Skilled and Less Skilled Estimators' Strategies for Estimating Discrete Quantities
Author(s): Terry Crites
Source: The Elementary School Journal, Vol. 92, No. 5 (May, 1992), pp. 601-619
Published by: University of Chicago Press
Stable URL: http://www.jstor.org/stable/1001741
Accessed: 01-10-2015 22:02 UTC
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Skilled and Less
Skilled Estimators'
Strategies for
Estimating Discrete
Quantities
Terry Crites
NorthernArizonaUniversity
The Elementary School Journal
Volume 92, Number 5
? 1992 by The University of Chicago. All rights reserved.
0013-5984/92/9205-0005$01.00
Abstract
This study describesstrategiesthat third-,fifth-,
and seventh-gradestudents used when making
estimates of discrete quantities. An estimation
test was administeredto 401 students from a
small, rural, midwestern district to stratify the
population into thirds. From each grade, 6 students who had test scores in the top one-third
and 6 studentswith scores fromthe bottomthird
(a total of 36 students) were interviewed individually about the strategiesthey used to solve
20 questions involving estimation of discrete
quantities.The most commonly used strategies
were benchmarkcomparison,eyeball, and decomposition/recomposition.Interviewdata suggested that (a) successful estimators tended to
use the decomposition/recompositionand multiple benchmark strategies; (b) less successful
estimators generally used perceptually based
strategies;(c) skilled estimatorswere more successful than less skilled estimatorson to-be-estimateditems that containedlarge numbers;and
(d) skilled estimatorsmade more acceptableestimates,tended to subdivideproblemsinto parts,
and guessed less often than less skilled estimators.
In recent years much emphasis has been
placed on students' skill in estimation. Organizations such as the National Advisory
Committee on Mathematical Education
(1975), National Council of Teachers of
Mathematics (1980, 1989), National Council of Supervisors of Mathematics (1989),
Conference Board of the Mathematical Sciences (1984), and National Research Council (1989) have suggested that estimation
should be an important component of a student's mathematical education. Because of
these recommendations, estimation is playing a larger role in the mathematics curriculum and is receiving new emphasis in
mathematics textbooks, such as Mathemat-
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602
THE ELEMENTARYSCHOOL JOURNAL
ics Unlimited (Fennell, Reys, Reys, & Webb,
1987).
However, some educators agree with
Trafton (1986) that current mathematics
curriculum and instruction do not devote
enough attention to estimation. For example, Paulos (1988, p. 74) bemoans the mathematical preparation of students:
According to Paulos (1988, p. 11): "To
get a handle on big numbers, it's useful to
come up with one or two collections (of objects) corresponding to each power of ten,
up to maybe 13 or 14. The more personal
you can make these collections, the better.
It's also good practice to estimate whatever
piques your curiosity: How many pizzas are
consumed each year in the United States?
How many words have you spoken in your
life? How many different people's names
appear in the New York Times each year?
How many watermelons would fit inside
the U.S. Capitol building?"
Other educators agree that children's inability to estimate quantities is related to
their limited conception of the numbers involved and therefore that teachers should
help students develop an experiential base
from which to make better estimates (Leutzinger, Rathmell, & Urbatsch, 1986; Turkel
& Newman, 1988). There is support for providing students with a wealth of opportunities to estimate, as well as suggestions for
activities that teachers can use (Bright, 1988;
Gronert & Marshall, 1979; Ross & Ross,
1986).
Research on estimation is also needed
badly. In a summary of research on this
topic, Benton (1986) called for more research and replication in the area of choice
of strategy and for research in grades 1-3.
Carter (1986, p. 80) cited areas in which
research is needed: "Case studies to determine the ability of young children to make
estimates and the extent to which they employ guesses or benchmarks [known standards] when making reasonable estimates
would be valuable." Motivated by the concerns expressed above, my study was an
initial attempt to understand what strategies
good and poor estimators use to estimate
discrete quantities.
Estimation is generally not taught ...
aside froma few lessons on roundingoff
numbers.The connectionis rarelymade
thatroundingoff and makingreasonable
estimateshave somethingto do with real
life. Gradeschool studentsaren'tinvited
to estimate the number of bricks in the
side of a school wall, or how fast the class
speedsterruns, or the percentageof students with bald fathers, or the ratio of
one's head's circumference to one's
height, or how many nickels are necessary to make a tower equal in height to
the EmpireState Building,or whetherall
those nickelswould fitin theirclassroom.
Paulos's point is that students do not receive the rich variety of experiences with
numbers that allow them to consider competently questions about the relative magnitude of two quantities (which is important
in determining probabilities), to realize incongruities when confronted with facts that
seem contrary to logic, and to know not
only how to perform an operation but also
when the performance of the operation is
appropriate.
Paulos believes that in order to be mathematically "numerate" it is necessary to understand both the actual and relative sizes
of large numbers. There is value in understanding that if one spent $1,000 per day it
would take about 3 years to spend $1 million, 3,000 years to spend $1 billion, and
3,000 years to spend the $1 trillion in our
country's annual budget. The relative significance of humans in the history of the
world can be illustrated by realizing that the
evolution of modern humans occurred only
around 1 trillion (1 X 1012) seconds ago as
compared to the estimated age of the planet
of at least 60 quadrillion (6 X 1016) seconds.
Background
The limited data available, such as the results of the fourth National Assessment of
Educational Progress (NAEP) (Brown et al.,
1989), indicate that students' performance
MAY 1992
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ESTIMATION
in all areas of estimation is poor. The NAEP
data provide evidence that "some students
have a lack of understanding of the relative
size of numbers greater than 100" (Kouba,
Carpenter, & Swafford, 1989, p. 37). Although 50% of third graders could correctly
estimate the number of birds in a picture,
25% responded, "I don't know," indicating
"a lack of feel for comparative sizes of quantities beyond 100, or a lack of estimation
strategies" (p. 38).
My review of the literature yielded only
one study that investigated strategies students used to estimate discrete quantities
(Siegel, Goldsmith, & Madson, 1982). Siegel et al. developed a model of the estimation process that distinguished two types
of estimation problems: those that required
the use of a benchmark to make an estimate
and those that required the decomposition
and then subsequent recomposition of the
to-be-estimated item.
