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

2. 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

3. 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

4. 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

5. 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., = 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

6. 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

7. 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

8. 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

9. 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

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

11. 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

12. Connections between Symbols and Conceptual Understanding
Figure 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

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

20. 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

21. 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