ZDM

, Volume 43, Issue 5, pp 709–722

Iranian students’ measurement estimation performance involving linear and area attributes of real-world objects

Authors

    • Faculty of Mathematical SciencesShahid Beheshti University
  • Leyla G. Khosroshahi
    • Faculty of Mathematical SciencesShahid Beheshti University
  • Anne R. Teppo
Original Article

DOI: 10.1007/s11858-011-0338-1

Cite this article as:
Gooya, Z., Khosroshahi, L.G. & Teppo, A.R. ZDM Mathematics Education (2011) 43: 709. doi:10.1007/s11858-011-0338-1
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Abstract

This article reports on an exploratory investigation of the measurement estimation performance of ten Iranian high school students on a set of real-world length and area measurement tasks. The results of a qualitative analysis of the data indicate that the students employed a variety of either mental or physically present Individual Frames of Reference as the non-tool units of measure in various estimation tasks. The analysis also found that a range of types of frames of reference was used across students in response to particular tasks and to the physical environments in which the tasks were situated. These results suggest that there is a complex interaction among a student’s individual preference for a particular type of Individual Frame of Reference, the nature of the estimation activity, and the physical context in which the activity takes place. These findings, which contribute to an understanding of the nature of the measurement unit that is employed during an estimation process, provide a different perspective from other studies that focus on categorizing estimation strategies, or processes.

Keywords

High school studentsMeasurement estimation tasksLinear measurementArea measurementIndividual Frame of Reference

1 Measurement

Most numbers that we see and work with in our daily lives are measures. To name a few—the distance between two cities, the temperature, capacity and weight of edibles that we buy and eat, car speed, and the time we spend traveling between home and workplace. In many cases, we respond to such situations by estimating. When we fill a bag with fruit, e.g., we judge the amount of fruit to buy by how heavy it is and how full we fill the bag.

“Measuring can be described as ordering our surrounding world through numbers in order to better control that world” (Buys & De Moor, 2008, p. 15). While measurement methods have expanded to include such areas as statistics (i.e., mean and variation), psychology (i.e., IQ scores), and social research (i.e., poverty levels), our focus in this article is on the measurement of physical phenomena.

Measurement is the process of assigning a numerical value to an attribute of an object by comparing the attribute to some preselected unit (Sowder, 1992; National Council of Teachers of Mathematics, 2000). The unit may be represented by a standardized tool (as in a meter stick) or by some kind of informal or personal non-standard unit (as in the length of one’s hand span). Measurement may be mediated by automated tools, such as a digital scale used to determine the weight of a bag of fruit, or by a car’s odometer that records cumulative distance traveled.

The fundamental processes that underlie measurements of physical attributes are the iteration or the accumulation of the unit of measure. In the case of weight, think of this process as placing a certain number of kilograms and grams on a balance scale to equal the weight of the required object. For distance, imagine a meter stick being successively laid down to cover the specified length. The quantity, or magnitude, that is recorded is the count of the number of units employed (e.g., to totally balance, or that are repeatedly applied). The value of the magnitude is a function of the size of the unit of measure. For example, the distance traveled will be a larger number if it is measured in meters rather than in kilometers.

Units of measure can be used in fractional ways as well as through iteration of the whole unit. Think of what is involved in using a meter stick to measure a floor tile. Here, the result is read directly from the markings on the stick, where the length of 1 m has been subdivided into 100 smaller, equal divisions. This partitioning into fractional amounts makes it possible to take increasingly more precise readings of the physical attributes of various objects.

The measurement tasks that competent individuals carry out on a daily basis are supported by an underlying knowledge of basic measurement concepts and skills. This knowledge, while developed through classroom instruction, is also honed by experience in everyday life and specialized within the workplace.

The Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000, p. 44) states that the goal of instructional programs “from prekindergarten through grade 12 should enable all students to—understand measurable attributes of objects and the units, systems, and processes of measurement; [and] apply appropriate techniques, tools, and formulas to determine measurements.”

Beyond formal schooling, the Cockcroft Report (1982) describes, “the mathematical needs of employment as a ‘feeling for measurement.’

This implies very much more than an ability to calculate, to estimate and to use measuring instruments, although all of these are part of it. It implies an understanding of the nature and purposes of measurement, of the many different methods of measurement which are used and of the situations in which each is found; it also implies an ability to interpret measurements expressed in a variety of ways” (Chapter 3, p. 85).

