Mathematical goals

This lesson unit is intended to help students to make sense of standard algorithms for addition, subtraction, multiplication, and division of positive integers. In particular it should assist them in the following areas:

• Improving conceptual understanding of why and how the algorithms work.
• Developing procedural fluency in carrying out the algorithms.
• Becoming more able to spot unreasonably sized answers and to debug errors in procedures.

Introduction

This lesson is structured in the following way:

• Before the lesson, students work individually on an assessment task, Getting it Wrong, designed to reveal their current understanding and difficulties. You review their solutions and create questions for students to answer in order to improve their work.
• The main lesson begins with a whole-class introduction, in which students critique systematic errors in standard algorithms. Students try to explain why what is being done does not work.
• Then students work in pairs or threes on a collaborative task to find errors in some sample student work. They are encouraged to go beyond ‘correcting’ the answers to conjecturing why the student might have made the error and explaining why the method used does not (always) work.
• In a whole-class discussion, students describe what they have learned from the task.
• Finally, students receive your comments on the assessment task and use these to attempt a similar task, approaching it with insights that they have gained from the lesson.

Materials required

• Each student will need a copy of the assessment tasks Getting it Wrong and Getting it Wrong (revisited), a mini-whiteboard, pen, and eraser.
• Each small group of students will need two cards from either Card Set (1) or Card Set (2) (already cut up) and some blank paper to work on.

Time needed

20 minutes before the lesson, an 85-minute lesson (or two shorter lessons), and 20 minutes in a follow-up lesson. Timings given are approximate and will depend on the needs of your class.