Computation Methods: Calculators, Mental Computation, and Estimation CHAPTER 10 Tina Rye Sloan To accompany Helping Children Learn Math9e, Reys et al. ©2009 John Wiley & Sons

Focus 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? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Number and Operations Standard: Compute fluently and make reasonable estimates. Grades Pre-K-2 Develop and use strategies for whole-number computations, with a focus on addition and subtraction. Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators. NCTM (2000). Principles and Standards for School Mathematics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Number and Operations Standard: Compute fluently and make reasonable estimates Grades 3-5 Develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results. Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tool. NCTM (2000). Principles and Standards for School Mathematics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Number and Operations Standard: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Grades Pre-K-2 Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers. NCTM (2000). Principles and Standards for School Mathematics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Number and Operations Standard: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Grades 3-5 Develop and use strategies to estimate the results of whole- number computations and to judge the reasonableness of such results. Develop fluency in adding, subtracting, multiplying, and dividing whole numbers. NCTM (2000). Principles and Standards for School Mathematics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Myths and Facts of 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Myths and Facts of Calculators (cont’d) 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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Myths and Facts of 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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

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 below, 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. (see next slide) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

QUESTION SHOULD DOES NOT MATTER SHOULD NOT MATTER A. 36 x 106 =SDMSN B. Explain a rule that generates this set of numbers:..., , 0.25, 1, 4, 16,... SDMSN C (8 - 2 x (4 + 3)) =. SDMSN Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

QUESTION SHOULD DOES NOT MATTER SHOULD NOT MATTER D. The decimal fraction most nearly equals: (a) 2/10 (b) 2/11 (c) 2/9 (d)2/7 (e) 2/8 SDMSN E. The number of students in each of five classes is 25, 21, 27, 29, and 28. What is the average number of students in each class? SDMSN Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

QUESTION SHOULD DOES NOT MATTER SHOULD NOT MATTER F. I have four coins; each coin is either a penny, a nickel, a dime, or a quarter. If altogether the coins are worth a total of forty-one cents how many pennies, nickels dimes, and quarters might I have? SDMSN Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

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 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Three-Step Challenge Solve this problem at least five other 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? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Mental Computation-Computation done internally without any external aid like paper and pencil or calculator. Often nonstandard algorithms are used for computing exact answers. Mental Computation Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Strategies and Techniques for Mental Computation Extend basic facts (i.e. extend 4+ 5=9 to 40+50=90 and =900) Use compatible or ‘friendly’ numbers (i.e … Students could recognize that 8 and 22 are compatible numbers; 7 and 13 are compatible numbers). Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

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 ways 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Guidelines for Teaching Mental Computation (continued) 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Guidelines for Teaching Mental Computation (continued) Encourage inventiveness. There is no one right way to do any mental computation. Make sure children are aware of the difference between estimation and mental computation. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Mental Computation You drove 42 miles, stopped for lunch, and then drove 34 more miles. How many miles have you traveled? Explain how you solved the problem Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Mental Computation You earned 36 points on your first project. Then you earned 28 points on your second project. How many points have you earned? Explain how you solved the problem Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Mental Computation You watched a video for 39 minutes. Then later you watch a second video for 16 minutes. How many minutes did you watch in all? Explain how you solved the problem Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

A Student's View of Mental Computation Interviews with students in several countries about their attitude toward mental computation produced surprising consistent responses. Here is a "typical" attitude of a middle grade student: (next slide) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

I 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) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

How would you respond to this student? If you had an opportunity to talk with the student's teacher, what would you tell her? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Computational Estimation-The process of producing an answer that is sufficiently close to allow decisions to be made. Computational Estimation Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Guidelines for Teaching Estimation Give your students problems that encourage and reward computational estimation. Make sure students are not computing exact answers and then rounding to produce estimates. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Guidelines for Teaching Estimation (cont’d) 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Computational Estimation Strategies Front-End Estimation Adjusting Compatible Numbers Flexible Rounding Clustering Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Computational Estimation You have $10 to buy detergent and a mop. Do you have enough? Explain how you solved the problem. $ Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

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. $ Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 $2.39 $0.68