Benchmark estimation is the application
of a known standard to the to-be-estimated
item, such as estimating the length of a
piece of paper that is about 1 foot long by
comparing it to a foot ruler. Fractional or
multiple benchmark estimation is defined
as the use of a benchmark that is some manageable multiple or fraction, respectively, of
the to-be-estimated item. An example of
fractional benchmark estimation is estimating the number of kernels of popcorn in a
container by comparing it with the known
number of beans in a similar container.
Since three kernels of popcorn are about the
same size as one bean, there must be approximately three times as many kernels of
popcorn as there are beans. Multiple benchmark estimation is typified by estimating
the height of a tin can by comparing it to
an inch (which is smaller by a factor of
about one-fourth).
Decomposition/recomposition is used
when no benchmark is available and the tobe-estimated item is subdivided into parts
small enough so that a benchmark can be
applied. The decomposition is said to be
regular if the decomposed pieces of the to-
603
be-estimated item are all the same size. For
example, the width of a tile floor could be
estimated by decomposing it into parts (all
tiles in a single row). The width of a tile
could then be estimated by comparing it to
some benchmark (a foot or an inch). An
estimate for the width of the floor could
then be made by recomposition (multiplying the width of a tile by the number of tiles
in a row).
If the to-be-estimated item cannot be
easily subdivided, or if it can be subdivided
but not into parts that are the same size, the
decomposition is said to be irregular. An
example of the first kind of irregular decomposition is estimating the number of
people attending a sporting event. An example of the second kind of irregular decomposition is estimating the total number
of coins (e.g., pennies, nickels, dimes, and
quarters) in a container. The number of each
kind of coin might be estimated separately
(decomposition) and then added (recomposition) to get the total estimate.
In the Siegel et al. (1982) study, subjects
were presented with 24 items from a pool
of problems classified as benchmark, fractional or multiple benchmark, or regular or
irregular decomposition. After interviewing
140 children from grades 2 through 8 and
10 college-educated adults, the researchers
identified the following "strategies": don't
know, guess, eyeball, range, benchmark
comparison, benchmark, fractional benchmark, multiple benchmark, pseudodecomposition, and decomposition/recomposition. Siegel et al. considered the first five
strategies listed above to share common features and labeled them as being "perceptually based." Similarly, the next three
strategies were said to be examples of
"benchmark" strategies. A description of
each strategy and an example of its use are
given in Appendix A.
The Siegel et al. (1982) study was the
foundation for the study reported here. The
goals of the Siegel et al. study were simply
to develop a model of the estimation process
and to identify what estimation strategies
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604
THE ELEMENTARYSCHOOL JOURNAL
students used. However, I extended their
study by quantifying strategy usage and by
identifying and characterizing strategies that
skilled and less skilled estimators used to
make acceptable estimates of discrete
quantity.
In related research, Baroody and Gatzke
(1991) interviewed 18 potentially gifted
preschool-kindergarten children about their
ability to perform three tasks: (a) estimation
tasks, where children were to estimate the
number (3-35) of dots in a set (Baroody and
Gatzke called them cookies); (b) numberreferent tasks, where children decided
whether a set of dots was larger or smaller
than given reference numbers (5, 10, or 20);
and (c) order-of-magnitude tasks, where
children decided where a set of dots fit in
relation to two reference numbers (e.g.,
Does the set of dots number less than 10,
between 10 and 20, or more than 20?). Results showed that with sets of 8 most estimates were within 25% of the actual value,
but with sets of 15 or more there were few
accurate estimates. (Baroody and Gatzke
used the actual value plus or minus 25% as
the primary criterion for an acceptable estimate, but they also used the actual value
plus or minus 50% as a criterion for some
estimates.)
A majority of children were successful
on the number-referent task, but performance varied on the order-of-magnitude task.
From further analyses the authors concluded that children could more accurately
place sets that were smaller than the reference number than place sets that were
larger. Also, children did not possess accurate mental benchmarks for 5, 10, and 20.
judgments including but not restricted to
number comparison, recognition of unreasonable results from calculations, and performing mental computation using nonalgorithmic forms (Sowder, 1988). Estimation
test refers to the 24-item instrument that I
developed and used to assess students' ability to estimate quantities. Skilled estimators
are students in each grade who scored in the
upper one-third of the distribution of scores
on the estimation test. Less skilled estimators
are students in each grade who scored in the
bottom one-third of the distribution of scores
on the estimation test.
Method
Definition of Terms
Several terms used throughout this
article are defined here. Estimation of discrete quantity is performed when the to-beestimated item is the cardinal number of a
set of objects. Number sense is an ability to
use number magnitude-relative and absolute-to make qualitative and quantitative
Instruments
I developed a multiple-choice and an
open-ended version of a 24-item estimation
test (see App. B). The questions on the two
versions were identical. They differed only
in that the open-ended version lacked the
foils contained in the multiple-choice version. Care was taken to develop items that
would test a variety of types of estimation
of discrete quantity. Some items involved estimating the number of objects in containers
(sometimes the objects were visible, at other
times they were not). Other items relied on
the possession of spatial visualization, measurement, mental computation, or numbersense skills. All problems, however, had
whole numbers as the correct answers and
were similar in nature and content to the
interview questions described below.
The content validity of the estimation test
was established by an examination of the
instrument by members of my doctoral committee. The reliability (internal consistency)
of the test was computed using Kuder-Richardson's formula 20, which uses a split-half
method for computing reliability. The reliabilities for the third, fifth, and seventh
grades on the multiple-choice version were
.26, .71, and .53, respectively. The reliabilities for the third, fifth, and seventh grades
on the open-ended version were .29, .60,
and .34, respectively.
I administered the estimation test to each
student in the population. I gave the openMAY 1992
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ESTIMATION
ended version of the estimation test to the
first class I met. Subsequently, as I met with
other classes, I alternately administered multiple-choice and open-ended versions. As a
consequence, 219 students received the
open-ended version and 182 students received the multiple-choice version of the estimation test.
In addition to the estimation test, I developed an interview protocol (see App. C)
consisting of 20 questions. In all but two
questions, the student was given a physical
prop and asked to estimate some characteristic of the prop. To understand better the
strategy a student used, appropriate probing
questions were asked after the student's initial answer.
The interview data were analyzed by
studying a typed transcription of the interviews along with notes I made during the
interviews. First, for each problem, the accuracy of the estimate was judged. As in the
Siegel et al. (1982) study, I considered an
estimate accurate if it was within 50% of the
exact answer. A wide margin of error was
chosen for at least three reasons. First, there
was no better alternative. In computational
estimation, one can make different estimates
using all the known strategies and take the
smallest and largest of these estimates as the
lower and upper bounds for an accurate estimate. The nature of the strategies used in
estimation of discrete quantity offers no such
opportunity. A percentage error is left as the
only reasonable choice. Second, previous
and current research support this criterion.
