CHAPTER 5 Process of Doing Mathematics Tina Rye Sloan To accompany Helping Children Learn Math9e, Reys et al. ©2009 John Wiley & Sons  

Focus Questions What five processes are identified in Principles and Standards for School Mathematics as key to an active vision of learning and doing mathematics? 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 elementary children be encouraged to communicate their mathematical thinking? What connections are important to aid elementary children in learning mathematics? What are three major goals for representation as a process in elementary school mathematics? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

NCTM Process Standards Problem Solving Reasoning and Proof Communication Connections Representations Principles and Standards for School Mathematics (NCTM, 2000) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

NCTM Process Standards Instructional programs from pre-kindergarten through grade 12 should enable students to: Problem Solving 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 Master 5-1: Process Standards Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Problem Solving Activity Rolling the Dice . Players take turns rolling the dice. The first player rolls the two dice and finds their sum. (For example, if 2 and 2 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

NCTM Process Standards Instructional programs from pre-kindergarten through grade 12 should enable students to: Reasoning and Proof 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 Master 5-1: Process Standards Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Reasoning and Proof Figure 5-4 Pictures of odds and evens can help students justify why the sum of two odd numbers is always even. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Mathematical Reasoning Leads to Mathematical Memory Built on Relationships Figure 5-6 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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

NCTM Process Standards Instructional programs from pre-kindergarten through grade 12 should enable students to: Communication 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 Master 5-1: Process Standards Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Communication Figure 5-2 Fourth-grade students’ writing about playing the dice game Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

NCTM Process Standards Instructional programs from pre-kindergarten through grade 12 should enable students to: Connections 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 Master 5-1: Process Standards Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Connections between Symbols and Conceptual Understanding Figure 5-10 Arranging dots in square patterns connects the number 1, 4, 9 and 16 to their reference as square numbers Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

NCTM Process Standards Instructional programs from pre-kindergarten through grade 12 should enable students to: Representations 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 Master 5-1: Process Standards Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

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

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, nonroutine problems Reasoning and Proof -encourage conjectures and explanation of ideas Master 5-2: Supporting Mathematics Learning with the Process Standards Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

How Can Teachers Support Mathematics Learning with the Process Standards? (cont’d) 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 Master 5-2: Supporting Mathematics Learning with the Process Standards Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Five Ways to Represent Mathematical Ideas Pictures Written symbols Spoken language Relevant situations Mani-pulatives Master 5-3: Five Ways to Represent Mathematical Ideas Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

3-5 Big Ideas for Teaching Mathematics Recommendations Specific Methods/ Reasons Why Materials Beneficial Master 5-4: Big Ideas for Teaching Mathematics Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Activity These are pentominoes Master 5-5: Pentominoes Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

These are not pentominoes: Activity (cont’d) These are not pentominoes: Master 5-5: Pentominoes Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Activity (cont’d) Write a definition of a pentomino. How many different pentominoes are there? Illustrate each of these. 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? Master 5-5: Pentominoes Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009