CHAPTER 2 Helping Children Learn Mathematics with Understanding Tina Rye Sloan To accompany Helping Children Learn Math9e, Reys et al. ©2009 John Wiley & Sons  

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 behaviorist 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? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

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 Master 2-1: Strategies that Support All Learners Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Procedural Knowledge-skillful use of mathematical rules or algorithms 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 Adding is putting together Master 2-2: Procedural and Conceptual Knowledge Add then subtract… Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

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. Master 2-2: Procedural and Conceptual Knowledge Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

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. Master 2-2: Procedural and Conceptual Knowledge …to find 25% I must… Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

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… Master 2-2: Procedural and Conceptual Knowledge Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Two Theories of Learning: Behaviorism and Constructivism HOW DO CHILDREN LEARN MATHEMATICS? Two Theories of Learning: Behaviorism and Constructivism Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Behaviorism Behavior can be shaped by reinforcement of drill and practice. Specific skills need to be learned in a fixed order. Master 2-3: Behaviorism and Constructivism Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Behaviorism (cont.) Clear objectives help students and teachers. Edward L. Thorndike B.F. Skinner Robert Gagne Master 2-3: Behaviorism and Constructivism Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Constructivism Learners actively create or invent (construct) their own knowledge.  Students create (construct) new mathematical knowledge by reflecting on their physical and mental actions. Master 2-3: Behaviorism and Constructivism Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

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 Master 2-3: Behaviorism and Constructivism Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

HOW CAN WE HELP CHILDREN MAKE 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. Master 2-4: How Children Learn Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

HOW CAN WE HELP CHILDREN MAKE SENSE OF MATHEMATICS? Learners are actively involved in the learning process. Recommendation #2: Teachers should actively involve students. Master 2-4: How Children Learn Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

HOW CAN WE HELP CHILDREN MAKE SENSE OF MATHEMATICS? Learning proceeds from the concrete to abstract. Recommendation #3: Teachers should move learning from concrete to abstract. Master 2-4: How Children Learn Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

HOW CAN WE HELP CHILDREN MAKE SENSE OF MATHEMATICS? Learners need opportunities for talking and communicating their ideas with others. Recommendation #4: Teachers should use communication to encourage understanding. Master 2-4: How Children Learn Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

The Staircase Problem Examine these staircases: Describe in words a relationship (formula) involving the sum of the first 4 counting numbers suggested. Master 2-5: The Staircase Problem Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

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? Master 2-5: The Staircase Problem Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

The Staircase Problem p. 3 Examine these staircases: The sum of the first n counting numbers is: 1 + 2 + 3 + 4 + . . . . + n = _________ Master 2-5: The Staircase Problem Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009