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Welcome to Module 2

Teaching ThroughProblem Solving

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Getting Started

  Module 2 focuses on the importance of problem solving, and on teaching through problem solving. Module 3 will explore teaching about problem solving.

Problem solving is a goal of learning mathematics and also a means of doing so. Problem solving allows students to explore, develop, and apply their understanding of mathematical concepts.

What do you think?

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Key Messages

The primary goal of problem solving is making sense of mathematics.

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Key Messages

Teachers have a responsibility to promote the experience of problem-solving strategies in their classrooms, and to foster in their students a positive attitude towards problem solving.

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Key Messages

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.

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Key Messages

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.

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Key Messages

Problems serve two main purposes: to explore, develop, and apply conceptual understanding of a mathematical concept (teaching through problem solving); andto guide students through the development of inquiry or problem-solving processes and strategies (teaching about problem solving).

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Key Messages

The teacher’s role as facilitator is crucial in the provision of an effective problem-solving experience.

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Let’s do a problem together…

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, also of equal strength. The result is a tie.Round 2: On one side is a fire-breathing dragon. It is pitted against two of the fairy godmothers and one handsome frog. Again, it’s a draw.Round 3: The fire-breathing dragon and three fairy godmothers are on one side and the four handsome frogs are on the other.Who will win the third round?

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Working on It

How did you feel?

Complete BLM 2.1, then share your thoughts with a partner.

 

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Think-Pair-Share

Think-Pair-Share is a cooperative learning strategy in which participants think about a question (issue, situation, idea, etc.), then share their thoughts with a partner. It is a simple strategy that can be easily incorporated into almost any form of instruction. 

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The Importance of Problem Solving

Children naturally and intuitively solve problems in their everyday lives. They seek solutions to sharing toys with friends and build elaborate structures in their play. Teachers who use problem solving as the focus of their mathematics class help their students to develop and expand these intuitive strategies.

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The Importance of Problem Solving

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.

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Routine and Non-Routine Problems

Routine problems: One- or two-step simple word problems

Non-routine

problems:

Problems that require mathematicalanalysis and reasoning;

many non-routine problems can be solved in more than one way,

and may have more than one solution.

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Attitude is Everything!

Since beliefs influence behaviour, students need to believe that they are capable of solving problems.

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When children become effective problem solvers, they reap the benefits of:

understanding and practising skills in context;

being able to make hypotheses, experiment, draw conclusions, and use trial and error;

using flexible representations to help them solve problems;

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When children become effective problem solvers, they reap the benefits of:

wondering about and questioning situations in their world;

persevering in tackling new challenges;

formulating and testing their own explanations;

communicating their explanations and listening to others’ explanations;

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When children become effective problem solvers, they reap the benefits of:

participating in open-ended experiences that have a clear goal but a variety of solution paths;

developing strategies that can be applied in new situations;

collaborating with others to develop new strategies; and…

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When children become effective problem solvers, they reap the benefits of:

finding enjoyment in mathematics!

I can count by 5’s while I’m skipping!

5, 10, 15, 20, 25, 30, 35, …

I can put my face in the water and count to

17! Watch!

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Problem solving should permeate our mathematics teaching.

Problem solving should be the mainstay of mathematical teaching and should be used as the basis for mathematical instruction.

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Teaching Through Problem Solving

Inside-Outside Circle

Form 2 concentric circles, one circle of people inside the other, with the inside circle facing the outside circle.

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Teaching Through Problem Solving

Explain to your partner what you think teaching through problem solving means.

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Teaching Through Problem Solving

Outside circle, rotate counter-clockwise past two people.

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Teaching Through Problem Solving

Describe to your new partner what instructional implications you think teaching through problem solving would have.

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Teaching Through Problem Solving

Outside circle, rotate counter-clockwise past two people.

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Teaching Through Problem Solving

Identify one advantage of and one obstacle to designing a program that teaches through problem solving.

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Teaching Through Problem Solving

Teaching Through Problem Solving

Getting Started

(preparing for learning)

Working on It

(facilitating learning)

Reflecting and Connecting

(reflecting on, extending,

and consolidating learning)

The Three-Part Problem-Solving Lesson

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Getting Started(preparing for learning)

The Getting Started part of the lesson allows the teacher to engage all students as they explore a new mathematical concept in a familiar context.

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Getting Started (preparing for learning)

The teacher presents the problem and any pertinent information needed to solve the problem. Initial instructions for completion of the task are clearly outlined. Before the students are sent off to work, the teacher checks to ensure that they understand the problem.

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Working on It (facilitating learning)

providing situations in which students try their own strategies;

The teacher facilitates learning by:

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Working on It (facilitating learning)

offering guidance and redirection through questioning;

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Working on It (facilitating learning)

providing assistance to those who require it, and allowing the other students to solve the problem independently.

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Reflecting and Connecting(reflecting on, extending, and consolidating learning)

The teacher leads a discussion in which students share their strategies and consider different solutions to the problem. Time is allocated for the sharing of several examples. This discussion validates the various strategies used, and consolidates learning for students.

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Reflecting and Connecting(reflecting on, extending, and consolidating learning)

The discussion, questioning, and sharing that occurs allows students to make connections with their own thinking and to internalize a deeper understanding of mathematical concepts.

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Problem-Solving Vignette

Read the vignette on pp. 5.9- 5.15 of the guide.In your section of the place mat, record important ideas about teaching through problem solving as illustrated in the vignette.Summarize your group’s ideas in the centre on your place mat.

Session1a.34

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The Teacher’s Role

A Jigsaw Activity1. Providing appropriate and

challenging problems (pp.5.16– 5.19)

2. Supporting and extending learning (pp. 5.19–5.20)

3. Encouraging and accepting students’ strategies (p. 5.21)

4. Questioning and prompting students (pp. 5.21–5.23)

5. Using think-alouds (p. 5.23)

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Reflecting and Connecting

Suppose…A new teacher has recently read the Guide to Effective Instruction in Mathematics, Kindergarten to Grade 3, 2004, and is eager to begin to teach using a problem-solving approach.

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Reflecting and Connecting

The teacher 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?

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In Your Classroom…

Find a closed problem from a textbook and revise it to make it into a richer, more open-ended problem.Try the problem with your class.(Use pp. 5.16–5.19 for advice on modifying problems.)