2 Measurement estimation

2.1 What is meant by measurement estimation?

Measurement competence includes understanding why, knowing how, and knowing that in relation to fundamental measurement concepts and principles, procedural skills, mathematical formulae and relationships, and measurement facts as well as possessing an appropriate vocabulary and set of benchmark units for measurement estimation and evaluation.

The aspect of measurement competence that is the focus of this article is that of measurement estimation. Measurement estimation can be described as making a measurement without using measurement tools (Bright, 1976). In such a situation, the unit of measure consists of some kind of mental referent unit that provides a “feel for the size of the unit” (Sowder, 1992). Adams and Harrell (2003, p. 229) describe this process as making “a judgment or [developing] an opinion about a particular measurement attribute.”

Estimation develops through experience. Skill in estimation grows with increased measurement competence, practice with particular measurement tools, the accumulation of a set of mental referent units, and the development of a range of estimation strategies that are related to the contexts in which the estimations take place (Cockcroft, 1982; Crites, 1992). Estimation competence is most developed in areas where individuals use particular forms of measurement as part of their everyday activities (Joram, Subrahmanyam & Gelman, 1998).

Measurement estimation is an important real-life skill. In interviews of adults, common reasons for workplace estimation is that it does not require tools, can be performed quickly, and, in many situations, it may be the only measurement option available (Adams & Harrell, 2003). The butcher estimates the thickness and weight of meat when a portion is cut for a customer, and it is only after the portion is prepared that it is actually weighed. When driving a car, we maintain an appropriate distance behind the vehicle in front of us to allow sufficient space for sudden breaking situations. The ability to estimate also provides a way to quickly judge the reasonableness of answers obtained by actual physical measurements or to assess the validity of a particular measurement method.

Estimation also plays an important role in learning both the principles and procedures of measurement. Because estimations are not made by the direct application of tools, students engaged in such tasks are freed from a purely procedural focus on tool use and must rely instead on more fundamental principles of measurement (Joram et al., 1998). Estimation tasks provide students opportunities to learn and practice measurement skills, and provide teachers opportunities to assess students’ measurement understandings. Practice with estimation also helps students develop a sense for the magnitudes of standard measurement units (Joram, Gabriele, Bertheau, Gelman & Subrahmanyam, 2005); e.g., to provide an answer to the following (Jones & Rowsey, 1990, p. 902), “Is a liter of gasoline more likely to fill an automobile gasoline tank or is it more likely to fill a lawn mowers gasoline tank?” Finally, estimation activities present occasions for students to become engaged in mathematical activities such as problem solving, the application of other areas of mathematics, and explorations of the link between abstract mathematical ideas and real-world applications (Hodgson et al., 2003).

Joram et al. (1998, p. 413) note that “measurement estimation has been identified as a critical area for mathematical development, yet little is known about how estimators make their judgments and how competency in measurement estimation can be supported through instruction.” The authors support this claim with a summary of trends in research on measurement estimation during the last half of the twentieth century, characterizing changes in focus through three “generations” of studies. Initially, researchers “were interested in simply documenting how well children and adults could estimate” (p. 417). In the next generation, studies from the 1960s through the 1970s shifted attention to examine individuals’ abilities to estimate specific attributes of physically present objects, and the degree of accuracy of such estimates. Beginning in the 1980s, researchers moved to investigations of the cognitive processes that people employed during the estimation tasks.

2.2 Measurement estimation strategies

Empirical research has also investigated the nature of the processes, or strategies, that individuals use in measurement estimation. Two studies in particular treat the same topics of estimation (length and area) as are the focus of our investigation—one by Hildreth (1983) and one by Castillo (2006, reported in Segovia & Castro, 2009). Hildreth identified a range of strategies for estimating length and area using a 24-item test that was administered to 24 students each in fifth and seventh grade and in the freshmen year of college. Castillo studied pre-service teachers’ and ninth grade students’ performance on estimation tasks for length, area, volume, and weight. Two of the four tasks included estimating the length of the teacher’s desk and the surface area of a blackboard.

The length and area estimation strategies identified by Hildreth (1983, pp. 50, 51) and Castillo (Segovia & Castro, 2009, p. 521) share both similarities and differences. Similar strategies include:
  • unit iteration (either with a length or an area unit);

  • comparison with referents (either equal to, larger than or smaller than the object) that were either physically present or absent;

  • squeezing (i.e., placing the estimate between two other measures);

  • formulas that employ estimated measures (e.g., A = L × W).

