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MODULE 9 – TEACHING BASIC FACTS AND MULTIDIGIT COMPUTATIONS

This module could be extended over two sessions to allow for a deeper exploration of the content.

Materials: BLM 9.1, BLM 9.2, BLM 9.3, BLM 9.4a, BLM 9.4b, chart paper, markers

Looking Back(based on Module 8)

Ask participants to conduct a round-table sharing of their implementation plans for a home connection.

Getting Started

Introduction

Ask participants to calculate 46 + 27 mentally. Ask volunteers to explain the methods they used to find the answer. Record the different methods on chart paper.

Possible Methods: Add the ones, regroup the ten from 13, and add the tens (similar to

traditional paper-and-pencil algorithm). Add the tens to get 60, add the ones to get 13, and add 60 and 13 to get

73. Add 3 to 27 to get 30, subtract 3 from 46 to get 43, and add 30 and 43 to

get 73. Add 4 to 46 to get 50, subtract 4 from 27 to get 23, and add 50 and 23 to

get 73.

Ask table groups to discuss the following questions: Is there one right way to solve this question? Why do people use different methods, and not just the traditional

algorithm, to solve questions like this? What are the implications of teaching basic facts and multidigit

computations?

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Module 9 – Teaching Basic Facts and Multidigit Computations

KEY MESSAGES

Learning the basic facts conceptually involves developing an understanding of the relationships between numbers and how these relationships can be evolved into strategies for doing computations in a meaningful and logical manner.

Providing students with problem-solving contexts that relate to the basic facts will allow them to develop a meaningful understanding of the operations.

Research evidence suggests that the use of conceptual approaches in computation instruction results in improved achievement, good retention, and a reduction in the time students need to master computational skills.

Students’ development of computational sense goes through several stages and is improved by exposure to a range of computational strategies, through guided instruction by the teacher and shared learning opportunities with other students.

Using word problems to introduce, practise, and consolidate the basic facts is one of the most effective strategies teachers can use to help students link the mathematical concepts to the abstract procedures.

Students who can work flexibly with numbers are more likely to develop efficient strategies, accuracy, and a strong foundation for understanding other standard algorithms.

When standard algorithms are being introduced, it is important that students develop an understanding of the operations rather than just memorize rules.

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Working on It

Definitions and Approaches

Basic facts include the addition, subtraction, multiplication, and division of numbers from 0 to 9.

By Grade 3, students are expected to develop proficiency in single-digit addition and subtraction, and in multiplication and division up to the seven times table.

Multidigit computations include all combinations of two or more digits in addition, subtraction, multiplication, and division.

By Grade 3, students are expected to develop efficiency with the addition and subtraction of multidigit numbers (up to three digits) and to use these computations in problem-solving situations.

In the past, the emphasis in teaching was on memorization of the basic number facts, sometimes to the exclusion of establishing a firm conceptual understanding of the underlying number structures.

Learning the basic facts conceptually involves developing an understanding of the relationships between numbers (e.g., 7 is 3 less than 10 and 2 more than 5) and of how these relationships can be developed into strategies for doing the computations in a meaningful and logical manner.

As well as teaching computations using a strategy-and-reasoning approach, it is also important to teach computations in problem-solving contexts. Providing students with problem-solving contexts that relate to the basic facts will allow them to develop a meaningful understanding of the operations.

Games, active learning experiences, and investigations provide students with opportunities to use manipulatives and to interact with their peers as they rehearse basic fact strategies and practise multidigit computations.

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Use a jigsaw strategy (see Module 1, p. 4) to investigate topics related to teaching the basic facts. Organize participants into home groups of six. Ask participants to number themselves from 1 to 3 and to join their expert group. Assign the following topics:

Expert Group 1 – Principles for Teaching the Facts (p. 10.13)

Expert Group 2 – Prior Learning / Developing Computational Sense (pp. 10.14–10.16)

Expert Group 3 – Worksheets / Timed Tests (pp. 10.16–10.18)

Instruct participants to read the material on their assigned topic, and to discuss, as a group, the main points of their reading. These points can be recorded on BLM 9.1.

When participants have finished their summaries, have them return to their home groups to share their main points. Participants can complete other sections of BLM 9.1.

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Addition and Subtraction Facts

Ask participants to find a partner and decide which person will investigate 'Using Models' (p. 10.19) and which person will investigate 'Strategies' (pp. 10.20–10.24).

Instruct participants to read their sections and to record a summary of main points on BLM 9.2, which they will share later with their partner. Inform participants that they will also need to conduct a brief activity with their partner to help him or her better understand the topic.

