Chapter 5 To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons...
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Transcript of Chapter 5 To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons...
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.