Chapter 5Chapter 5

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

 

Guiding Questions

• How is teaching mathematics through problem solving different from simply teaching students to solve problems?

• For young children, what does mathematical reasoning involve and how does it help them make sense of mathematical knowledge and relationships?

• How can you encourage elementary children to communicate their mathematical thinking?

• What connections are important to help elementary children learn mathematics?

• What are three major goals for representation as a process in elementary school mathematics?

NCTM Process Standards

• Problem Solving• Reasoning and Proof• Communication• Connections• Representations• Principles and Standards for School

Mathematics (NCTM, 2000)

NCTM Process Standard:Problem Solving

Instructional programs from pre-kindergarten through grade 12 should enable students to: – build new mathematical knowledge through

problem solving – solve problems that arise in mathematics and in

other contexts – apply and adapt a variety of appropriate strategies

to solve problems– monitor and reflect on the process of mathematical

problem solving

Problem Solving ActivityRolling the DicePlayers place 12 counters on a 1-12 game board. Players take turns rolling the dice. The first player rolls the two dice and finds their sum. (For example, if 2 and 3 are rolled, the sum is 5.) Each player may remove one counter from his or her 5 space. Even if there is more than one counter on that space, only one may be removed. If there are no counters on that space, no counters may be removed from any space. The next player rolls the two dice and finds their sum (e.g., 4 + 4 = 8). Each player now removes on counter from his/her 8 space, and so on. The goal of the game is to empty your board. The first player with no counters left on his/her board is the winner.

NCTM Process Standard:Reasoning and Proof

• Instructional programs from kindergarten through grade 12 should enable students to: – Recognize reasoning and proofs as fundamental

aspects of mathematics – Make and investigate mathematical conjectures– Develop and evaluate mathematical arguments

and proofs – Select and use various types of reasoning and

methods of proof

Reasoning and Proof

Figure 5-5 Pictures of odds and evens can help students justify why the sum of two odd numbers is always even.

Mathematical Reasoning Leads to Mathematical Memory Built on Relationships

Figure 5-7 A 10-by-11 rectangle built with two staircases from 1 to 10 can help you remember the formula for the sum of a series of numbers

NCTM Process Standard: Communication

• Instructional programs from kindergarten through grade 12 should enable students to: – organize and consolidate their mathematical thinking

through communication– communicate their mathematical thinking coherently and

clearly to peers, teachers, and others– analyze and evaluate the mathematical thinking and

strategies of others– use the language of mathematics to express mathematical

ideas precisely

NCTM Process Standard: Connections

• Instructional programs from kindergarten through grade 12 should enable students to: – recognize and use connections among

mathematical ideas– understand how mathematical ideas interconnect

and build on one another to produce a coherent whole

– recognize and apply mathematics in contexts outside of mathematics

Connections between Symbols and Conceptual Understanding

Arranging dots in square patterns connects the number 1, 4, 9 and 16 to their reference as square numbers

NCTM Process Standard: Representations

• Instructional programs from kindergarten through grade 12 should enable students to: – create and use representations to organize,

record and communicate mathematical ideas– select, apply and translate among mathematical

representations to solve problems– use representations to model and interpret

physical, social and mathematical phenomena

Which graph best represents the height of students in the class?

Note that the circle graph does not order the heights as clearly as either the bar or line graph. The line graph incorrectly gives the impression that there are children of heights between the measurement points.

Representation

How Can Teachers Support Mathematics Learning with the

Process Standards?

• For each standard, list specific instructional practices you plan to include in your classroom.

• Problem Solving -encourage sense making, non-routine problems

• Reasoning and Proof -encourage conjectures and explanation of ideas

How Can Teachers Support Mathematics Learning with the

Process Standards? (cont.)• Communication-work individually and in small

groups, use whole class discussion, and writing

• Connections-connect to real life and other subjects

• Representations-provide a variety of materials, have students use objects, symbols, pictures and look for various representations/solutions

Five Representations of Mathematical Ideas

(Source: adapted from Lesh and Landau, 1983, p.271.)

Activity

These are pentominoes

Activity (cont.)

These are NOT pentominoes.

Activity (cont.)

• Write a definition of a pentomino.• How many different pentominoes are there? • Illustrate each of them.• What is the area of each pentomino?• What is the perimeter of each pentomino?• What can you conclude about shapes with the

same area? Do these always have the same perimeter? Why or why not?

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.