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MEASUREMENT ESTIMATION SKILLS AND STRATEGIES

OF LOWER GRADE STUDENTS

Lisa Stinken and Stefan Heusler

Westfälische-Wilhelms Universität Münster, Germany

Abstract: Measurement estimation is an important part of everyday live and a higher-level

competence in science and mathematics education. In order to improve estimation skills, at

first, estimation abilities and strategies have to be examined. In this study, a questionnaire and

an interview survey are combined in order to determine measurement estimation skills and

strategies used by German students. So far, over 800 students in the grades eight to ten and 30

college juniors participated in the questionnaire survey. First results show no significant

improvement of the estimation abilities for higher grades. Both; pupils and students have a

lack in estimation skills. We found that estimates of physical quantities which are used

quantitatively in everyday live, and/or perceptible quantities (such as temperature) were more

accurate than others like force or acceleration. In addition to the questionnaire, first interviews

revealed that students are untrained estimators, but also that they have too high confidence in

their own estimates. Besides this, a whole number of different estimation strategies could be

identified, confirming those known from previous estimation studies in mathematics, but

expanding the range to physical quantities such as force or velocity, where new strategies like

‘physical decomposition’ were observed.

Keywords: measurement estimation, accuracy, strategies

INTRODUCTION

Estimation ability is an interdisciplinary competence. It has a great relevance in mathematics

and natural sciences. It is not only used in the laboratory or the classroom, but is essential also

in everyday live. Furthermore, estimation is often the only possible way to achieve a result,

for example if measurements are impossible, or if not all data are known, or if the situation is

too complex for exact calculations. Making estimates is timesaving and can give a quick

overview of the situation. It can also serve as a basis for decision making, not only in science,

but also in economics and even politics. For these reasons, one important aspect of natural

science education is to enable students to make accurate estimates. However, in physics

education, measurement estimation is mostly taught implicitly. In Germany, measurement

estimation is even only part of elementary school education in mathematics. But is this

sufficient? How can estimation skills be improved? Which estimation strategies emerge,

which are most successful?

State of Research

Studies in the field of mathematical education over the last 60 years have shown that both

students and adults have great deficits in their ability of measurement estimation (Crawford &

Zylstra, 1952; Reys et al. 1982; Hildreth, 1983; Crites, 1992; Joram et al., 2005). But almost

all of the studies existing so far have focused on quantities like numbers, length or area, which

play an important role in mathematics. Only a few studies included some physical quantities

like velocity, time or temperature (e. g. Corle, 1960; 1963). The principal aim of this study is

to fill in this gap and to investigate the estimation ability of students concerning quantities

which are commonly used in the physics classroom. Additionally, estimation strategies will

be analyzed.

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METHODS

Given the aim of investigating the estimation abilities and strategies used by students for

different physical quantities, a broad range of estimation tasks has been developed. First, a

questionnaire to determine the accuracy in measurement estimation was designed; second,

interview questions were formulated to identify the estimation strategies. Additionally, some

of the questionnaire and interview tasks involved the request to quote the accuracy of the own

estimation just made. The study deals with physical quantities length, mass, time,

temperature, area, volume, density, acceleration, speed and force. A first run of the

questionnaire survey also included more abstract quantities like energy, power and current.

For each quantity, there were at least four different everyday life objects or activities to

estimate (TEO: ‘to estimate object’) in order to determine the accuracy of the given

estimations for the given quantity. The estimation tasks were structured in five everyday

situations. In Figure 1 the first estimation situation is shown as an example.

Figure 1. Everyday life situations as shown in the example give the context for the

estimation task for the different physical quantities.

The aim of the interview survey was to identify strategies used by students to generate

estimates. There were two different types of estimation tasks: measurement estimation

problems in which the TEO was physically present and problems in which it was physically

absent. Again, the accuracy of the estimates was determined; and afterwards the estimation

strategies were analyzed.

Pilot tests of the questionnaire and the interview guideline were conducted on a small number

of students to ensure that children are familiar with TEOs and are able to understand all

involved tasks.

By now over 800 grammar school pupils from North Rhine-Westphalia and Lower Saxony in

the grades eight to ten (14-16 years) participated in the questionnaire. In addition over 30

first-year college students took part as comparison group.

RESULTS

The results are separated in two sections: (1) the accuracy of the measurement estimations for

different quantities and (2) the used measurement estimation strategies.

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Accuracy

In order to determine the accuracy of the given estimates the mean ratio of each estimate and

the related TEO was calculated for each student and physical quantity.