Additional strategies identified by Hildreth include:
  • subdivision clues (e.g., width of doorways used to measure hallway);

  • prior knowledge (e.g., known area for floor or ceiling tile).

Additional strategies identified by Castillo:
  • rough guess;

  • breaking down into either equal, or different-sized parts;

  • readjusting (i.e., estimating and then making adjustments to that value).

2.3 Importance of an Individual Frame of Reference

Measurement estimation is a complex activity that can be viewed in different ways, through different investigative lenses. While much interest has focused on examining estimation strategies, researchers and mathematics educators have also focused on clarifying the nature of the measurement unit that is employed during an estimation process. It is this later perspective that we draw upon in our study.

Individuals develop, through everyday experiences, a set of “personal referents” or “benchmarks” as stand-ins for traditional units of measure. These referents not only give meaning to the relative magnitudes of standard units, but also provide mental models for measurement estimations (Joram et al., 2005). In one study, for example (Joram, 2003, p. 57), a response similar to “4 times around my old high school track” was given by three quarters of a group of 36 undergraduates to the question “When you think about a mile, what do you think of?” As Joram explains, “these individuals thought of a mile as a distance that was personally known to them, and this translation process apparently helped them represent the magnitude of the mile unit.”

At the elementary level, the construction of “personal reference measures” can help children develop an understanding of different measurement systems. Such a set might consist of the following referents (Buys & De Moor, 2008, p. 21).
  • the width of a finger as a reference measure for the centimeter;

  • the thickness of a fingernail as a reference measure for the millimeter;

  • a big step as a reference measure for the meter;

  • a bottle of milk as a reference measure for the liter;

  • a pack of sugar as a reference measure for the kilogram.

The use of some kind of mental referent unit is central to the measurement estimation process. The unit may be non-standard or an internalized version of a standard unit, reflecting the influence of both personal and shared experiences on the development of an individual’s measurement estimation performance. Besides the terms “benchmark” and “reference point,” such a unit has also been described as a “mental ruler” in the case of estimating length (Clements, 1999), and as an “Individual Frame of Reference” (Crites, 1992). In the discussions that follow, we have chosen to refer to this non-tool measurement estimation unit using the phrase Individual Frame of Reference (IFR).

3 Investigation of high school students’ use of Individual Frames of Reference in measurement estimation tasks

3.1 Multi-part study

Our study was motivated by informal observations of individuals’ everyday estimation activities. We noticed that people regarded such tasks as challenging, and that many times they struggled with measurement estimations. These informal observations suggested this as an appropriate area for more systematic research.

To this end, we developed a multi-part investigation that included (1) a pilot study of informal interviews to elicit information about adults’ use of measurement estimation strategies, (2) the development of a set of components that constitute a framework for school measurement instruction (Khosroshahi, 2007; Gooya & Khosroshahi, 2008; Gooya & Khosroshahi, 2009) and (3) a content analysis of grades 1 through 8 Iranian mathematics textbooks’ treatment of topics related to components of measurement instruction.1

A fourth, and final, component of the research was motivated by the pilot studies and informed by our review of relevant literature. In particular, in a number of the pilot studies, we found that those who usually gave “good” estimations also possessed “good” Individual Frames of References and that, on the contrary, the lack of an ability to employ such references led to unsuccessful measurement estimations. We, therefore, designed a qualitative study to investigate the nature of the Individual Frames of Reference that high school students employ to estimate measurements of various attributes of real-world objects.

While the focus of this paper is on the final component of the study, we mention a few observations here from our textbook analysis to briefly indicate the nature of the measurement instruction to which Iranian primary students are exposed. We found that the main emphasis across the texts in grades 1 through 8 is on subject matter knowledge regarding measurement processes. A moderate amount of instruction is given to real-world measurement activities and formal measurement tools, with standard and non-standard units of measurement and estimation receiving even less attention. We found minimal treatment of Individual Frames of Reference across the eight grades. In light of these findings, we were interested in investigating the nature of high school students’ measurement estimation performance.

3.2 Research procedures for real-world estimation activities

In order to explore whether students used Individual Frames of Reference for measurement estimation and, if so, how they were used, we designed a sequence of activities related to estimations of weight, length, and area. One of the authors of this paper and two co-researchers administered the tasks to ten female high school students2 who volunteered to be participants. The students were 15–16 years old and in grades nine and ten from an all-girls gifted school affiliated with the National Organization for Development of Exceptional Talents. Students completed the research tasks individually in different settings (classrooms A & B and the schoolyard). In each setting, one student and one researcher at a time carried out three activities designed for that setting and the whole process was video-taped.