Participants who are investigating strategies could make a list of strategies along with a brief explanation or example of each one. As an activity, they could model one of the examples with their partner.

Participants can continue to record ideas on BLM 9.2 when partners present their main points.

Consolidating Facts

To help students practise selecting and using strategies for addition and subtraction facts, teachers should try the following approaches:

Use problem solving as the route to practising the facts. Model problems (e.g., using counters) when needed. Recognize that the level of strategy development for recalling the facts is

rarely the same for all students. Use games, repetition of worthwhile activities or songs, and mnemonic

devices to individualize strategy development. Ensure that any drill practice is focused on using strategies and not just on

rote recall. Cluster facts and practice around strategies. Have students make their own strategy list of the facts they find hardest. Help students make connections between the facts (e.g., by using

triangular flashcards).

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Multiplication and Division Facts

Thinking About Multiplication and DivisionIt is important to realize that there are several different ways to think about these operations:

Multiplication can be thought of as repeated addition, as an array, and as a collection of equal groups.

Properties and strategies that help with conceptual understanding of multiplication are:– the identity property (a X 1 is always a);– the zero property (0 X a is always 0);– the commutative property (2 X 3 = 3 X 2);– the distributive property (4 X 6 = 2 X 6 + 2 X 6);– the associative property (5 X 12 is the same as 5 X 6 X 2);– the inverse relationship with division.

Division can be thought of as repeated subtraction, equal partitioning, or sharing.

Properties and strategies that help with conceptual understanding of division are:– the use of 1 as a divisor (6 ÷ 1 = 6);– the relationship of division to fractional sense (4 candies divided into 2 groups represents both 4 ÷ 2 and the whole divided into 2 halves);– the inverse relationship with multiplication.

Have participants choose a different partner for the next activity. Ask them to decide which person in their pair will investigate 'Using Models' (pp. 10.26–10.27) and which person will investigate 'Strategies' (pp. 10.28–10.32).

Each person should prepare a summary of main points of their topic and develop a brief activity based on the material in the Guide.

Participants who are investigating strategies could make a list of strategies along with a brief explanation or example of each one. As an activity, they could model one of the examples with their partner.

After the pairs have completed their work, have them present their information to their partner. Participants can record ideas on BLM 9.3.

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Multidigit Whole Number Calculations

“There is mounting evidence that students both in and out of school can construct methods for adding and subtracting multidigit numbers without explicit instruction.” – Carpenter, Franke, Jacobs, Fennema, & Empson, A Longitudinal Study of

Invention and Understanding of Children’s Multidigit Addition and Subtraction, Journal for Research in Mathematics Education, 1998, p. 4

Many teachers learned only one way to solve multidigit computations – using the standard North American algorithm taught in schools. But solving a multidigit computation can be done in many ways, depending on the context of the problem.

Consider making change from a $10 bill for a $7.69 purchase. Is the standard algorithm efficient in this situation?

Students who can work flexibly with numbers are more likely to develop efficient strategies, accuracy, and a strong foundation for understanding other standard algorithms.

The standard North American algorithms were established to help make calculations fast. Such shortcuts are practical and useful for those who understand the algorithm.

For students who have not been taught the underlying concepts, memorizing the abstract algorithm is often the beginning of their belief that mathematics 'doesn’t make sense' and is dependent on memorizing rules or routines. (See example on p. 10.34.)

Students who are encouraged to use their own flexible strategies for computing multidigit numbers develop the following:

A better sense of number More flexibility in solving problems A stronger understanding of place value Greater facility with mental calculations More ease in linking meaning to symbols in the traditional algorithms

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Ask participants to form groups of five. Each group member is responsible for reading about one of the following topics and recording important ideas on BLM 9.4a or BLM 9.4b:

Teaching Algorithms (pp. 10.37–10.40) Student-Generated Algorithms (pp. 10.40–10.42) An Investigative Approach (pp. 10.42–10.43) Standard Algorithms (pp. 10.49–10.51) Estimation (pp. 10.51–10.53)

When participants have finished their research, they report what they have learned to their group. Participants can record what they learn from other group members on BLM 9.4a and BLM 9.4b.

Point out that there are a number of activity ideas for teaching multidigit computations (pp. 10.43–10.49), but that they are not covered in this module, owing to time constraints. Appendices 10-1 and 10-2 provide instructions and backline masters for the activities.

Reflecting and ConnectingIn Your Classroom

Encourage participants to try a new strategy or activity for teaching basic facts or multidigit computations.

Find an appropriate way to celebrate the completion of the modules and participants’ learning (e.g., provide social time).

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