Helping Children LearnMathematics with

Understanding

Chapter 2Chapter 2

To accompany Helping Children Learn Math Cdn Ed, Reys et al.©2010 John Wiley & Sons Canada Ltd.

 

Focus Questions

1. How can we create a supportive classroom climate for the diverse learners in our classroom?

2. What is procedural knowledge and how is it different from conceptual knowledge?

3. How do behaviourist approaches to learning differ from constructivist approaches to learning?

4. What are four recommendations for helping children make sense of mathematics based on what is known about how children learn mathematics?

How Can We Support the Diverse Learners in our Classrooms?

• Create a positive learning environment• Avoid negative experiences that increase

anxiety• Establish clear expectations• Treat all students as equally likely to have • Aptitude for mathematics• Help students improve their ability to retain • Mathematical knowledge and skills

How Can We Help Children Acquire Both Procedural Knowledge and Conceptual

Knowledge?• Procedural Knowledge-skillful use of

mathematical rules or algorithms• Conceptual Knowledge-understanding

meaning of mathematical concepts

Add then subtract…

Adding is putting together

Which is it?

• To divide 23 candies among four friends, Steve knows each must receive an equal amount and there may be some left.

• To take 23 divided by 4, Steve knows to take 5 x 4 and subtract the result from 23.

Which is it?

• Jill knows that to find 25% of a price she can cut the price in half, then half again to find one-fourth.

• Jill knows that to find 25% of a price she can multiply the price by .25.

…to find 25% I must…

Which is it?

• Nancy knows that to find the area of a rectangle, she must find out how much space it covers.

• Nancy knows that to find the area of a rectangle, she must multiply the length times the width.

I must multiply…

How Do Children Learn Mathematics?

Two Theories of Learning:

Behaviourism and Constructivism

Behaviourism

• Behaviour can be shaped by reinforcement of drill and practice.

• Specific skills need to be learned in a fixed order.

Behaviourism (cont.)

• Clear objectives help students and teachers.

• Edward L. Thorndike• B.F. Skinner • Robert Gagne

Constructivism

• Learners actively create or invent (construct) their own knowledge.

• Students create (construct) new mathematical knowledge by reflecting on their physical and mental actions.

Constructivism (cont.)

• Learning reflects a social process in which children engage in dialogue and discussion with themselves as well as others as they develop intellectually.

– William Brownell, – Jean Piaget, – Jerome Bruner, – Zoltan Dienes

How Can We Help ChildrenMake Sense of Mathematics?

• Several characteristics and stages of thinking exist; children progress through stages as they mature.

Recommendation #1: Teachers should teach to the developmental characteristics of students.

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How Can We Help ChildrenMake Sense of Mathematics?

• Learners are actively involved in the learning process.

Recommendation #2: Teachers should actively involve students.

How Can We Help ChildrenMake Sense of Mathematics?

• Learning proceeds from the concrete to abstract.

Recommendation #3: Teachers should move learning from concrete to abstract.

How Can We Help ChildrenMake Sense of Mathematics?

• Learners need opportunities for talking and communicating their ideas with others.

Recommendation #4: Teachers should use communication to encourage understanding.

The Staircase Problem

• Examine these staircases:

• Describe in words a relationship (formula) involving the sum of the first 4 counting numbers suggested.

The Staircase Problem p. 2

• Examine these staircases:

• Describe in words a relationship (formula) suggested. How many counting numbers are involved? What is their sum?

The Staircase Problem p. 3

• Examine these staircases:

• The sum of the first n counting numbers is: 1 +

2 + 3 + 4 + . . . . + n = _________

Copyright

Copyright © 2010 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (The Canadian Copyright Licensing Agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.