Chapter 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 Activity Rolling the Dice Players 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., = 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?

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