Siegel et al. (1982), on which my study was
based, used a plus or minus 50% criterion.
Baroody and Gatzke (1991) also used this
criterion as one measure of an acceptable estimate. Third, once I decided to use a percentage error, I thought it best not to have
it too restrictive since, particularly with large
numbers, a wide range of estimates is acceptable (e.g., scientists estimate the age of
the universe at 2-4 billion years). If the percentage error is set too narrowly, adverse
results can occur. If 25% error is acceptable,
would 20% be better? If 20% is used, would
605
15% be a better criterion? A narrow range
of acceptable estimates implies that the best
estimate is the one that is closest to the exact
answer. This is contrary to all that is known
about estimation.
After the accuracy of the estimate was
judged, I attempted to determine the method
of solution or primary thought process the
student used. Each method was classified as
one of the Siegel et al. strategies previously
mentioned. The methods of solution were
then searched for common themes, and the
frequency with which students with estimation test scores in the top and bottom
thirds used each method was determined.
The reliability of the classification of the
strategies used in making estimates of discrete quantities was measured by the following cross-checking procedure. After classifying the strategies used in all interview
protocols, I chose four interview excerpts for
each of the eight estimation strategies students used during the interviews (for a total
of 32 interview excerpts). Each excerpt was
independently classified by another doctoral
student in mathematics education. The two
lists of strategies were then examined for
agreement. Four strategies had 100% agreement, three had 75% agreement, and one
strategy had 50% agreement.
Subjects
The students who participated in this
study were drawn from a small, rural, midwestern community with a population of
about 10,000. Parental permission slips were
sent home with all students in the third, fifth,
and seventh grades before the study began.
All students in these grades were included
in the sample unless their parents returned
the form and denied their permission. The
final sample pool consisted of 401 students
(94% of the student population) and contained 196 females and 205 males. The students had received no formal instruction in
estimating discrete quantities prior to the
study.
Results of the estimation test were used
only to stratify the population into thirds. In
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THE ELEMENTARYSCHOOL JOURNAL
each grade, six students from the top stratum
and six students from the bottom stratum
were chosen to be interviewed. At each
grade, the top performer on each version of
the estimation test was automatically included in the pool of interviewees. Ten other
students, four from the top one-third and six
from the bottom one-third, were randomly
selected from all students in each grade. No
effort was made to stratify the interviewees
by gender. Table 1 gives the estimation test
scores of the students who were selected to
be interviewed.
were tape-recorded and transcribed later. I
first explained the goals of the interview to
each student by reading the following: "Estimation is what you do when you want to
know how big something is or how many
things there are, but you don't measure or
you don't count. You estimate to try to determine about how big something is or about
how many there are. I will ask you several
questions in which I want you to make an
estimate. I will also ask you to explain what
thinking you did in order to determine your
estimate. If you can, please feel free to think
aloud while you are making your estimate."
I then read aloud each question and presented any appropriate prop. After a student
made an initial estimate, I used probing
questions to gain a better understanding of
Procedure
In November and December of 1988, I
interviewed students individually in a quiet
study room at their schools. The interviews
TABLE
1. Scores on Estimation Test, Gender, and Version of Estimation Test Completed of
Students Interviewed, by Grade and Estimation Ability
Student
Skilled and Less Skilled
Groups by Grade
Grade 3:
Skilled:
Score
Gender
Version"
Less skilled:
Score
Gender
Version
Grade 5:
Skilled:
Score
Gender
Version
Less skilled:
Score
Gender
Version
Grade 7:
Skilled:
Score
Gender
Version
Less skilled:
Score
Gender
Version
1
2
3
4
5
6
14
M
MC
12
M
OE
11
F
MC
11
F
MC
10
M
OE
10
M
MC
8
F
MC
7
M
OE
6
M
OE
6
M
OE
4
F
OE
2
M
OE
23
F
MC
17
M
MC
16
M
MC
14
M
OE
13
F
OE
10
M
OE
8
F
OE
8
M
OE
8
F
MC
8
M
OE
6
F
OE
6
M
OE
19
F
MC
15
M
OE
13
F
MC
13
F
OE
12
F
OE
12
F
OE
8
M
OE
8
F
OE
7
F
OE
7
F
OE
7
F
OE
5
M
OE
aMC = multiple choice; OE = open ended.
MAY 1992
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ESTIMATION
the student's thinking and strategy use.
Other than asking follow-up questions, I remained unobtrusive during the interview. I
did not try to guide students' reasoning or
methods of solution, nor did I indicate that
an answer or procedure was correct or incorrect. Each interview lasted about 30 minutes.
Results
Accuracy
The open-ended version of the 24-item
estimation test was administered to 76 third
graders, 72 fifth graders, and 71 seventh
graders. The mean number of acceptable estimates and the standard deviation of the
estimation test scores for the third, fifth, and
seventh grades were 6.04 (2.28), 7.78 (3.16),
and 9.00 (2.48), respectively. The multiplechoice version was administered to 87 third
graders, 68 fifth graders, and 27 seventh
graders. The mean number of acceptable estimates and the standard deviation of the
estimation test scores for the third, fifth, and
seventh grades were 10.79 (2.61), 12.21
(3.10), and 13.33 (3.14), respectively.
On the interview questions, the mean
number of acceptable estimates for third
graders was 6.7 out of 20. At least 75% of
the third-grade students correctly answered
four of the 20 questions (numbers 1, 7, 18,
and 20). However, for 11 questions, no
more than 25% of the students could make
an acceptable estimate. There were five
questions (numbers 8, 9, 12, 13, and 17) that
no third grader could correctly estimate.
Also among the third-grade students,
there was not a discernible difference between the skilled and less skilled estimators
in the mean number of acceptable estimates
given in response to the interview questions. The skilled estimators had a higher
mean number of acceptable estimates than
the less skilled estimators on seven of the
20 questions (numbers 5, 7, 14, 16, 18, 19,
and 20). In contrast, skilled estimators' performance on seven problems (numbers 1,
2, 4, 6, 10, 11, and 15) was lower than that
607
of the less skilled estimators in the third
grade.
