Dindin Abdul Muiz Lidinillah, S.Si., S.E., M.Pd.
Elementary School Teacher Education Program
Education University of Indonesia – Tasikmalaya Campus

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)

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

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.

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
Moderately
Structured
Problems
Ill Structured
Problems
Closed Problem Open Problem

Routine Problems
Real life problem
Involve one mathematical operation
Basic skills and sequence steps
Need understanding, retrieve information, choose the operation and algorithms
Solving through story telling and relate it to real situation
Non Routine Problems
Complex problem
Require more than one mathematical operation
Using critical and creative thinking Skills
Need understanding, retrieve information, choose the operation and algorithms
Various strategies and methods to solve it

Type of
Problem
Well Structured
Problems
Moderately
Structured
Problems
Ill Structured
Problems
Definition Problems that always use the same step-bystep solution.
Problems that require varying strategies and adaptations 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

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 welldefined and agreed-upon solution; There may not be a fully satisfactory solution at all.

Needed information often statement must be Gathered

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

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.
Hanlie Murray, Alwyn Olivier and Piet Human, Learning Through Problem
Solving, University of Stellenbosch, South Africa

•
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
Hanlie Murray, Alwyn Olivier and Piet Human, Learning Through Problem
Solving, University of Stellenbosch, South Africa

Teaching Problem Solving
(as a Goal or a Basic Skill)
Teaching Mathematics Through (via)
Problem Solving (as a Process of Learning or Thinking Process)

”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.”


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

Consider extreme case

Generalize
Sickafus, Heuristics for Technical Problem Solving, 2004

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

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


Reading
 Understanding

Analisys
 Exploration

Planning

Implementation

Verification


Analyzing and understanding a problem
 Designing and planning a solution

Exploring solution to difficult problem
 Verifying a solution

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

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


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


Pattern recognition

Working backwards

Guess and test

Simulation or experimentation

Reduction and expansion
 Organized listing/ exhaustive listing

Logical deduction
 Divide and conquer


Estimate

Use computational skills

Use algebraic skills

Use geometric skills

Use a calculator when appropriate


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

Reys, et.al. (1989)
 Observation

Inventory and Checklist
 Paper and Pencil Test
Krulik dan Rudnik (1995)

Observation

Metacognitive Journal

Summary Paragraph

Test

Portofolio

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