process of doing mathematics chapter 5 tina rye sloan to accompany helping children learn math9e,...

Process of Doing Mathematics

CHAPTER 5

Tina Rye SloanTo 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)


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



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.


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

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.

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




• 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

Communication


Figure 5-2 Fourth-grade students’ writing about playing the dice game




•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

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




•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

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, nonroutine problems

•Reasoning and Proof -encourage conjectures and explanation of ideas


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


Five Ways to Represent Mathematical Ideas


3-5 Big Ideas for Teaching Mathematics

Recommendations Specific Methods/ Reasons Why

Materials Beneficial


These are pentominoes


Activity

These are not pentominoes:


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?


Activity (cont’d)

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