The estimation activities began with weight-related tasks, followed by linear and area measurements of real-world objects. We report here only the results of the linear and area estimation activities. (A brief summary of the research procedures for the weight tasks is given in Appendix 1.) The research procedures for the linear and area measurement estimation tasks included the following:
  1. 1.

    The students were given a Form (Appendix 2) describing the research project and the procedures for individually completing tasks. They were instructed not to use any formal measuring tools including a ruler, meter stick, or such. The students were also asked to think aloud when prompted by the researcher.

     
  2. 2.

    Following the weight tasks, each student entered classroom B and was asked to estimate the length and the width of the chalkboard. The researcher recorded the student’s estimated numbers in Form 1 (Appendix 2) as well as any verbal explanations produced in response to a request to explain the mental processes that the student used to arrive at those estimations.

     
  3. 3.

    Each student then entered the schoolyard, where the researcher asked her to estimate the height of the school building, the area of schoolyard, and the height of a pine tree, which was outside the schoolyard but could be seen from inside the yard. The researcher recorded on Form 2 (Appendix 2) all the estimated numbers and the verbal explanations students gave when requested to explain their mental processes.

     

A qualitative analysis was carried out to identify the types of Individual Frames of Reference that the students used. Data consisted of the researchers’ descriptions of the students’ observed estimation behavior and the written records of the students’ explanations. In all, there were 50 measurement events—generated by the ten students’ completion of five tasks each. Classification categories were developed by sorting these data into groups based on similarities in behavior and explanations.

4 Results

4.1 Students’ responses to measurement estimation tasks

The data collected from each student regarding the length and area measurement estimation tasks are presented in Tables 1 and 2. The students’ names are pseudonyms. In Table 1, the strategies for estimating the chalkboard dimensions are summarized for each student. Table 2 presents more extended information based on the students’ own explanations of their estimation strategies for the schoolyard height and area measurements. In the cases where students reported the use of some kind of imagined meter stick, this is indicated by the phrase “mental meter.”
Table 1

Summary of responses from schoolroom length and width estimation tasks

Student

Est.

Chalkboard length 3.90 m

Est.

Chalkboard width 1.55 m

Sahel

3.25

Compared with the tiles on wall

1.5

Used mental meter

Houra

2.7

Compared with length of her hand

1.1

Compared with length of her hand

Shafagh

4

Compared with the tiles on floor

1.5

Compared with the tiles on floor

Arefe

4

Used mental meter

2

Used mental meter

Farnaz

4

Compared with the tiles on wall

2

Compared with the tiles on wall

Mahtab

3.15

Compared with the tiles on floor

1.5

Compared with the tiles on floor

Rahele

3.25

Compared with the tiles on floor

1.7

Compared with the chalkboard length

Saede

3

Compared with the tiles on wall

1.5

Used mental meter

Mitra

5

Compared with the chalkboard width

1.5

Compared with her height

Minou

3.3

Used mental meter

1.3

Used mental meter

Table 2

Summary of responses from the schoolyard height and area measurement estimation tasks

Student

Height of building 10.5 m

Height of tree 12 m

Area of yard 620 m2

Sahel

9.5

Mental meter

10

Tree is slightly higher than the school buildinga

360

Width: 15 tiles of 1 m side length; length: 28 of the same tiles. Area: 420. Then subtracting the building beside, it is about 360 m2

Houra

9

Each story is 3 m (a fact in constructing buildings)

6

Equal to the height of the building beside itb

240

Length is equal to 4 rooms, each room is 6 m = 24 m. Width: compared with the height of the building = 10 m

Shafagh

6

Each story is 2 m (using mental meter)

6

As tall as the [school] building

300

Two basketball fields (120) + one volleyball field (60) = 300 m2

Arefe

10

Mental meter

10

First: mentally 13 m. Then: maybe equal to the height of the [school] building

240

Used mental meter for estimating the length and width of the yard

Farnaz

10.5

Each story is 3 m (I know) + ground story which is taller + 1 meter top of the building

8.5

Height of the yard wall is 3 m, height of the pine tree above the yard’s wall is 5 m, plus branches and leaves are 0.5 m