Fifth- and seventh-grade students performed better than third graders. Among
the fifth graders, the mean number of acceptable estimates was 9.3 out of 20, with
at least a 50% solution rate on nine problems. At least a 75% solution rate was observed on six problems (numbers 1, 3, 7, 12,
18, and 20). Fifth-grade students showed
solution rates of 25% or lower on eight
problems. There was one problem (number
17) that no fifth grader could accurately estimate. The skilled fifth-grade estimators
performed better than the less skilled estimators on 10 of the 20 problems. Skilled
estimators were outscored by the less
skilled estimators on three problems (numbers 1, 12, and 18).
The mean number of acceptable estimates for seventh graders was 8.5 out of 20.
Although seventh graders solved only two
problems with at least a 75% solution rate
(numbers 14 and 20), they had at least a
50% solution rate on eight problems. Seventh graders had solution rates of 25% or
lower on seven problems. At least one seventh-grade student correctly answered each
of the 20 questions. Seventh-grade skilled
estimators outscored the less skilled estimators on 10 questions, with the opposite
holding true on five questions (numbers 1,
3, 4, 7, and 18). The proportions of students,
classified by grade and estimation group,
who gave acceptable estimates to each interview question are given in Table 2.
Strategy Use
A cumulative frequency for each strategy used, as well as the number of acceptable estimates for each strategy, is reported
in Table 3. The table shows that, for students in all three grades, three strategies
(benchmark comparison, eyeball, and decomposition/recomposition) accounted for
two-thirds of the strategies used.
Differences in strategy use occurred
across the three grades. Third graders used
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TABLE
2. Interview Problems and Proportions of Skilled and Less Skilled Estimators Who
Gave Acceptable Estimates
Problem
1. About how many pipes does this pipe
organ have?
Skilled estimators
Less skilled estimators
2. About how many words are on this
piece of paper?
Skilled estimators
Less skilled estimators
3. About how many M&M's are in this
small bag?
Skilled estimators
Less skilled estimators
4. About how many M&M's are in this
large bag?
Skilled estimators
Less skilled estimators
5. About where on this page would you
find the seventieth name?
Skilled estimators
Less skilled estimators
6. About how many days do you think
you have been alive?
Skilled estimators
Less skilled estimators
7. Are there more or fewer than 1,000
X's on this piece of paper?
Skilled estimators
Less skilled estimators
8. About how many kernels of popcorn
are in this jar?
Skilled estimators
Less skilled estimators
9. About how many beans are in this
jar?
Skilled estimators
Less skilled estimators
10. About how many M&M's are in this
jar?
Skilled estimators
Less skilled estimators
11. About how many lots-a-links are in
this jar?
Skilled estimators
Less skilled estimators
12. About how many links would it take
to form a chain that is 3 feet long?
Skilled estimators
Less skilled estimators
13. About how many links are there in a
mile?
Skilled estimators
Less skilled estimators
Grade 3
Grade 5
Grade 7
.67
.83
.50
1.0
.50
.67
.0
.50
.33
.17
.50
.17
.67
.67
1.0
.67
.50
.83
.0
.33
.17
.0
.17
.33
.50
.17
.67
.33
.33
.33
.0
.17
.33
.33
.67
.0
1.0
.67
1.0
1.0
.50
.67
.0
.0
.33
.17
.33
.17
.0
.0
.17
.17
.50
.17
.17
.33
.33
.17
.33
.0
.33
.50
.67
.17
.17
.17
.0
.0
.83
1.0
.67
.50
.0
.0
.17
.0
.17
.17
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609
ESTIMATION
TABLE2. (continued)
Problem
14. Pick out an object that is around 25
multilink cubes long.
Skilled estimators
Less skilled estimators
15. About how many beans would it take
to fill this jar?
Skilled estimators
Less skilled estimators
16. About how many beans would it take
to fill a gallon jug?
Skilled estimators
Less skilled estimators
17. Give me around 50 beans.
Skilled estimators
Less skilled estimators
18. About how many dots are enclosed
by the shape?
Skilled estimators
Less skilled estimators
19. About how many beans would it take
to fill up the shape?
Skilled estimators
Less skilled estimators
20. About how many letters are in the
word "adrenocorticotrophic"?
Skilled estimators
Less skilled estimators
Grade 3
Grade 5
Grade 7
.67
.33
.50
.50
1.0
.67
.0
.17
.50
.0
.33
.0
.33
.0
.50
.50
.67
.50
.0
.0
.0
.0
.17
.17
1.0
.67
.67
1.0
.50
.83
.50
.33
.67
.0
.50
.17
1.0
.83
1.0
1.0
1.00
1.00
NoTE.-Refer to App. C for more information about the interview problems.
the less sophisticated strategies (guessing
and eyeball) more frequently than older students. Conversely, fifth and seventh graders
used the more sophisticated strategies
(benchmark comparison and decomposition/recomposition) more often than did
third graders. The response "I don't know"
occurred more frequently among third graders than among older pupils.
A close examination of strategy use and
associated success rates revealed that the
use of certain strategies was associated with
a higher percentage of acceptable estimates.
The strategies that led to acceptable estimates at least 50% of the time are shown
in Table 4.
The strategies that skilled and less
skilled estimators used most frequently are
presented in Table 5. These data show that,
with one exception, skilled estimators
tended to use the higher-order strategies
(benchmark, multiple benchmark, bench-
mark comparison, and decomposition/recomposition), whereas less skilled estimators were more likely to say "I don't
know," to use strategies that relied on
guessing, or to use "false" strategies (pseudodecomposition).
Discussion
Overall, students did an adequate job of explaining their reasoning when making estimates, although students' estimates often
were not accurate. I generally understood
the essence of the strategies students described. However, third graders had more
difficulty describing their thinking than did
students in the fifth or seventh grades.
