Problem Solving
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The problem is given at the beginning of the learning rather
than at the end.
The problem is non-routine. The students can not solve it
immediately. But,
Every student has an entry.
The problem is engaging and compelling so students will
persevere.
The problem invites multiple methods of solution.
The problem fosters discussion and debate.
The problem is at an appropriate level for the audience.
The best ones allow the students to come up with their own
problem to solve.
A compilation from a variety of sources –
Marilyn Burns, Marian Small, John Van de
Walle, Dan Meyer, and me.
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“The definition of a good mathematical
problem is the mathematics it generates
rather than the problem itself.”
“It's fine to work on any problem, so long as
it generates interesting mathematics along
the way - even if you don't solve it at the end
of the day.”
“I loved doing problems in school, I'd take
them home and make up new ones of my
own.”
Another Example of Learning
Through Problem Solving
Laura has trained her pet rabbit,
Jack, to hop up a flight of 7
steps. Jack can hop up one
step at a time or two steps at a
time. How many different ways
can Jack hop up the flight of
seven steps?
ERLC – Learning Through Problem Solving
Rabbit Problem Debrief
Debrief. Discuss.
What mathematical concept is addressed?
Could students arrive at this concept on their own?
Where will they struggle?
How will you scaffold for those who struggle?
ERLC – Learning Through Problem Solving
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What does your classroom look like while
students are working on problems?
How do you debrief the problems with
students?
Do you tell them the right answer?
Where do you find these problems?
When a problem is not
particularly compelling,
how do we engage
students?
ERLC – Learning Through Problem Solving
An example of learning
THROUGH
problem solving.
From Dan Meyer’s Blog (dy/dan)
http://blog.mrmeyer.com/?p=5983
From worst to better…
ERLC – Learning Through Problem Solving
Worst
•The problem claims to represent the
real-world but its illustration is only
clip-art or a line drawing.
•The problem specifies the exact
method of its own solution, usually in
a series of substeps labeled "a, b, c,
d.“
•The problem only gives information
that the student will use in the
solution.
ERLC – Learning Through Problem Solving
Bad
The same as worst except:
The real-world problem
presents itself as it exists
in the real world.
ERLC – Learning Through Problem Solving
Good
The same as bad except:
The problem reveals no information about
itself — no measurements, especially —
forcing the student to decide for herself
what information is relevant to the
solution.
The problem doesn't hint at its own
solution method with sub-steps. The
student can develop that solution socially,
in conversation with her teacher or her
classmates.
ERLC – Learning Through Problem Solving
An example of learning
THROUGH
problem solving.
http://vimeo.com/9552977
… to Best
ERLC – Learning Through Problem Solving
Best
The same as good except:
The problem hangs itself on a single, concise, intuitive question, one that
any student can answer, regardless of mathematical ability. The teacher
solicits guesses and records them publicly, investing the students in the
outcome of the exercise, and refers back to them later, perhaps introducing
the concept of percent error.
The solution to the problem isn't read from an answer key. Instead, it's
observed by the class together in a second multimedia artifact. The class
compares the answer derived from their theoretical model to this practical
answer. This is scary. The class will almost certainly be wrong but the
conversation about sources of error should be embraced, not feared.
ERLC – Learning Through Problem Solving
http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html
ERLC – Learning Through Problem Solving
Math 10 Applied – Angle of Elevation and
Depression
1. Students measured heights of their eyes.
2. Students found one object in the classroom that
was the same height as their eyes.
3. Students moved around the room and listed
objects they would have to look up to see
(elevation) and down to see (depression).
ERLC – Learning Through Problem Solving
Math 10 Applied – Angle of Elevation and
Depression
4. I took the class to the atrium and asked them,
in teams of three, to estimate the height of the
second floor railing above the main floor.
5. We returned to the classroom and recorded
each team’s guess in a chart on the SMART ™
Board.
6. We made clinometers, and each team had a
clinometer and a tape measure.
ERLC – Learning Through Problem Solving
Math 10 Applied – Angle of Elevation and
Depression
7. They returned to the atrium to calculate the
actual height of the atrium.
8. Each team recorded their calculated height next
to their estimate.
9. We revealed the actual height, and celebrated
the best estimate and the best calculation.
ERLC – Learning Through Problem Solving
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A quick question that promotes thought and
exploration is valuable.
ERLC – Learning Through Problem Solving
From Andrew Shores at http://andysunknownquantity.blogspot.com/
Looking at this illustration, what questions do you want to answer?
Can the grasshopper start on any particular number (not counting shaded
boxes) and be safe by taking hops of 4?
What if it took hops of 5? 6? 7?
What if the spider webs were on perfect cubes? What if they were on
triangular numbers? Fibonacci numbers? etc.
John Scammell can be reached at:
E-mail
john.scammell@epsb.ca
Twitter
@thescamdog
Blog
www.thescamdog.wordpress.com
ERLC – Learning Through Problem Solving