H3 Mathematics Problem Solving Lecture 1 (What Is the Problem?)

H3 Mathematics Problem Solving Lesson 1: What Is the Problem? Ho Weng Kin Nanyang Technological University 2026 1 / 32

How this deck connects to the book Lesson design principle Slides create the experience; the book consolidates the ideas. ▶Chapter 1:What counts as a genuine mathematical problem? ▶Chapter 1:First experience with the four stages through examples. ▶Chapter 2:Make the stages explicit so that our thinking becomes visible and improvable. Book link:Chapter 1; Chapter 2, Section 1 2 / 32

Lesson objectives By the end of this lesson, you should be able to: 1 distinguish a routine exercise from a genuine problem; 2 describe Pólya’s four problem-solving stages; 3 analyse a problem using the language of unknowns, data, assumptions and required conditions; 4 recognize that problem solving does not take place in a vacuum, but draws on mathematical resources. Book link:Chapter 1, Section 1; Chapter 2, Sections 1–4 3 / 32

A routine task and a genuine problem Question 1 Multiply23451by2. Question 2 Which smallest whole number, when multiplied by2, results in the last digit being shifted to the first digit? ▶Question 1 is routine because a standard algorithm is readily available. ▶Question 2 is a genuine problem if you have not seen it before. Book link:Chapter 1, Section 1 4 / 32

A routine task and a genuine problem Question 1 Multiply23451by2. Question 2 Which smallest whole number, when multiplied by2, results in the last digit being shifted to the first digit? ▶Question 1 is routine because a standard algorithm is readily available. ▶Question 2 is a genuine problem if you have not seen it before. Book link:Chapter 1, Section 1 4 / 32

A routine task and a genuine problem Question 1 Multiply23451by2. Question 2 Which smallest whole number, when multiplied by2, results in the last digit being shifted to the first digit? ▶Question 1 is routine because a standard algorithm is readily available. ▶Question 2 is a genuine problem if you have not seen it before. Book link:Chapter 1, Section 1 4 / 32

What makes Question 2 a problem? Working definition A problem is a task for which there is no readily accessible algorithm that completely determines the method of solution. This working definition is modified from Lester’s original definition in [1]. For problem solving to begin meaningfully, three things matter: 1 there is no immediate procedure; 2 the solver wants to know the answer; 3 some real effort is made. Book link:Chapter 1, “What is a mathematical problem?”; Chapter 1, “Where are we heading to?” 5 / 32

What makes Question 2 a problem? Working definition A problem is a task for which there is no readily accessible algorithm that completely determines the method of solution. This working definition is modified from Lester’s original definition in [1]. For problem solving to begin meaningfully, three things matter: 1 there is no immediate procedure; 2 the solver wants to know the answer; 3 some real effort is made. Book link:Chapter 1, “What is a mathematical problem?”; Chapter 1, “Where are we heading to?” 5 / 32

The first problem-solving trajectory For the digit-shift problem, the book walks us through four recurring phases: Understand the problem Devise a plan Carry out the plan Look back Today we firstexperiencethese stages, then we make them explicit. Book link:Chapter 1, Sections 1.1–1.4; Chapter 2, Section 3 6 / 32

The first problem-solving trajectory For the digit-shift problem, the book walks us through four recurring phases: Understand the problem Devise a plan Carry out the plan Look back Today we firstexperiencethese stages, then we make them explicit. Book link:Chapter 1, Sections 1.1–1.4; Chapter 2, Section 3 6 / 32