Introduction to (Mathematical) Problem Solving

1. Introduction to (Mathematical) Problem Solving Dindin Abdul MuizLidinillah, S.Si., S.E., M.Pd. Elementary School Teacher Education Program Education University of Indonesia – Tasikmalaya Campus

2. A Problem is a situation, quantitative or otherwise, that confront an individuals or groups of individual, that requires resolution, and for which the individual sees no apparent or obvious means or path to obtaining a solution (Stephen Krulik and Jesse A. Rudnick, Teaching Reasoning and Problem Solving in Elementary School, 1995) What Is Problem ?

3. You (personally) have a problem if the following four conditions are satisfied: • You have a clearly defined given initial situation. • You have a clearly defined goal (a desired end situation). Some writers talk about having multiple goals in a problem. However, such a multiple goal situation can be broken down into a number of single goal problems. Dave Moursund, Improving Math Education in Elementary Schools: A Short Book for Teachers, 2005 What Is Problem ?

4. 3. You have a clearly defined set of resources that may be applicable in helping you move from the given initial situation to the desired goal situation. These typically include some of your time, knowledge, and skills. Resources might include money, the Web, and the telephone system. There may be specified limitations on resources, such as rules, regulations, guidelines, and timelines for what you are allowed to do in attempting to solve a particular problem. 4. You have some ownership—you are committed to usingsome of your own resources, such as your knowledge, skills, time, and energy, to achieve the desired finalgoal. What Is Problem ?

5. Mathematical tasks or activities come in a variety of guises: investigations, projects, traditional story sums, real-life problems, abstract problems, puzzles, etc. Were all of these suitable for learning through solving problems, or were some more suitable than others? Hanlie Murray, Alwyn Olivier and Piet Human, Learning Through Problem Solving, University of Stellenbosch, South Africa What Is The Kind of (Math) Problem ?

6. Non Routine Routine Ill Structured Problems Well Structured Problems Moderately Structured Problems Open Problem Closed Problem What Is The Kind of (Math) Problem ?

7. What Is The Kind of (Math) Problem ?

8. What Is The Kind of (Math) Problem ?

9. It (problem solving) is the means by wich an individual uses previously acquired knowledge, skills, and understanding to satisfy the demands of an unfamiliar situation. (Stephen Krulik and Jesse A. Rudnick, Teaching Reasoning and Problem Solving in Elementary School, 1995) What Is Problem Solving ?

10. Problem solving means engaging in a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understandings. Solving problems is not only a goal of learning mathematics but also a major means of doing so. NCTM 2000, Principle and Standards for School Mathematic. Virginia : NCTM. What Is Problem Solving ?

11. Problem-solving : being able to solve mathematical problems occurring in daily life, workplace and in other subject-matters; being able to use athematical language to express, communicate and form Mathematical thinking. (MOE 2000), In Dave Moursund, Improving Math Education in Elementary Schools: A Short Book for Teachers, 2005 What Is Problem Solving ?

12. Problem Solving as a Goal • Problem Solving as a Process • Problem Solving as a Basic Skill What Is Problem Solving ?

13. The Goals of Mathematical Education (Polya, 1969) is a talk that he gave to a group of elementary school teachers. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. For the higheraimsabout which I am now talking are some general tactics of problems—to have the right attitude for problems and to be able to attack all kinds of problems, not only very simple problems,whichcan be solved with the skills of the primary school, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. What Is The Goals of Problem Solving ?

14. But the foundations should be started in the primary school. And so I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems. Polya, George (1969). The goals of mathematical education. In Dave Moursund, Improving Math Education in Elementary Schools: A Short Book for Teachers, 2005 What Is The Goals of Problem Solving ?

15. Learning occurs when students grapple with problems for which they have no routine methods. Problems therefore come before the teaching of the solution method. The teacher should not interfere with the students while they are trying to solve the problem, but students are encouraged to compare their methods with each other, discuss the problem, etc. Hanlie Murray, Alwyn Olivier and Piet Human, Learning Through Problem Solving, University of Stellenbosch, South Africa What Is The Goals of Problem Solving ?

16. The role of the teacher • The classroom culture • Interaction patterns among students • The kind of problem posed • The mathematical structure of the problem • Sustained learning • The type of response elicited from the student • Teacher awareness, understanding and co-operation • Informing the larger community HanlieMurray, Alwyn Olivier and Piet Human, Learning Through Problem Solving, University of Stellenbosch, South Africa Some of Issues about Teaching PS ?

17. Teaching Problem Solving (as a Goal or a Basic Skill) Teaching Mathematics Through (via) Problem Solving (as a Process of Learning or Thinking Process) Some of Issues about Teaching PS ?

18. ”Heuristic will be used here to mean a general suggestion or strategy, independent of any particular topic or subject metter, that helps problem solver approach and understand a problem and efficiently marshal their resources to solve it.” Definition of Heuristic

19. Search for Pattern • Draw a Figure • Formulate an equivalent problem • Modify the problem • Choose effective notation • Exploit symmetry • Divide into cases • Work backward • Argue by contradiction • Check for parity • Considerextreme case • Generalize Sickafus, Heuristics for Technical Problem Solving, 2004 Types of Heuristic in Math

20. Understanding The Problem (SEE) • Devising a Plan (PLAN) • Carring Out The Plan (DO) • Looking Back (CHECK) Four Steps Polya’s Model

21. Reading • Analisys • Exploration • Planning/Implementation • Verification Schoenfeld’s Model

22. Reading • Understanding • Analisys • Exploration • Planning • Implementation • Verification Artzt & Armour-Thomas’s Model

23. Analyzing and understanding a problem • Designing and planning a solution • Exploring solution to difficult problem • Verifying a solution Wickelgren’s Model

24. Read and Think • Explore and Plan • Select a Strategy • Find an Answer • Reflect and Extend KrulikdanRudnik’s Model

25. Identify the facts • Identify the question • Visualize the situation • Describe the setting • Restate the action Read and Think

26. Organize the information • Is there sufficient information ? • Is there to much information ? • Draw a diagram or construct a model • Make a chart, a table, a graph, or a drawing Explore and Plan

27. Pattern recognition • Working backwards • Guess and test • Simulation or experimentation • Reduction and expansion • Organized listing/ exhaustive listing • Logical deduction • Divide and conquer Select a Strategy

28. Estimate • Use computational skills • Use algebraic skills • Use geometric skills • Use a calculator when appropriate Find an Answer

29. Check your answer 1) Is the computation correct ? 2) Is the question answered ? 3) Is the answer reasonable ? 4) How does the answer compare with your estimate ? • Find alternate solution • What if… ? • Extend to either : 1) generalization; 2) a mathematical consept • Discuss the solutions • Create interesting variations on the original problem Reflect and Extend

30. Reys, et.al. (1989) • Observation • Inventoryand Checklist • Paper and Pencil Test Krulik dan Rudnik (1995) • Observation • Metacognitive Journal • Summary Paragraph • Test • Portofolio How to Asses Math Problem Solving ?

31. Mathematical Investigation • Mathematical Exsploration • Problem Posing • Scaffolding • Problem Based Learning • Open Ended Problem • Metacognitive Teaching Models that Related with PS

32. LETS TO DISCUSS