Introduction to (Mathematical) Problem Solving

Introduction to (Mathematical) Problem Solving

Dindin Abdul Muiz Lidinillah, S.Si., S.E., M.Pd.

Elementary School Teacher Education ProgramEducation University of Indonesia – Tasikmalaya Campus

What Is Problem ?

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 ?

You (personally) have a problem if the following four conditions are satisfied:1.You have a clearly defined given initial

situation.2.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 ?

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 using some of your own resources, such as your knowledge, skills, time, and energy, to achieve the desired final goal.

What Is The Kind of (Math) Problem ?

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


Routine Non Routine

Well Structured Problems

Ill Structured Problems

Moderately StructuredProblems

Closed Problem Open Problem


Routine Problems Non Routine Problems

Real life problem Complex problemInvolve one mathematical operation

Require more than one mathematical operation

Basic skills and sequence steps

Using critical and creative thinking Skills

Need understanding, retrieveinformation, choose the operation and algorithms

Need understanding, retrieveinformation, choose the operation and algorithms

Solving through story telling and relate it to real situation

Various strategies and methods to solve it


Type of Problem

Well Structured Problems

Moderately StructuredProblems

Ill Structured Problems

Definition Problems that always use the same step-bystep solution.

Problems that require varying strategies andadaptations to fit particular Contexts

Problems with vague and unclear goals. Solution strategies least Constrained

Characteristics

Solution strategy is usually predictable

Convergent (one right answer)

All starting information is usually part of the problem statement

Often more than one acceptable solution strategy.

Convergent (one right answer).

Needed information often must be gathered.

Solution is not well defined or predictable. Multiple perspectives, goals, and solutions.

There is no single well-defined and agreed-upon solution; There may not be a fully satisfactory solution at all.

Needed information often must be Gathered

What Is Problem Solving ?

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)


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.


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


Problem Solving as a Goal

Problem Solving as a Process

Problem Solving as a Basic Skill

What Is The Goals of Problem Solving ?

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 higheraims about 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,which can 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.


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


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.


Some of Issues about Teaching PS ?

• 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


Some of Issues about Teaching PS ?

Teaching Mathematics Through (via) Problem Solving (as a Process of Learning

or Thinking Process)

Teaching Problem Solving (as a Goal or a Basic Skill)

Definition of Heuristic

”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.”

Types of Heuristic in Math

Search for PatternDraw a FigureFormulate an equivalent problemModify the problemChoose effective notationExploit symmetryDivide into casesWork backwardArgue by contradictionCheck for parityConsider extreme caseGeneralizeSickafus, Heuristics for Technical Problem Solving, 2004

Four Steps Polya’s Model

Understanding The Problem (SEE)Devising a Plan (PLAN)Carring Out The Plan (DO)Looking Back (CHECK)

Schoenfeld’s Model

• Reading• Analisys• Exploration• Planning/Implementation• Verification

Artzt & Armour-Thomas’s Model

ReadingUnderstandingAnalisysExplorationPlanningImplementationVerification

Wickelgren’s Model

Analyzing and understanding a problemDesigning and planning a solutionExploring solution to difficult problem Verifying a solution

Krulik dan Rudnik’s Model

Read and Think Explore and Plan Select a Strategy Find an Answer Reflect and Extend

Read and Think

Identify the facts Identify the question Visualize the situation Describe the setting Restate the action

Explore and Plan

Organize the informationIs there sufficient information ?Is there to much information ?Draw a diagram or construct a modelMake a chart, a table, a graph, or a

drawing

Select a Strategy

Pattern recognitionWorking backwardsGuess and testSimulation or experimentationReduction and expansionOrganized listing/ exhaustive listingLogical deductionDivide and conquer

Find an Answer

EstimateUse computational skillsUse algebraic skillsUse geometric skillsUse a calculator when appropriate

Reflect and Extend

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 solutionWhat if… ?Extend to either : 1) generalization; 2) a

mathematical conseptDiscuss the solutionsCreate interesting variations on the

original problem

How to Asses Math Problem Solving ?

Reys, et.al. (1989) Observation Inventory and Checklist Paper and Pencil Test

Krulik dan Rudnik (1995) Observation Metacognitive Journal Summary Paragraph Test Portofolio

Teaching Models that Related with PS

Mathematical Investigation Mathematical Exsploration Problem Posing Scaffolding Problem Based Learning Open Ended Problem Metacognitive

LETS TO DISCUSS

Documents

Introduction to (Mathematical) Problem Solving