MODULE 2 – TEACHING THROUGH PROBLEM
SOLVING
Modules 2 and 3 focus on Chapter 5 – Problem Solving. In Module 2,
participants learn about the importance of problem solving and of teaching
through problem solving. In Module 3, they will explore teaching about problem
solving.
Materials: BLM 2.1, BLM 2.2, chart paper, markers
Looking Back
(based on Module 1)
Have participants share the strategies they used in
their classrooms to address the principles of
mathematics instruction. Encourage participants to
discuss the impact of the strategies they used.
10 min
#1 - 2
Getting Started
Introduction
Problem solving is both a goal of learning mathematics and a means of doing so.
Problem solving allows students to explore, develop, and apply their
understanding of mathematical concepts.
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Module 2 – Teaching Through Problem Solving
KEY MESSAGES
 The primary goal of problem solving is
making sense of mathematics.
5 min
#3 - 8
 Teachers have a responsibility to promote
the experience of problem-solving processes in their classrooms, and to
foster in their students a positive attitude towards problem solving.
 Problem solving is not only a goal of learning mathematics but also a
major means of doing so. Problem solving should be the mainstay of
mathematical teaching and should be used daily as a basis for
mathematical instruction.
 The problem-solving processes that Kindergarten students will use look
very different from those that Grade 3 students use. Initially, students will
rely on intuition. With exposure, experience, and shared learning, they will
formalize an effective approach to solving problems by developing a
repertoire of problem-solving strategies that they can use flexibly when
faced with new problem-solving situations.
 Problems serve two main purposes: to explore, develop, and apply
understanding of a mathematical concept (teaching through problem
solving), and to guide students through the development of inquiry or
problem-solving processes and strategies (teaching about problem
solving).
 The teacher’s role as facilitator is crucial in the provision of an effective
problem-solving experience.
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Module 2 – Teaching Through Problem Solving
Working on It
20 min
Have participants work in groups of three or four to
solve the problem below.
#9 - 11
FROGS, DRAGONS, AND FAIRY GODMOTHERS
Who will win the third round of tug-of-war?
Round 1: On one side are four handsome frogs, each of equal strength. On the
other side are five fairy godmothers, each of equal strength. The result is dead
even.
Round 2: On one side is a fire-breathing dragon. It is pitted against two of the
fairy godmothers and one handsome frog. Again, the result is a draw.
Round 3: The fire-breathing dragon and three of the fairy godmothers are on one
side, and the four handsome frogs are on the other. Who will win the third round?
(See a possible strategy on p. 4.)
Adapted from Van de Walle, Elementary and Middle School Mathematics:
Teaching Developmentally, Fourth Edition, Pearson Longman, 2001, p. 22
After the activity, ask volunteers to explain their strategies and solutions to the
whole group.
Use a think-pair-share strategy to encourage participants to reflect on and
discuss their feelings about the problem-solving experience. Ask participants to
complete BLM 2.1 individually, and then to share their thoughts with a partner.
Think-Pair-Share is a cooperative learning strategy in which participants
privately think about a question (issue, situation, idea, etc.), and then share their
thoughts with a partner.
As a large group, discuss the kinds of emotions that students might experience
during problem-solving activities in the mathematics program.
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Module 2 – Teaching Through Problem Solving
The Importance of Problem Solving

10 min
Children naturally and intuitively solve problems
in their everyday lives.
#12 - 20

With exposure, experience and shared learning,
children will develop a repertoire of problemsolving strategies that they can use flexibly when faced with new problemsolving situations.

Children should learn to solve both routine and non-routine problems.

Since beliefs influence behaviour, students need to believe that they are
capable of solving problems.

When children become effective problem solvers, they reap many benefits
(e.g., they learn to persevere, collaborate, communicate).

