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.
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.
Module 2 –Teaching Through Problem Solving
KEY MESSAGES
The primary goal of problem solving is making sense of mathematics.
Teachers have a responsibility to promote the experience of problem-solving processes in their classrooms, and to foster in their studentsa positive attitudetowards 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.
Module 2 – Teaching Through Problem Solving
Working on It
Have participants work in groups of three or four to solve the problem below.
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.
Module 2 –Teaching Through Problem Solving
The Importance of Problem Solving
- Children naturally and intuitively solve problems in their everyday lives.
- With exposure, experience and shared learning, children will develop a repertoire of problem-solving strategies that they can use flexibly when faced with new problem-solving 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 forthe ' 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.
Module 2 – Teaching Through Problem Solving
Use an Inside-Outside-Circle strategy to discuss the following questions:
- What do you think 'teaching through problem solving' means?
- What are some instructional implications of teaching through problem solving?
- What isan 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
The three-part problem-solving lesson involves the following phases:
- Getting Started (preparing for learning)
- Working on It (facilitating learning)
- Reflecting and Connecting (reflecting on, extending, and consolidating learning)
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 discussthe important ideas about teaching through problem solving that are illustrated in the vignette. Ask participants to record their ideas using a place-mat strategy.
Place Matis 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 inhis 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.
Module 2 –Teaching Through Problem Solving
The Teacher’s Role
Use a jigsaw strategy (see Module 1, p. 4) to explore the teacher’s role in teaching through problem solving.Have participants form home groups of five, and number themselves 1 to 5 to 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.
Module 2 –Teaching Through Problem Solving
Reflecting and Connecting
Discussion
Ask participants to discuss the following situation with their table group:
Anew teacher has recently read the Guide to Effective Instruction in Mathematics, Kindergarten to Grade 3, 2004, and is eager to use a problem-solving approach in her teaching. The teacher plans and teaches a problem-solving 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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