160

Width and length: 10 × 16 m, used mental meter

Mahtab

6

Each story is about 2 m (mental meter)

7

A bit higher than the [school] building

910

Width: compared with yard height (6 + 4 = 10), Length is equal to 6 rooms each room is 6 m = 36 m. Area: First 360 m2, then changed to 910 because I made a mistake in my calculations

Rahele

8

Each story is about 2.70 m. I know

9

A bit higher than the [school] building

400

Compared with a mental image of a 120 m2 house: the area of schoolyard is a bit more than 3 times

Saede

30

Intuitively. (she did not have a good estimation of far distances) Went nearer to building and compared it with her own height. She changed it to 7.5 m

35

5 m higher than the school building

500

Length: 30, width: 20, area: 600. It seemed too big, then I said 500

Mitra

10

5 or 6 times as tall as me (170 cm)

12

A bit higher than the school building

200

2.5 or 3 times as big as our house

Minou

10

Each story is 3 m high. I had thought about it before. + There is a 25 cm gap between each two stories

12.5

4 or 5 times as tall as yard wall, which is about 3 m. (I wanted to compare it with the height of the building, but I did not! because the whole thing was video-taped)

225

Estimated a small part of yard (12 m2), and mentally covered the yard with it

aReference to “school” building is to the building that was used in the first yard estimation task

bThe building shown in the right-hand corner in Fig. 4

4.2 Individual Frame of Reference

The categories that emerged from the analysis describe the students’ behavior in terms of their use of three different types of IFR. The specific classification characteristics that comprise each of these categories illuminate the complexity of students’ measurement estimation behavior within particular task environments.

Three different kinds of IFR were identified; students used mental images, made comparisons with physically present objects, and applied prior knowledge. Mental images consisted of a meter length and some kind of experientially familiar unit of area. Physical referents included, e.g., the length of wall or floor tiles, the height of a school building, and the length of a student’s hand. Students also used their prior knowledge of the fact that a one-story building is about 3 m high. These frames of reference were applied both mentally and through direct comparisons with physical objects.

4.2.1 Mental images

Many students used a mental image of a meter as their referent. In five instances, when asked to estimate the length or width of the chalkboard, students employed this image as the basis of their estimate (instead of counting nearby wall or floor tiles). In a typical response, Arefe explained that for estimating the length of the chalkboard, she first developed a mental image for 1 m and then used that to give an estimate of 4 m.

Many students also used an imagined meter as their referent in the schoolyard tasks. Four girls mentally applied this length to estimate the height of the school building. Four students estimated the length and width of the schoolyard in meters and then multiplied these values together to calculate its area. Farnaz estimated the tree’s height by adding different estimated lengths, taking the sum of the wall (3 m), the amount of tree showing above the wall (5 m), and the amount of leaves above this (0.5 m).

The other type of mental referent employed was based on some kind of experientially familiar non-standard unit of area. Two students compared the area of the schoolyard to an image of their own house. For example, Rahele, knowing that her house had an area of 120 m2, explained that “the area of the school yard is about three times bigger than the area of my house, so the area of the yard is roughly about 400 square meters.” One student envisioned the yard as composed of two basketball fields and one volleyball field. Another student estimated that a small part of the yard was about 12 m2 and mentally iterated this referent across the whole area. Two other students also used a mental image of a room in a house. However, these girls employed the length of one side of this room (6 m) as a non-standard mental measure to estimate the length and width of the schoolyard prior to calculating its area.

4.2.2 Physical referents

The students employed a range of objects that were next to or near the task objects as physical referents for the estimation activities. The appropriate dimension of the physical referent was first estimated and then compared to or mentally iterated against the target object.

In nine instances, students used the tiles on the wall or floor of the classroom to estimate either the length or width (or both dimensions) of the chalkboard. The wall tiles touched the sides of the chalkboard, while the exposed wall between the bottom of the board and the floor tiles was blank plaster. (See Fig. 3, Appendix 3 for a picture of the chalkboard and floor tiles.) The close proximity of the tiles to the board made it possible to directly compare, e.g., the length of the chalkboard with an equally long section of floor tiles. As Shafagh explained, “the length of the chalkboard is roughly equal to 13 tiles and the length of each tile is about 30 cm, so the length of the chalkboard is 390 and a bit more, that is, 400 cm.”