Sometimes third-grade students who said
that they were guessing or that it "just
looked like it" seemed to have actually used
some other process-one they could not adequately describe. It may be that the questions on the estimation test were too difficult
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THE ELEMENTARYSCHOOL JOURNAL
610
TABLE
3. Cumulative Frequency and Number of Acceptable Estimates for Each Strategy Used
by Skilled and Less Skilled Estimators
Grade 3
Grade 5
Grade 7
Number
Number
Number
Frequency Acceptable Frequency Acceptable Frequency Acceptable
of Use
Estimates
of Use
Estimates
of Use
Estimates
Strategy
"'Don't know":
Skilled
Less skilled
Guess:
Skilled
Less skilled
7
11
1
4
0
5
0
2
2
3
0
0
20
13
5
1
4
9
0
2
15
18
3
5
Eyeball:
Skilled
Less skilled
43
52
20
19
22
28
13
17
11
22
4
9
1
0
1
0
2
3
1
2
0
1
0
1
28
26
4
9
45
34
19
6
33
40
12
10
4
8
1
3
16
8
12
6
19
10
14
5
Pseudodecomposition:
Skilled
Less skilled
11
7
4
1
6
17
1
2
8
12
1
3
Decomposition/recomposition:
Skilled
Less skilled
6
3
5
2
25
16
16
12
32
14
33
12
Range:
Skilled
Less skilled
Benchmark comparison:
Skilled
Less skilled
Multiple benchmark:
Skilled
Less skilled
NOTE.-No student used the benchmark and fractional benchmark strategies.
for third-grade students, or the interview situation may have been too stressful. For
whatever reason, students in the third grade
did not contribute as much information to
this study as did older students.
With the exception of question 7, all
problems on which the third graders
showed any aptitude (at least a 50% solution rate) involved quantities less than 250.
This result suggests that third graders have
not developed a "feel" for large number
quantities, an explanation that is consistent
with the NAEP data (Kouba et al., 1989, p.
37). This deficiency would impede their estimation ability and may help explain why
most third graders could not estimate the
number of words in a one-page story, the
number of objects in a quart jar, the number
of days they had been alive, or the number
of links it would take to make a 1-mile-long
chain.
The fact that fifth graders had a higher
mean number of acceptable estimates on
the interview questions than did the seventh graders was unexpected. I have no logical explanation of this finding other than
the natural variation that can occur in random samples. The seventh graders had a
higher mean number of acceptable estimates than did the fifth graders on both
versions of the estimation test. However,
although the difference in mean scores was
statistically significant (p < .01) on the
open-ended version of the estimation test,
no significant difference (p > .05) was
found between the mean scores on the mulMAY 1992
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ESTIMATION
4. Estimation Strategies with a Minimum
TABLE
50% Success Rate
Success
Rate (%)
Grade 3:
Skilled estimators:
Range
Decomposition/recomposition
Less skilled estimators:
Decomposition/recomposition
Grade 5:
Skilled estimators:
Multiple benchmark
Decomposition/recomposition
Eyeball
Range
Less skilled estimators:
Multiple benchmark
Decomposition/recomposition
Range
Eyeball
Grade 7:
Skilled estimators:
Multiple benchmark
Decomposition/recomposition
Less skilled estimators:
Range
Decomposition/recomposition
Multiple benchmark
100.0
83.3
66.7
75.0
64.0
59.1
50.0
75.0
75.0
66.7
60.7
73.7
71.9
100.0
85.7
50.0
tiple-choice version. Also, the highest score
on both versions of the estimation test came
from a fifth-grade student. Students who
took the multiple-choice version of the estimation test performed better (p < .001)
than students who took the open-ended
version. This may explain why less skilled
estimators were disproportionately identified by the open-ended version of the test.
Although the difference in estimation
ability on the interview questions between
skilled and less skilled estimators was not
as great as I first predicted, one common
theme emerged: skilled estimators generally
performed better than less skilled estimators
on questions that involved large quantities
(greater than 1,000). This trend was confirmed by a t test that compared the two
types of estimators on problems that involved large numbers and those that involved smaller quantities. A significant difference (p < .001) in performance was found
611
on problems that involved large numbers,
but no significant difference (p > .50) was
found on those that used smaller numbers.
Robert E. Reys (personal communication,
June 28, 1989) reached a similar conclusion
from his research on computational estimation. Good estimation ability may be
more apparent or useful when the to-be-estimated quantity is relatively large rather
than when it is smaller.
Interview Excerpts
Insight into students' thinking can be
achieved by examining excerpts from the interviews. These excerpts illustrate both the
good and bad strategies students used in
making estimates. In the excerpts that follow, "I" represents the interviewer and "S"
the student. All names are pseudonyms.
Problem 6, which asked for estimates of
how many days the student had been alive,
resulted in a variety of solution methods.
Juanita, a third-grade student and the subject of the excerpt below, gave a reasonable
explanation for her answer and clearly demonstrated that she could use appropriate
strategies to make estimates:
I: About how many days have you been
alive?
S: 3,586.
I: How did you get your answer?
S: I went by years, and there are 365
days in a year, and my age.
In contrast, David, a seventh grader,
stretched the bounds of logic in coming up
with a mathematical way of solving this
problem:
I: About how many days have you been
alive?
S: 12,000.
I: How did you arriveat your estimate
of 12,000?
S: I timesed 76 by 88.
I: Why did you multiplythose numbers?
S: I was born in '76 and this is '88.
Some students made their estimates by
guessing and saying the first large number
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612
THEELEMENTARY
SCHOOLJOURNAL
TABLE5.
StrategiesUsed Moreby Skilled and Less Skilled Estimators
Skilled
Less Skilled
Benchmarkcomparison
Guessing
Pseudodecomposition
Decomposition/recomposition
Range
Benchmarkcomparison
Decomposition/recomposition
Multiplebenchmark
Grade3
Grade5
Grade7
Decomposition/recomposition
Multiplebenchmark
that came to their heads. Chris is in the
seventh grade:
I: Abouthow many days have you been
alive?
S: Around 1,000,000-no 1,000.
I: How did you arriveat your estimate?
Why 1,000 and not 1,000,000?
S: That [1,000,000]is a long time, and I
know thatI have not been alive that
long.
I: Has anyone been alive 1,000,000
days?
S: Yes.
I: How old would you have to be to
have lived 1,000,000 days?
S: Around 30 years.
I: So, how did you get 1,000 days for
your age?
S: I thought that there were 396 days in
a year.
I: How old is someone who has been
alive for 10,000 days?
S: About 20 years.
I: So, you're 12 years old and you've
been alive for 1,000 days. Someone
20 years old has been alive for
10,000 days and someone 30 has
Eyeball
"Don't know"
Multiplebenchmark
Eyeball
Pseudodecomposition
Guess
"Don't know"
Range
Benchmarkcomparison
Eyeball
Guess
Pseudodecomposition
"Don't know"
Range
S: 10 billion.
I: How did you arrive at your answer?