=�

=1

An accuracy value of one represents perfect estimates, a value above one indicates that the

student tents to overestimate this quantity and a value beneath one indicates an

underestimation of this student on average. In Figure 2 the resulting boxplots for each

quantity are shown. It can be seen that lower grade students are rather good in estimating

length and temperature. Over 50% of the given estimates are within the range of -50% to

+100% of the TEO. In mean, the quantities velocity, force, area and volume were

underestimated by the students, especially force and volume with over 50% of the estimates

less than half the size of the TEO. In contrast, more than half of the students tend to

overestimate the quantities mass, time, acceleration and density. Particularly striking are the

estimates concerning time and density. Over 75% of the students overestimate times by a

factor of two. The large variations of the estimation ability can be seen especially in the

accuracy of the estimates concerning density. The range of the given estimates varies

strongly, as can be seen on the length of the box, which represents the range of the middle

50% of the estimates. Again, length and time are the quantities for which not only the highest

accuracy in the estimates could be identified, additionally almost all students show similar

estimation abilities (small box length).

Figure 2. Box-plots of estimates given by the students in the questionnaire survey.

Presented are the mean ratio of the estimates to the TEOs, the dashed lines represent a

deviation of -50% to +100% to the TEO.

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The questionnaire survey showed almost no increase in the measurement estimation ability

between the grades eight to ten and first-year college students. There were no significant

differences between the grades eight and ten and only significant differences in the estimation

ability concerning the quantities mass, acceleration (8th grade vs. students), Temperature (8th,

10th grade vs. students) and force (10th grade vs. students).

Additionally, only the minority of expected correlations between the estimation ability of

interconnected quantities like length and area could be verified. For this purpose, the

Spearmans correlation coefficient was determined for each combination of investigated

physical quantities. Small positive significant correlations (**p ≤ 0.01) could only be found in

the combinations of area and volume (rs=.214** to rs=.337**) and acceleration and velocity

(rs=.267** to rs=.302**).

Also of interest is, that for the majority of students no significant correlation between the

physical expertise and their estimation ability was determined.

Strategies

The analysis of the 31 interviews confirmed the questionnaire results concerning the accuracy

of the estimation ability of students for different quantities. Here again the estimates of

Length and Temperature were most accurate.

The analysis of the used strategies showed that students use various strategies when

estimating physical quantities. Strategies that have been described in previous mathematical

education studies, for instance by Forrester et al. (1990), Hildreth (1983), Joram et al. (1998)

and Siegel et al. (1982), could be recovered. Additionally, some new adequate strategies like

‘Physical Decomposition’ could be discovered. In total, students applied 31 different

estimation strategies separated in four main categories (see Figure 3).

Figure 3. Estimation strategies used by the students. The percentages are normalized to

the total number of estimation strategies used by all students in the interviews.

The most commonly used strategies were Benchmarks (23%), Argumentation (14%), Physical

Decomposition & Calculation (14%), Mental / Real Measurement (11%) and Limits (9%).

The use of Benchmarks means in general the comparison of a known standard or familiar

object with known size to the TEO. For example, in order to estimate the height of a door one

could imagine the own body height and compare this height with the door to generate an

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estimate. By using the Argumentation strategy one justifies the estimation on every day or

physical knowledge. For instance the temperature of a pocket warmer has to be below 100°C,

since otherwise the user would burn himself. The Physical Decomposition & Calculation

simply describes the decomposition of the TEO into subTEO, which can be estimated easier.

Subsequently, the subTEOs are combined with the help of a known formula. One example for

this strategy is the decomposition of a rectangle in its length and width. After estimating these

subTEOs, the area can be determined by multiplying the length and width. When using a

Mental / Real Measurement students imagine either a measurement device next to the TEO

reading the value mentally or they use an object with known size as measurement instrument

(e.g. finger range). Another often used strategy is the statement of Limits. In this case students

indicate instead of a specific estimate a range with lower and upper limit in which they

suspect the size of the TEO.

The choice of the applied strategy is independent of the age of the asked student, 8th grade

pupils and first-year students show no significant differences in their strategy choice.

The applied strategy is also in the majority of the cases independent of the physical presence

or absence of the TEO. Only exceptions are the quantities length and area. If the TEO is

present while a length or area estimation, students tend to apply more real measurements, for

example by using their finger range as measurement instrument.

Strategies, which led to most accurate and most inaccurate estimates, are listed in Table 1.

Nearly always one of the most used strategies leads to accurate estimates concerning at least

one physical quantity. Only exception is the Argumentation, although Argumentation is in

total the second most used strategy, the use of this strategy does not imply an accurate

estimate. Quite the contrary seems to be true. Estimates, which base mainly on Everyday

Argumentation, are often inadequate. For five of the ten physical quantities the use of

Everyday Argumentation leads with a high probability to inaccurate estimates. Apart from

Everyday Argumentation the use of Benchmarks, Real Measurement, Pseudo Physical

Decomposition and Calculation (inadequate) increase the probability of inadequate estimates.

