Process Standards for School Mathematics Principles and Standards for School Mathematics National Council of Teachers of Mathematics, 2000

Process Standards for School Mathematics

Principles and Standards for School Mathematics

National Council of Teachers of Mathematics, 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 problemsmonitor and reflect on the process of mathematical problem solving


Reasoning and ProofRecognize 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


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


Connectionsrecognize 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


Representationscreate 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

Problem TypesProblem-involves a situation in which the solution route is not immediately obvious Exercise-a situation in which the solution route is obvious Routine problem-the application of a mathematical procedure in the same way it was learnedNon-routine problem-the choice of mathematical procedures is not obvious

Which terms are synonyms?

Suppose students have been multiplying whole numbers.

56 77 48 287 793

x2 x4 x5 x7 x8

Are these problems?

15 rows of stamps. 8 stamps in each row. How many stamps?

24 packs of baseball cards.

8 cards in a package.

How many baseball cards?

Try this. Is this a problem for you?

Use the numerals 0,1,2,3,4,5,6,7,8 to form a 3 by 3 square. The sum of the numbers in every row is 12. The sum of the numbers in every column is 12.

___ ___ ___ ___ ___ ___ ___ ___ ___

Use 1,2,3,4,5,6,7,8,9 ___ ___ ___ + ___ ___ ___

_________________

___+___+___= 18

Try this. Is this a problem for you?

Teaching Problem Solving Effectively

Instruction should build on what children already know.

Engaging students in problem solving should not be postponed until after they have “mastered” computational skills.

Problem-solving strategies can be specifically taught.

Hembree and Marsh 1993, Kroll & Miller 1993, Suydam, 1982


No one strategy is optimal for solving all problems.

Teaching a variety of strategies provides children with a repertoire from which they can draw.

Students need to be faced with problems in which the way to solve them is not apparent, and they need to be encouraged to test many alternative approaches.



Children’s problem-solving achievements are related to their developmental level. Thus, they need problems at appropriate levels of difficulty.

Factors which contribute to students’ difficulties with problem solving include knowledge, beliefs and affects, control, and sociocultural factors.


Planning for Problem Solving

Consider including problems that: contain superfluous or insufficient information.

involve estimation.

require students to make choices about the degree of accuracy required.

involve practical applications of mathematics to consumer or business situations.


Consider including problems that: require students to conceptualize very large or very small numbers.

are based on students' interests or events in their environment.

involve logic, reasoning, testing of conjectures, and reasonableness of information.


Consider including problems that: are multi-step, or require the use of more than one strategy to attain a solution.

require decision making as a result of the outcome.

Problem-Solving Strategies

Act It Out

Make a Drawing or Diagram

Look for a Pattern

Construct a Table

Identify All Possibilities

Problem-Solving Strategies

Guess and Check

Work Backward

Write an Open Sentence

Solve a Simpler or Similar Problem

Change Your Point of View

The Horse Problem

A man buys a horse for $60, sells it for $70, buys it back for $80, and sells it for $90. How much does the man make or lose in the horse trading business?

Assessing Problem Solving The assessment of students' ability to use mathematics in solving problems should provide evidence that they can:

Formulate problemsApply a variety of strategies to solve problemsSolve problemsVerify and interpret resultsGeneralize solutions

These are pentominoes

These are not pentominoes:

Write a definition of a pentomino.

How many different pentominoes are there?

Interviewer, "Paco had 13 cookies. He ate 6 of them. How many cookies does Paco have left?“ Meredith: Fifth Month of Kindergarten

Interviewer, "Carla has 7 candies. How many more candies does she need so that she will have 11 candies to

share with her friends?“ Allan: Fifth Month of First Grade

Interviewer, "Nineteen children are going to the circus. Five children can ride in each car. How many cars will be

needed to get all 19 children to the circus?" Clint: Fifth Month of Second Grade

Interviewer, "Robin has 3 packages of gum. There are 6 pieces of gum in each package. How many pieces of gum

does Robin have altogether?“ Bill: Third Month of Third Grade

Interviewer, "Tad had 15 guppies. He put 3 guppies in each jar. How many jars did Tad put guppies in?“

Darla: Third Month of Third Grade

Interviewer, "Tad had 15 guppies. He put 3 guppies in each jar. How many jars did Tad put guppies in?“

Ellen: Third Month of Third Grade

Interviewer, "Nineteen children are taking a mini-bus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit three

to a seat, and how many can sit two to a seat?“ Allison: Fifth Month of Fourth Grade

Documents

Process Standards for School Mathematics Principles and Standards for School Mathematics National Council of Teachers of Mathematics, 2000