Chapter 10Chapter 10

To accompany Helping Children Learn Math Cdn Ed, Reys et al.©2010 John Wiley & Sons Canada Ltd.

 

Guiding Questions

1. What computational methods should students use?2. What are some myths and facts about using

calculators?3. What are some strategies for mental computation,

and how can teachers encourage mental computation?

4. What are some strategies for computational estimation, and how can teachers encourage estimation?

Balancing Your Instruction

• More than 80% of all mathematical computations in daily life involve mental computations.

• A better balance of instructional time for mental computation, estimation, and written computation is needed.

Balancing Your Instruction (cont.)

Children need to learn a variety of computational methods and when to use each. Students should be taught that:

1. All computation begins with a problem and with the recognition that computation is needed to solve the problem.

2. Certain decisions must be made when doing computation.

3. Estimation is always used to check the reasonableness of a result.

Myths and Facts About Calculators• Myth: Using calculators does not require thinking.• Fact: Calculators do not think for themselves. Students

must still do the thinking.• Myth: Using calculators lowers mathematical achievement.• Fact: Calculators can raise students’ achievement.• Myth: Using calculators always makes computations

easier.• Fact: It is sometimes faster to compute mentally.• Myth: Calculators are useful only for computation.• Fact: Calculators are also useful as instructional tools.

Myths and Facts About Calculators

A calculator should be used as a computational tool when it:– facilitates problem solving– eases the burden of doing tedious computation– focuses attention on meaning– removes anxiety about doing computation

incorrectly– provides motivation and confidence

Myths and Facts About Calculators

A calculator should be used as an instructional tool when it:– facilitates a search for patterns– supports concept development– promotes number sense– encourages creativity & exploration

Calculator Test Items

Suppose that you are an elementary school teacher that is involved in constructing questions for a test. You want each question used to measure the mathematical understanding of your students. For each proposed test item on the following slide, decide if students should (S) use a calculator, it doesn't matter (DM) if the students use a calculator, or students should not (SN) use a calculator in answering the test item presented.

Please take time to discuss your thinking with others.

Calculator Test Items

QUESTION SHOULD DOES NOT MATTER

SHOULD NOT

MATTER

A. 36 x 106 = S DM SN

B. Explain a rule that generates this set of numbers: …, 0.0625,

0.25, 1, 4, 16, ...S DM SN

C. 12 - (8 - 2 x (4 + 3)) = . S DM SN

Three-Step Challenge • Use the , , =, and numeral keys on your

calculator to work your way from 2 to 144 in just three steps.

• For example:– Step 1: 2 12 = 24– Step 2: 24 12 = 288– Step 3: 288 2 = 144

• Now find a different way to get the answer 144.• Share and compare your answers with others.

Three-Step Challenge

• Solve this problem at least five different ways. Record your solutions.

• Choose your own beginning and ending numbers for another three-step challenge. Decide if you must use special keys or all the operation keys.

• Challenge a classmate.• How did you use estimation, mental

computation, and calculator computation?

Mental Computation

• Mental Computation is computation that done “all in the head”—that is without tools such as calculator, or paper and pencil.

• Through mental math, young children often develop logical ways of computing before they can write. If one glass of lemonade

is 10 cents, 3 glasses willbe 10… 20… 30 cents

Strategies and Techniques for Mental Computation

• Extend basic facts Extend 4 + 5 = 9 to 40 + 50 = 90 or 400 + 500= 900

• Use compatible or ‘friendly’ numbers 8+7+22+5+13

Students could recognize that 8 and 22 are compatible numbers; 7 and 13 are compatible numbers, so 8 + 22 = 30 and 7 + 13 =20 so the question becomes 30 + 20 + 5

Encouraging Mental Computation

• Always try mental computation before using paper and pencil or a calculator.

• Use numbers that are easy to work with.• Look for an easy way.• Use logical reasoning.• Use knowledge about the number system.

Why Emphasize Mental Computation?

• Mental computation is very useful.• Mental computation is the most direct and

efficient way of doing many calculations.• Mental computation is an excellent way to help

children develop critical-thinking skills and number sense and to reward creative problem solving.

• Proficiency in mental computation contributes to increased skill in estimation.

Guidelines for Teaching Mental Computation

• Encourage students to do computations mentally.

• Learn which computations students prefer to do mentally.

• Find out if students are applying written algorithms mentally.

Guidelines for Teaching Mental Computation (cont.)

• Plan to include mental computation systematically and regularly as an integral part of your instruction.

• Keep practice sessions short, perhaps 10 minutes at a time.

• Develop children's confidence.

Guidelines for Teaching Mental Computation (cont.)

• Encourage inventiveness. There is no one right way to do any mental computation.

• Mental computation or estimation? Make sure children are aware of the difference between estimation and mental computation.

Mental Computation

You drove 42 kilometres, stopped for lunch, and then drove 34 more kilometres. How many kilometres have you travelled? Explain how you solved the problem.

Mental Computation

On Monday you walked your dogs for 39 minutes. On Wednesday you only fit in a 17 minute dog walk. How much time did you walk your dogs altogether?

A Student's View of Mental Computation

• Interviews with students in several countries about their attitude toward mental computation produced surprisingly consistent responses.

• The following slide shows a "typical" attitude of a middle school student.

A Student’s View of Mental ComputationI learn to do written computation at school, and spend more time at school doing written computation than mental computation. I find mental computation challenging, but interesting. I enjoy thinking about numbers and trying to come up with different ways of computing. It helps me to understand things better when I think about numbers in my head. Sometimes I need to write things down to check to see if what I have been thinking is okay. I think it is important to be good at both mental and written computation, but mental computation will be used more as an adult and so it is more important than written computation. Although I learned to do some mental computation at school I learned to do much of it by myself.

(McIntosh, Reys & Reys)

A Student’s View of Mental Computation

• How would you respond to this student?• If you had an opportunity to talk with the

student's teacher, what would you tell her?

Computational Estimation

• Computational Estimation: The process of producing answers that are close enough to allow for good decisions without performing elaborate calculations.

When Should Students Use Estimation?

Encouraging Estimation

• Begin by making students aware of what estimation is about, so they acquire a tolerance for error.

• Give students immediate feedback on their estimates.

• Encourage them to be flexible when thinking about numbers.

Computational Estimation Strategies

• Front-End Estimation• Adjusting• Compatible Numbers • Flexible Rounding• Clustering

Choosing Estimation Strategies

• Give your students problems that encourage and reward estimation.

• Make sure students are not computing exact answers and then rounding to produce estimates.

Choosing Estimation Strategies (cont.)

• Ask students to tell how they made their estimates.

• Fight the one-right-answer syndrome from the start.

• Encourage students to think of real-world situations that involve making estimates.

Computational Estimation

• You have $5 to buy a soft drink, hamburger, and an order of French fries. Do you have enough? Explain how you solved the problem.

$2.39$2.39

$0.68

Copyright

Copyright © 2010 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (The Canadian Copyright Licensing Agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.