The use of Benchmarks can lead to accurate estimates as well as to inaccurate estimates. The

crucial factor when applying Benchmarks is the correct size of the known standard or familiar

object. Although students, who imagine the comparison object with an incorrect size, may

apply the Benchmark strategy correctly, they still achieve an inaccurate estimate due to the

deviation between the real and their assumed size of the used comparison object. The same

problem appears when using the Real Measurement strategy. As long as the assumed size of

the known object, which is used as measurement instrument, is not identical with its real size

this strategy will unavoidable lead to inaccurate estimates. Another not promising strategy is

the Pseudo Physical Decomposition. Just as in the Physical Decomposition students start to

break down the TEO into subTEOs, but instead of combining the subTEOs subsequently, they

break off at this point. This may be due to the unawareness of the appropriate formula. Often

the students switch after this failed estimation try to the Everyday Argumentation strategy.

When using the Physical Decomposition & Calculation strategy main causes for inaccurate

estimates are the use of an inappropriate formula (inadequate calculation) and the proceeding

calculation of the TEO with previously inaccurate estimated subTEOs (adequate calculation).

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Table 1. Strategies leading with high probability to accurate and inaccurate estimates

for each physical quantity.

Physical

quantity

Strategies used for most accurate

estimates

Strategies used for most

inaccurate estimates

Length

Benchmarks

Benchmarks (incorrect assumed size

of Benchmark)

Mass

Benchmarks

Everyday Argumentation & Sense

of Proportion

Time

Mental / Real Measurement

Real Measurement (incorrect

assumed size of measurement

instrument)

Temperature

Benchmarks & Limits

Everyday Argumentation

Area

Physical Decomposition &

Calculation (adequate)

Physical Decomposition &

Calculation (adequate, but incorrect

size of subTEOs or inadequate)

Volume

Benchmarks

Extension

Velocity

Physical Decomposition &

Calculation (adequate)

Pseudo Physical Decomposition &

Everyday Argumentation

Force

Mental / Real Measurement

Everyday Argumentation

Density

Physical Decomposition &

Calculation (adequate)

Pseudo Physical Decomposition &

Everyday Argumentation

Acceleration

Physical Decomposition &

Calculation (adequate)

Benchmarks, Physical

Decomposition & Calculation

(inadequate)

DISCUSSION AND CONCLUSIONS

The study showed that pupils as well as college students are pretty good in estimating length

and temperature. In contrast, they are poor estimators for abstract quantities such as

acceleration or density. This is partly due to the lack of accurate Benchmarks, as well as due

to the unawareness of appropriate formulas for calculating these quantities based on

subTEOs.

No significant correlation between the estimation ability and mathematical and physical

knowledge is detectable. A significant correlation of estimation abilities could only be

verified for the directly related quantities “area and volume” and “acceleration and velocity”.

Since there is no significant increase in the estimation ability between the eighth grade and

graduation, it can be assumed that teaching estimation skills implicitly does not work well.

Our results also showed no differences in the strategy use for lower grade and first-year

college students. This indicates, that students do not develop new estimation strategies.

Therefore, it is necessary to improve the education concerning estimation abilities using those

strategies which lead to the most accurate estimation results. The importance of a good

estimation ability in daily live is obvious. Students must be given sufficient occasion to

develop and practice their skills. The training of measurement estimation beyond the

elementary school in mathematics and science could further increase the estimation ability of

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students, especially since many physical quantities are first introduced in higher grades.

One possible approach to increase the estimation ability of students is the development of an

adequate benchmark system and the explicit training of different estimation strategies, since

results up to now indicate that students are good estimators if they have a distinctive

repertoire of adequate benchmarks for basic quantities (length, mass, time and temperature)

and know how to determine derived physical quantities (are, volume, velocity, etc.) from

basic quantities. In contrast, if students use Benchmarks with an incorrect assumed size or do

not have the physical knowledge to calculate physical quantities based on underlying

subTEOs, the probability for inaccurate estimates increases strongly (see. Tab. 1).

REFERENCES

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Arithmetic Teacher, 7, 333-340.

Corle, C. (1963). Estimates of quantity by elementary teachers and college juniors. The

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Crawford, B. & Zylstra, E. (1952). A Study of High School Seniors Ability to Estimate

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Crites, T. (1992). Skilled and Less Skilled Estimators' Strategies for Estimating Discrete

Quantities. The Elementary School Journal, 92, 601-619.

Forrester, M., Latham, J. & Shire, B. (1990). Exploring Estimation in Young Primary School

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Hildreth, D. (1983). The Use of Strategies in Estimating Measurements. The Arithmetic

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