S: I just thought of a large number.
I: Do you know how many days there
are in a year?
S: Yes, 365.
I: There are 365 days in a year. You are
11 years old. Therefore you are
about 10 billion days old.
S: Yes.
Julie is a third grader:
I: Abouthow many days have you been
alive?
S: 30,000.
I: How did you arriveat your answer?
S: I just guessed.
I: Do you know how many days there
are in a year?
S: No.
I: If I told you that there were 365 days
in a year, would you feel comfortable with your answerorwould you
change it?
S: I would change it to around 800.
Don is in the fifth grade and is 11 years old:
Other students made unreasonable estimates, sometimes based on the use of a
logical process, and did not seem to be bothered when confronted with obvious contradictions. Heather is a fifth-grade student:
I: Abouthow many days have you been
alive?
I: Abouthow many days have you been
alive?
been alive for 1,000,000 days.
S: Yes.
MAY1992
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ESTIMATION
S: 70. No, 7 days in a week.
I: Do you know how many days are in
a year?
S: No.
I: Would that be useful?
S: Yes. Are there 50?
I: Thereare 52 weeks in a year and 365
days in a year.
S: Well, I've been alive 10 years and
there are 320 days in a year. So,
about 800.
Harry is in the third grade:
I: Abouthow many days have you been
alive?
S: About 60.
I: How did you get that answer?
S: I don'tknow. Itjust seemed like a reasonable number.
I: If I told you that there were 365 days
in a year, would you still be comfortablein saying that you've been
alive for 60 days?
S: Yes.
Less skilled estimators often demonstrated a lack of number sense. Donna, a
fifth-grade less skilled estimator, is the subject of the excerpt below:
613
I: About how many kernels of popcorn
are in this container?
S: 1,000.
I: About how many beans are in this
container?
S: 1,000.
I: Do you think that there are the same
number of kernels of popcorn as
there are beans?
S: No. There are about 1,008.
I: There are 1,000 kernels of popcorn
and 1,008 beans?
S: No. 1,010 popcorn and 1,080 beans.
Less skilled estimators also tended to use
irrelevant aspects of problems, as in the case
of Bridget, a fifth grader. When asked to
pour about 50 beans out of a container, she
filled a measuring cup up to the 50 millimeter mark, thinking that 50 millimeters of
liquid was also a measure of 50 beans.
In contrast, skilled estimators usually
used more sophisticated processes and focused on the relevant details of the problem. Larry is a skilled estimator who is in
the fifth grade:
I: About how many words are on this
piece of paper?
S: 250.
I: About how many links are there in a
chain 1 mile long?
S: 40.
I: How did you make your estimate?
S: Well, I just thought of a reasonable
number.,
I: You said that there were about 25
links in a 3-foot chain and 40 in a
1-mile chain. If I told you that there
were 5,280 feet in a mile, would you
change your estimate any?
S: I would change it to about 100.
Pat is a seventh-grade less skilled estimator:
I: About how many words are on this
piece of paper?
S: About 110.
I: How did you get your estimate?
S: There are about 100 on half of the
page. So there are about 110 on the
whole page.
Fred is a less skilled estimator who is in the
fifth grade:
I: How did you get your answer?
S: I looked at the sentences. Therewere
about 10 words in each one. Then
I estimated how many sentences
there were.
I: About how many M&M'sare in this
large bag?
S: 250.
I: How did you get your estimate?
S: I thought there were about 45 in the
small bag, and the big bag is five or
six times as large.
Jane is a skilled estimator who is in the seventh grade:
I: Abouthow many days have you been
alive?
S: About 4,800.
I: How did you get your estimate?
S: Eachyear is about 400 days long, and
I'm almost 12, so that would be
about 4,800 days.
Manuel is a fifth-grade skilled estimator:
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614
THE ELEMENTARYSCHOOL JOURNAL
I: About how many M&M'sare in this
large bag?
S: About 400.
I: How did you get your estimate?
S: I thoughttherewere about 50 M&M's
in the small bag. The large bag
weighs about eight times as much
as the small one, and 8 times 50 is
400.
Maria is also a fifth-grade skilled estimator:
I: About how many links would it take
to make a chain that is 3 feet long?
S: About 40.
I: How did you arriveat your estimate?
S: The links look like they are about 1
inch long.
From a synthesis of all the interviews,
several conclusions can be drawn. Students
from all three grades and both estimation
ability groups varied in their ability to make
estimates. Students who made accurate estimates tended to have good number sense
and metacognitive skills, isolated important
components of a problem while ignoring irrelevant information, were aware of the reasonableness of their answers, and were flexible in their thinking. Less skilled estimators
frequently guessed, did not adjust their estimates even when confronted with contradictory evidence, and were not adept at verbalizing their thinking strategies. As in the
Baroody and Gatzke study (1991), many of
the third-grade students did not seem to
have developed accurate mental benchmarks (frames of reference) to use when
making estimates.
Classroom Implications
The results of both the estimation test
and interview questions suggest that students' ability to make estimates of discrete
quantity is generally poor. Students' weakness in this area could be due to a variety
of factors: poor number sense, inability to
comprehend large-number quantities, or
undeveloped computational estimation or
mental computation skills. However, the
primary reason may simply be that, con-
trary to the intentions of documents such
as the Standards (National Council of
Teachers of Mathematics, 1989), the systematic study of estimation, and particularly
estimation of discrete quantity, is generally
not included in the elementary school
mathematics curriculum.
In order for students to acquire skill in
estimation, they must have practical experiences in making estimates so that they can
develop their own individual frames of reference for estimating the quantity of various
types of measurement (numerousness,
time, length, etc.) and different quantities
such as 100 (the height, in inches, of a basketball rim), 10,000 (the number of kernels
of popcorn in a quart), 100,000 (the number
of people who will fit into a large football
stadium), and 1,000,000 (the number of seconds in 2 weeks). Once these points of reference are established, specific estimation
strategies involving the use of benchmarks
and decomposition/recomposition should
naturally evolve.
Verbalization and being able to determine whether an answer is reasonable are
important components of understanding
and therefore should play a significant role
in mathematics classrooms. My interviews
indicated that students often changed their
initial estimates after I questioned them
about their thinking strategies. Students
sometimes found a more reasonable estimate after being questioned about whether
an estimate made sense in the context of
the problem. Much of the lack of success in
estimating among third graders may be attributable to their inability to verbalize their
thinking processes. All students should regularly be given opportunities to verbalize
their problem-solving strategies and should
be asked constantly to consider and defend
the reasonableness of their estimates.
