PROCESS STANDARDS FOR MATHEMATICS PROBLEM SOLVING The Purpose

PROCESS STANDARDS FOR MATHEMATICS

PROBLEM SOLVING • The Purpose of the Problem Solving Approach • The problem solving approach fosters the development of mathematical thinking. • Characteristics • The focus is on the processes of mathematical inquiry including problem-solving, reasoning, and communicating. • Students may or may not have received formal instruction on the content involved for the problem presented. • Students have an active role and teacher guides rather than directs instruction. • • Instructional Strategies Use of flexible grouping patterns Students discuss the problem, solution strategies and find a solution. Content instruction is completed incidentally as needed.

REASONING AND PROOF • Reasoning is about making generalizations and requires analytical thinking, flexibility, creativity, and practical thinking. It builds upon the student's mathematical ideas and helps students to connect to their own sense-making abilities. A fundamental component of children's reasoning is metacognition or the complex ability to think about one's own thinking. A mathematical proof is a formal way of communicating particular kinds of reasoning and justification which uses various methods such as representation, concrete materials or models, oral communication, and symbolic writing.

According to NCTM's Curriculum and Evaluation Standards, mathematical reasoning is demonstrated when students: • Use discovery, trial & error, and work backwards to solve a problem Make and test conjectures - a conjecture is an informed guess • Search for evidence to prove or disprove their conjectures • Create inductive and deductive arguments • Look for patterns to arrive at generalizations • Use spatial and logical reasoning

The teacher's role is to facilitate student exploration and encourage reasoning about mathematical relationships. The following are some strategies that can promote reasoning. • 1) Questioning Techniques - Ask open-ended questions. What is a pattern? Why does this work? Does this always work? Can you find an exception? Does anyone think the answer is different & why do you think so? How do you know this is true? • 2) "What Number Makes Sense? " (Krulik & Rudnick, 1999) - Rewrite a word problem with the solution but remove all the numbers. Students fill in the missing numbers and explain why they chose each number. • 3) "What's Missing? " (Krulik & Rudnick, 1999) - Present word problems with too little information and have students determine what information they need to solve the problem. • http: //www. learner. org/courses/teachingmath/gra des 3_5/session_04/section_04_d. html

COMMUNICATION • Communication is an essential part of mathematics education and includes listening, discussing, reading and writing. It is a means for supporting students' learning as they act out a situation, draw, use objects, give verbal accounts and explanations, use diagrams, write, and use mathematical symbols and numeration.

IDEAS For Communication • Students. . . *are encouraged to ask questions, paraphrase, or clarify ideas expressed by other classmates. *need to feel free to express ideas in a safe environment. *express their thinking using supportive vocabulary that reflects the "why, because, if, then" of their reasoning. *are asked to justify responses or solutions. • Teachers. . . • formulate questions that engage students and encourage discussion. * require students to use mathematical terms by having them available for student use. * model the "think out loud" strategy to share own thinking in response to the problem. *model self assessment: Does the answer make sense?

CONNECTIONS • Students should be able to: • 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 math. • 1 st – fractions, dec. , percents • 2 nd – symbolism 3 squared (3 x 3) • 3 rd – real world

REPRESENTATION • The term representation refers to both the process and to product of showing a mathematical concept or relationship in some form or the form itself. Representations provide students with the tools to expand their mathematical thinking and understanding. Some forms of representation are diagrams, pictures, graphical displays, symbolic expressions, and a variety of new forms associated with electronic technology.

According to NCTM's Principles and Standards for School Mathematics the Representation Standard enables students to: • create and use representations to organize, record, and communicate mathematical ideas • select, apply, and translate mathematical representations to solve problems • use representations to model and interpret physical, social, and mathematical phenomena

INSTRUCTIONAL TECHNIQUES • Teachers can begin to address this standard by asking students to use concrete materials (model) to prove their solutions and explain their thinking. To begin writing about their solutions, it may also be helpful for students to trace or pictorially display their results using concrete materials. Students should also be given lots of opportunities to collect, record, interpret and display data.

NOW IT IS YOUR TURN • Students will create a poster, bulletin board or some sort of representation that illustrates each of the five process standards and how they will be utilized in the classroom. • Students may create a bulletin board that will be used in the classroom at a later date.