Problem solving should be the mainstay of mathematical teaching and should
be used as the basis for mathematical instruction.
Possible Strategy for the ' Frogs, Dragons, and Fairy Godmothers'
Problem (p. 3)
Represent the three rounds symbolically:
Round 1: 4 F = 5 G
Round 2: 1 D = 2 G + 1 F
Round 3: 1 D + 3 G ? 4 F
Restate Round 3 as “1 D + 3 G ? 5 G”, since 4 F and 5 G were equally matched
in Round 1.
Eliminate 3 G from both sides of 1 D + 3 G ? 5 G: 1 D ? 2G
The dragon will win in Round 3, because in Round 2, the dragon was equally
matched with two fairy godmothers and one frog.
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Module 2 – Teaching Through Problem Solving
Use an Inside-Outside-Circle strategy to discuss
the following questions:
10 min
 What do you think 'teaching through problem
solving' means?
#21 - 26
 What are some instructional implications of
teaching through problem solving?
 What is an advantage of teaching through problem solving? What is an
obstacle to teaching through problem solving?
Inside-Outside Circle is a cooperative learning structure that allows participants
to discuss questions, issues, or topics with different partners.
Step 1 – Participants form two circles, one inside the other. Both circles have the
same number of people. Each participant in the inside circle faces a
partner in the outside circle.
Step 2 – Participants discuss a question with their partner.
Step 3 – The leader allows a minute for participants to discuss the question, and
then instructs the outside circle to rotate counter-clockwise past two
people.
Step 4 – Participants discuss the next question with their new partner.
The Three-Part Problem-Solving Lesson
10 min
The three-part problem-solving lesson involves the
following phases:
#27 - 34
 Getting Started (preparing for learning)
 Working on It (facilitating learning)
 Reflecting and Connecting (reflecting on, extending, and consolidating learning)
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Module 2 – Teaching Through Problem Solving
Problem-Solving Vignette
Arrange participants in groups of four. Have
participants read the vignette on pp. 5.9–5.15 of the
Guide, and discuss the important ideas about
teaching through problem solving that are illustrated
in the vignette. Ask participants to record their ideas
using a place-mat strategy.
20 min
#35
Place Mat is a cooperative learning strategy in which members of a small group
brainstorm ideas about a topic individually, and then summarize the group
members’ common points.
Step 1 – Participants create a place mat by drawing a circle in the centre of a
large rectangular sheet of paper (chart paper) and by dividing the area
around the circle into sections, one for each group member.
Step 2 – Each group member records personal thoughts about the topic or
question in his or her own outer section of the mat.
Step 3 – Participants share their ideas with the group. Common points are
summarized and recorded in the centre area of the place mat.
Lead a whole-group discussion by asking groups to share ideas they recorded in
the centre areas of their place mats.
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Module 2 – Teaching Through Problem Solving
The Teacher’s Role
Use a jigsaw strategy (see Module 1, p. 4) to
20 min
explore the teacher’s role in teaching through
problem solving. Have participants form home
groups of five, and number themselves 1 to 5 to
#36
identify their expert group number. Home-group
members join their expert groups to investigate the
following sections of the guide. Participants can record information on BLM 2.2.
Expert Group 1 –
 Providing Appropriate and Challenging Problems (pp. 5.16–5.19)
Expert Group 2 –
 Supporting and Extending Learning (pp. 5.19–5.20)
Expert Group 3 –
 Encouraging and Accepting Students’ Strategies (p. 5.21)
Expert Group 4 –
 Questioning and Prompting Students (pp. 5.21–5.23)
Expert Group 5 –
 Using Think-Alouds (p. 23)
Participants return to their home group to report on their discoveries and to
complete other sections of BLM 2.2.
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Module 2 – Teaching Through Problem Solving
Reflecting and Connecting
10 min
Discussion
#37 - 39
Ask participants to discuss the following situation
with their table group:
A new teacher has recently read the Guide to Effective Instruction in
Mathematics, Kindergarten to Grade 3, 2004, and is eager to use a problemsolving approach in her teaching. The teacher plans and teaches a problemsolving lesson and comes back to discuss it with you. She is upset and
disappointed that the lesson did not work and feels that the more traditional
approach to teaching mathematics would be a better idea. How would you
respond?
Ask participants to summarize their discussion for the whole group.
In Your Classroom
Ask participants to find a closed problem (i.e., a problem with one solution that
can be solved in one way) from a textbook, and revise it to make it into a richer,
more open-ended problem. Participants can refer to pp. 5.16–5.19 in the Guide
for advice on modifying problems. Invite participants to try the revised problem
with their students.
For Next Time
Ask participants to read 'Teaching about Problem Solving' and 'Observing and
Assessing Students as They Solve Problems' in Chapter 5 – Problem Solving
(pp. 5.24–5.39) of the Guide.
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