The results of this study identified two
primary strategies, multiple benchmark and
whose use
decomposition/recomposition,
tended to lead to accurate estimates. It is
clear from the data that these two strategies
are most valuable and should be the basis
MAY 1992
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ESTIMATION
of instruction in estimation
of discrete
quantity.
Limitations
This study involved only 36 students
from one small, rural, midwest district.
Generalizations of the results to other populations of students should be made with
caution. If students from another school district or age group had been studied, or if
students with average estimation skills had
participated, different results might have
been obtained.
Another limitation of the study was the
low reliability of the estimation test. The
difficulty in devising questions that tested
single mathematical concepts and were
suitable for students in grade 3 as well as
those in grade 7 may have contributed to
the low reliability. However, the low reliability of the estimation test should affect
only conclusions regarding the relative abilities of skilled and less skilled estimators
and not the more important findings involving strategy use.
The interview as a means of data collection also has limitations. It is possible
that students do not always verbalize their
strategies accurately or react well to the
pressure of being in an interview situation.
The questions I used may not have been
free of biases related to gender, age, interests, and background, or I may not have
properly interpreted students' responses
during the interviews.
A final limitation is that any interpretation of the strategies students employed
when answering the interview items must
be made with the realization that all but two
questions were open ended. The benchmarks that characterize multiple-choice,
number-referent, or order-of-magnitude
questions may affect the choice of strategy
used to make an estimate. Therefore, different results may have been observed if the
interview
items had been worded
differently.
615
Directions for Future Research
Although this study provided new insight into children's thinking processes and
strategy usage when making estimates of
discrete quantity, many more questions remain unanswered. For example, researchers
need to determine the grade at which estimation instruction should begin. Third
graders did not contribute as much information to the results of this study as did
fifth and seventh graders. Whether this was
a weakness of the interview, attributable to
the younger students' immaturity (lack of
prior training and experience, undeveloped
metacognitive skills, absence of the mental
constructs that are prerequisites for estimation, etc.), or due to students being nervous
or intimidated when being interviewed is
unknown. Further research that attempts to
explain why fifth graders performed better
than seventh graders on the interview questions would also be beneficial.
A second question for future research
concerns the most effective ways of developing students' ability to estimate discrete
quantities. Although in this study I identified strategies that led to more accurate estimates, educators need to know the most
effective ways of developing these strategies in school-aged children. For example,
what content and activities should be included in the curriculum and how much
time should be devoted to these activities?
Should estimation be taught independently
of other mathematical topics or integrated
with these topics? In what order should topics be presented to maximize students' estimation skills?
Previous research suggests that estimation is not an independent construct but
seems to be related to other mathematical
skills such as mental computation, spatial
visualization, and measurement. Research
exploring the association between estimation of discrete quantity and other mathematical skills would yield a better understanding of this complex mathematical
topic.
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616
THE ELEMENTARYSCHOOL JOURNAL
Appendix A
Categories and Examples of Strategies
Students Used to Estimate Discrete
Quantities during Interviews
"Don't know": Student was unable to explain
the estimate
I: Are there more or fewer than 1,000
X's on this piece of paper?
S: More.
I: How did you get your estimate?
S: I don't know.
Guess: Student indicated that the estimate was
a guess or that the student just thought of an
answer
I: About how many M&M's are in this
bag?
S: 20.
I: How did you get your estimate?
S: I just guessed.
Eyeball: Student gave a perceptual description
as the explanation for the estimate
I: About how many kernels of popcorn
are in this jar?
S: 150.
I: How did you get your estimate?
S: It just looks like there should be that
many.
Range: Student indicated boundaries of a range
for locating the estimate
I: About how many words are on this
page?
S: 125.
I: Why did you say 125?
S: There are at least 100, and I didn't
think there were 150.
Benchmark comparison: Student explained estimate by comparing the stimulus to another
object or distance
I: About how many beans are in this jar?
S: 4,000.
I: How did you get your estimate?
S: I estimated that there were 5,000 kernels of popcorn in that other jar,
and the beans are larger than the
kernels.
Benchmark: Student mentioned unit of measure
or measuring instrument that was carried
mentally or used in a gesture
I: About how long is this ant's head?
S: 1 cm.
I: How did you figure out 1 cm?
S: It's really tiny like a centimeter; it
looks like a centimeter on a ruler.
Fractional benchmark: Student mentioned a fractional unit or measure or started with a unit
and during explanation broke it down into a
fractional unit
I:
S:
I:
S:
About how tall is this letter "a"?
A 1/4 inch.
How did you get 1/4 inch?
It looks like four a's would make 1
inch.
Multiple benchmark: Student applied a unit of
measure multiple times
I: About how many links would it take
to make a chain 3 feet long?
S: 36.
I: How did you get your estimate?
S: A link is about 1 inch long, and there
are 36 inches in 3 feet.
Student showed an
Pseudodecomposition:
awareness that the problem could be broken
down into parts but clearly did not use this
information to formulate an estimate
I: About where on this page would you
find the seventieth name?
S: Halfway down the first column.
I: How did you decide that?
S: I counted a little bit and there are a
lot of names.
Decomposition/recomposition: Student divided
problem into parts and then recombined them
to arrive at an estimate
I: About how many dots are on the inside of this shape?
S: 70.
I: How did you get your estimate?
S: I looked at how many were in a row,
how many rows there were, and
multiplied.
NoTE.-Strategies and benchmark and fractional benchmark excerpts are from Siegel et al.
(1982). All other excerpts are from interviews
conducted by author.
Appendix B
Estimation Test (Multiple-Choice
Version)
1. About how many M&M's would it take to
make a line that is 1 inch long? (Student is
shown an M&M.)
A) 2
b) 20
c) 200
d) 2,000
2. About how many M&M's are in a small bag
of M&M's? (Student is shown a small bag of
M&M's.)
A) 50
b) 500
c) 5,000
d) 50,000
3. About how many M&M's are in a 1-pound
bag of M&M's? (Student is shown a 1-pound
bag of M&M's.)
