Expectations from the Number and Operations Standard

Grades Pre-K-5

Principles and Standards for School Mathematics

National Council of Teachers of Mathematics

2000

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. 

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.

Understand numbers, ways of representing numbers, relationships among numbers,

and number systems.

Grades Pre-K-2Develop a sense of whole numbers and

represent and use them in flexible ways, including relating, composing, and decomposing numbers.

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.

A calculator should be used as a computational tool when it:

facilitates problem solvingrelieves tedious computationfocuses attention on meaningremoves anxiety about computational

failuresprovides motivation & confidence

A calculator should be used as an instructional tool when it:

facilitates a search for patternscreates problematic situationssupports concept developmentpromotes number senseencourages creativity & exploration

Calculator Test Items

 Suppose that you are a 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)

QUESTION SHOULDDOES 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

 

QUESTION SHOULDDOES NOT MATTER

SHOULD NOT

MATTER

D. The decimal fraction 0.222 most nearly equals:

(a) 2/10 (b) 2/11 (c) 2/9 (d)2/7 (e) 2/8

S DM SN

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?

S DM SN

QUESTION SHOULDDOES 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?

S DM SN

 

Guidelines for Teaching Mental Computation

Encourage students to do computations mentally.

 Check to learn what computations students prefer to do mentally.

Check to learn if students are applying written algorithms mentally.

Guidelines for Teaching Mental Computation

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

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.

Guidelines for Teaching Estimation

Provide situations that encourage and reward computational estimation 

Check to learn if students are computing exact answers and then "rounding" to produce estimates

Guidelines for Teaching Estimation

Ask students to tell how their estimates were made.

Destroy the one-right-answer syndrome early.

Encourage students to think carefully about real-world applications where estimates are made.

Computational Estimation Strategies

Front-End EstimationAdjusting or CompensatingCompatible Numbers Flexible Rounding Clustering Rounding

Numbers

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

You drove 42 miles, stopped for lunch, then drove 34 miles. How many miles have you traveled? Explain how you solved the problem.42 + 34

Mental Computation

You earned 36 points on your first project. Then earned 28 points on your second project. How many points have you earned? Explain how you solved the problem. 36 + 28

Mental Computation

You watched a video for 39 minutes. You watch a second video for 16 minutes. How many minutes did you watch in all? Explain how you solved the problem. 39 + 16

Computational Estimation-The process of producing an answer that is sufficiently close to allow decisions to be made.

Computational Estimation

You have $10 to buy detergent and a mop. Do you have enough? Explain how you solved the problem.

$ 3.98

+ 5.98

Computational Estimation

You have $5 to buy a soft drink, sandwich, and a slice of pie. Do you have enough? Explain how you solved the problem.

$ . 68

2. 39

+2. 29

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

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?

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)

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)

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

student's teacher, what would you tell her?