Transcript calculators, mental computation, and estimation

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-2
Develop 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 solving
relieves tedious computation
focuses attention on meaning
removes anxiety about computational
failures
provides motivation & confidence
A calculator should be used as
an instructional tool when it:
facilitates a search for patterns
creates problematic situations
supports concept development
promotes number sense
encourages 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)
SHOULD
NOT
MATTER
QUESTION
SHOULD
DOES 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
SHOULD
NOT
MATTER
QUESTION
SHOULD
DOES NOT
MATTER
D. The decimal fraction
0.222 most nearly equals:
S
DM
SN
S
DM
SN
(a) 2/10 (b) 2/11 (c) 2/9
(d)2/7 (e) 2/8
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?
QUESTION
SHOULD
DOES 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
SHOULD
NOT
MATTER
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 Estimation
Adjusting or Compensating
Compatible 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?