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 problems • monitor and reflect on the process of mathematical problem solving
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
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
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
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
Problem Types • Problem-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 learned • Non-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 x2x4x5 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 3square. The sum of the numbers in every row is 12.The sum of the numbers in every column is 12. ___ ___ ___ ___ ___ ___ ___ ___ ___
Try this. Is this a problem for you? • Use 1,2,3,4,5,6,7,8,9 ___ ___ ___ + ___ ___ ___ _________________ ___+___+___= 18
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
Teaching Problem Solving Effectively • 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. Hembree and Marsh 1993, Kroll & Miller 1993,Suydam, 1982
Teaching Problem Solving Effectively • 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. Hembree and Marsh 1993, Kroll & Miller 1993,Suydam, 1982
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.
Planning for Problem Solving • 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.
Planning for Problem Solving • 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 problems • Apply a variety of strategies to solve problems • Solve problems • Verify and interpret results • Generalize solutions
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