A) 500
b) 5,000 c) 50,000 d) 500,000
4. About how tall is the average two-story
house?
a) 10 feet B) 30 feet c) 90 feet d) 270 feet
MAY 1992
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ESTIMATION
5. About how many people are in your school?
C) 1,000 d) 3,000
a) 90
b) 490
6. About how many people live in the state of
Missouri?
a) 5,000 b) 50,000 c) 500,000 D) 5,000,000
7. About how many days have you been alive?
B) 5,000 c) 50,000 d) 500,000
a) 500
8. About how many multilink cubes would you
have to stack on top of each other to make
a stack that is 6 feet tall? (Student is shown
a multilink cube.)
b) 900
A) 90
c) 9,000
d) 90,000
9. About how many lots-a-links would it take
to form a chain that is 3 feet long? (Student
is shown a lots-a-link.)
a) 3
B) 30
c) 300
d) 3,000
10. About how many lots-a-links would it take
to form a chain that is 1 mile long? (Student
is shown a lots-a-link.)
a) 500
b) 5,000 C) 50,000 d) 500,000
11. About how many words are on this piece of
paper? (Student is shown a story typed on a
piece of paper.)
a) 30
B) 300
c) 1,030
d) 10,030
12. About how many kernels of popcorn are in
this container? (Student is shown a jar filled
with popcorn.)
a) 90
b) 900
C) 9,000 d) 90,000
13. About how many beans are in this container?
(Student is shown a jar filled with beans.)
a) 30
b) 300
C) 3,000 d) 30,000
14. About how many M&M's are in this container? (Student is shown a jar filled with
M&M's.)
a) 10
b) 100
C) 1,000 d) 10,000
15. About how many lots-a-links are in this container? (Student is shown a jar filled with
lots-a-links.)
a) 20
B) 200
c) 2,000
d) 20,000
16. About how many beans would it take to fill
a gallon jug?
a) 1,000 B) 10,000 c) 100,000 d) 1,000,000
17. In which column would you find around the
seventieth name? (Student is shown a page
from the phone book.)
A) 1st
b) 2nd
c) 3rd
18. About how many names are on this page
from the phone book? (Student is shown a
page from the phone book.)
a) 50
B) 300
c) 1,000
d) 10,000
19. About how many pegs are enclosed by the
following figure? (Student is shown a rectangular array of dots with a polygon enclosing a certain number of them.)
A) 60
b) 260
c) 860
d) 1,460
20. About how many lots-a-links would it take
to fill this container? (Student is shown an
empty jar.)
617
B) 200
c) 2,000 d) 20,000
a) 20
21. About how many M&M's would it take to
fill this container? (Student is shown an
empty jar.)
C) 1,000 d) 10,000
a) 10
b) 100
22. About how many beans would it take to fill
this container? (Student is shown an empty
jar.)
C) 3,000 d) 30,000
a) 30
b) 300
23. About how many kernels of popcorn would
it take to fill this container? (Student is
shown an empty jar.)
a) 90
b) 900
C) 9,000 d) 90,000
24. About how many dots are there on this piece
of paper? (Student is shown a paper full of
dots.)
A) 3,000 b) 13,000 c) 23,000 d) 53,000
NOTE.-Correct answers are denoted by the
usage of a capital letter in the foil. The openended version of the estimation test is identical
to the multiple-choice version except for the absence of the foils.
Appendix C
Interview Questions
1. About how many pipes does this pipe organ
have? (Student is shown a picture of a pipe
organ.)
Acceptable estimate: 40-120
2. About how many words are on this piece of
paper? (Student is shown a story typed on a
piece of paper.)
Acceptable estimate: 150-450
3. About how many M&M's are in this bag?
(Student is shown a small bag of M&M's.)
Acceptable estimate: 25-75
4. About how many M&M's are in this large
bag? (Student is shown a large bag of
M&M's.)
Acceptable estimate: 250-750
5. About where on this page would you find
the seventieth name? (Student is shown a
page from the phone book.)
Acceptable estimate: Middle of the first column to middle of second column. Approximately 300 names were on the page.
6. About how many days do you think you
have been alive?
Acceptable estimate: 8 yrs.: 1,460-4,380
9 yrs.: 1,642-4,928
10 yrs.: 1,825-5,475
11 yrs.: 2,007-6,023
12 yrs.: 2,190-6,570
13 yrs.: 2,375-7,118
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618
THE ELEMENTARYSCHOOL JOURNAL
7. Are there more or fewer than 1,000 X's on
this piece of paper? (Student is given a piece
of paper with 600 X's typed on it.)
Acceptable answer: Fewer
8. About how many kernels of popcorn are in
this jar?(Student is given a clear jar in which
many kernels of popcorn are visible.)
Acceptable estimate: 4,500-15,500
9. About how many beans are in this jar? (Student is given a clear jar in which many beans
are visible.)
Acceptable estimate: 1,500-4,500
10. About how many M&M's are in this jar?
(Student is given a clear jar in which many
M&M's are visible.)
Acceptable estimate: 500-1,500
11. About how many lots-a-links are in this jar?
(Student is given a clear jar in which many
lots-a-links are visible.)
Acceptable estimate: 100-300
12. About how many links would it take to form
a chain that is 3 feet long? (Student is given
a lots-a-link.)
Acceptable estimate: 15-45
13. About how many lots-a-links would it take
to form a chain that is 1 mile long? (Student
is given a lots-a-link.)
Acceptable estimate: 25,000-75,000
14. Pick out an object that is around 25 multilink
cubes long. (Student is given a multilink cube
and four wooden dowels of length 6, 12, 20,
and 36 inches.)
Acceptable answer: the 20-inch dowel
15. About how many beans would it take to fill
this jar?(Student is shown a large empty jar.)
Acceptable estimate: 1,500-4,500
16. About how many beans would it take to fill
a gallon jug?
Acceptable estimate: 5,000-15,000
17. Give me around 50 beans. (Student is given
a clear container with several hundred beans
in it and a 1-cup measuring container.)
Acceptable estimate: 25-75
18. About how many dots are enclosed by the
shape? (Student is given a rectangular array
of dots with a polygon enclosing a number
of dots.)
Acceptable estimate: 30-90
19. About how many beans would it take to fill
up the shape? (Student is given the array of
dots from problem 18.)
Acceptable estimate: 60-190
20. About how many letters are in the word "adrenocorticotrophic?" (Student is shown the
word "adrenocorticotrophic.")
Acceptable estimate: 10-30
Note
This article is based on the author's doctoral
dissertation completed at the University of Missouri-Columbia in 1989 under the direction of
Robert E. Reys.
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