M ATHEMATICAL P ROBLEM S OLVING What is it? Why encourage it? How is teaching like a problem-solving endeavor?

W HAT T YPES OF Q UESTIONS A RE A SKED IN M ATH C LASSROOMS ? recognition/recall: a situation that can be resolved by recalling specific facts from memory drill/exercise: a situation that involves following a step-by- step procedure or algorithm simple translation/application: traditional textbook word problems which involve translating the words into a single equation, the format of which has typically been taught directly complex translation/application: similar to a simple translation, but it involves at least two steps

nonroutine/process problem: solutions require the use of thinking processes; a situation that does not have an apparent solution path available–there is no set format available for finding a solution applied problem: real-world, or at least realistic, situations where solutions require the use of mathematical skills, facts, concepts, and procedures. open-ended problem: a situation in which the student is often required to identify interesting questions to pursue and/or to identify...not only is there no apparent solution path available, often there is no direct question to purse

S O... WHICH OF THESE TYPES OF QUESTIONS ARE “ PROBLEMS ?”

W E FIRST NEED TO ASK : W HAT IS A PROBLEM ? A problem is a difficult question; a matter of inquiry, discussion, or thought; a question that exercises the mind..... and

I T IS A SITUATION OR TASK FOR WHICH : the person confronting the task wants or needs to find a solution. the person has no readily available procedure for finding the solution. the person makes an attempt to find the solution.

S O... NOT ALL CATEGORIES OF QUESTIONS ARE PROBLEMS... W HEN IS THE MATH OR TASK “P ROBLEMATIC ?”

W HAT I T M EANS FOR M ATH TO BE “P ROBLEMATIC ” “Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students.” “Problem Solving as a Basis for Reform in Curriculum and Instructions: The Case of Mathematics” Hiebert, et al. Educational Researcher May, 1996 (p. 12)

W HY ENCOURAGE PROBLEM SOLVING ?

T HE V ALUES OF T EACHING W ITH P ROBLEMS Places the focus of attention on ideas and sense making rather than on following the directions of the teacher. Develops the belief in students that: they are capable of doing mathematics. mathematics makes sense. Provides ongoing assessment data so that: instructional decisions can be made. students can be helped to succeed. parents can be informed.

Develops “mathematical power.” All five + 1 process standards are likely to be employed. Allows entry points for a wide range of students. Good problems have multiple entry points. Engages students so that there are fewer discipline problems. It’s just a lot of fun... for both teachers and students!!!!!

A N I DEA W ORTH C ONSIDERING : Most mathematical ideas can be taught via problem solving. That is— Tasks or problems are posed to engage students in thinking about and developing the intended mathematical ideas.

W HAT IS G EORGE P OLYA ’ S P ROBLEM S OLVING F RAMEWORK ? 1. Understand the Problem  Involves not only applying the skills necessary for literary reading, but determining what is being asked.

2. Devise a Plan  Involves finding or devising a strategy to help in solving the problem and answering the question.

3. Carry Out the Plan  Involves attempting to solve the problem with some chosen strategy, recording thinking to keep an accurate record of work.

4. Look Back  Involves interpreting the solution in terms of the original problem... Does it make sense? Is it reasonable?

5. Look Forward  Involves asking the “next question,” generalizing the solution, or determining another method of finding the solution.

H OW IS TEACHING MATHEMATICS LIKE A “P ROBLEM -S OLVING E NDEAVOR ?” Understand goals for teaching—be clear about what you want to accomplish. Key question: What mathematics do I want the children to understand and develop?

Plan the tasks—select problems, explorations, or activities that will most likely require children to wrestle meaningfully and thoughtfully with the math you identified as important. Key question: How can this problem (or activity) be used to help promote reflective thinking about the math I want children to learn?

Implement your plan—facilitate activities and focus appropriately on the needs and strengths of your individual students. Key question: What “teacher moves” and “math talk” might I use to help develop understanding?

Look back—through active listening in a student-centered classroom, you can learn what students know and how they understand the ideas being discussed. Key question: Based on what I learn about what my students understand, what direction do I need to go in my future planning for instruction?

P ROBLEM -S OLVING S TRATEGIES Guess and check Look for a pattern Make a list Solve a simpler version Draw a picture

Use symmetry Make a table or chart Do a simulation Look for a formula Consider cases

Model the situation Work backward Write an open sentence Use logical reasoning Consider all possibilities Change your point of view

S O HOW DO “ TRADITIONAL PROBLEMS ” DIFFER FROM THOSE THAT ARE TYPICALLY MORE AUTHENTIC — MORE CLOSELY RELATED TO REAL LIFE ?

D IFFERENCES B ETWEEN T RADITIONAL W ORD P ROBLEMS AND M ANY R EAL -L IFE P ROBLEMS Typical Textbook Problems The problem is given. All the information you need to solve the problem is given. There is always enough information to solve the problem. Typical Authentic Problems Often, you have to figure out what the problem really is. You have to determine the information needed to solve the problem. Sometimes you will find that there is not enough info to solve the problem.

There is typically no extraneous information. The answer is in the back of the book... There is usually a right or best way to solve the problem. Sometimes there is too much information--must decide what is needed. You, or your team, decides whether your answer is valid. There are usually many different ways to solve the problem. Typical Textbook Word Problems Typical Authentic Problems

G OALS FOR P ROBLEM S OLVING Strategy and Process Goals Develop problem analysis skills Develop and be able to select strategies Justify solutions Extend and generalize problems

Metacognitive Goals Monitor and regulate actions Determine when to try a new strategy Determine when a solution leads to a correct solution

Attitudinal Goals Gain confidence and belief in abilities Be willing to try and to persevere Enjoy doing mathematics