Students also used took advantage of a referent’s close proximity to estimate the height of the tree in the schoolyard. Instead of comparing equal lengths, as with the chalkboard and the specified number of tiles, this task was based on estimating both the height of the wall and how much taller the tree was than this value. (See Fig. 4, Appendix 3 for a picture of the tree behind the school wall.) For example, Minou expressed her estimation as a multiplicative comparison, stating that the tree was “four or five times as tall as the yard wall, which is about 3 m.”

Other students estimated the tree height by comparing it to the height of one of several school buildings. Three girls explained that the tree was as tall as the building, four said that the tree was slightly higher, and one girl, being more specific in her explanation, stated that the tree was 5 m higher.

Students also made comparisons using body parts. Houra held her palm close to the chalkboard and used this as an iterated unit to estimate both its length and width. Mitra explained that “I compared the width of the chalkboard with my height and found that the board is shorter than me.” She also judged the height of the school building by estimating that it was “five or six times as tall as me.” Saede walked closer to the building and used her own height to facilitate her estimation.

Two students used a previously estimated measurement to determine the second dimension of the chalkboard. Mitra compared the length to that of the width, estimating that it was “about three times its width.” Rahele estimated the width as a proportion of the length measurement. In her strategy, she “thought that the width is approximately what fraction of the length.”

4.2.3 Prior knowledge

Several students used prior knowledge to generate an estimated value. Four girls came up with the height of the three-story school building using this strategy. Three students claimed that one story of a building is 3 m, while the fourth used 2.7 m for this dimension. (See Fig. 5, Appendix 3 for a picture of the building.) Their final estimates were then based on multiplying their “known fact” by 3. Two of these girls also made adjustments to this value, adding on a bit more height using a mental meter (see below).

Note that above we classified one’s body height as a physical referent. It could be argued that this particular frame can also be considered as prior knowledge. We did not collect sufficient follow-up information from Mitra and Saede to make such a distinction clear. It may be that, when Mitra was standing in front of the chalkboard, her physical presence was instrumental in the estimation that the board was “shorter” than she was. On the other hand, Saede might have been using her height in a more abstract fashion as a known height when she estimated the building to be “five or six times as tall as me.”

4.2.4 Combination of frames of reference

A total of nine estimations were made using a combination of two different kinds of referent. Seven of these results combined the use of a mental unit of measure with a comparison to a physical referent present in the task environment. The other two estimations mixed the use of a mental meter with prior knowledge.

Five students estimated the height of the tree in the yard by first comparing it to the school building (whose height they had already estimated) and then estimating the difference in height using a mental meter. For example, Rahelel stated that the tree was a bit higher than the building, and added one extra meter to her building’s estimated height of 8 m.

Two other instances of a mixture of a mental and a physical referent were employed to estimate the area of the schoolyard. Both Houra and Mahtab used the length of an imagined room (6 m) to estimate one of the dimensions of the yard. Mahtab then compared this dimension to the other one and estimated the difference in terms of her mental room-length unit. Houra estimated the second yard dimension by equating it to the height of the school building. Both girls then multiplied their estimated lengths and widths together to calculate the area.

Two students combined prior knowledge with the use of a mental meter to estimate the building’s height. Farnaz explained that she knew that each story was 3 m tall. She then added an extra 1.5 m to adjust for a slightly taller ground story and an additional 1 m at the top of the building.

4.3 Estimation strategies and measurement processes

While not the main focus of our research, we briefly comment here about the range of estimation strategies we observed being used by the ten students. These processes fit well with the descriptions of those identified by Hildreth (1983) and Castillo (Segovia & Castro, 2009).

Various strategies included: iteration, e.g., by using the repeated movement of a hand across the chalkboard’s length; tiling, by counting the number of floor tiles commensurate with the length of the chalkboard; comparison, e.g., by estimating the height of the tree as being equal to that of the school building; breaking down the target object into smaller, unequal parts, e.g., by estimating the height of the tree as separate segments of wall, trunk and branches; using prior knowledge to establish a unit of comparison, such as that of the height of one story of a building; and the use of the area formula with estimated lengths and widths of the schoolyard.

Two students readjusted their initial estimates, switching from one strategy to another to arrive at a more accurate value. Saede, who first guessed that the school building’s height was 30 m, arrived at an estimate of 7.5 m by comparing its height to her own. Arefe originally used a mental meter to estimate the tree to be 13 m tall and then readjusted this value by stating that it was equal to the height of the building.

Several students combined different frames of reference in one estimate using, e.g., a strategy of near equivalence with adjustment as they estimated the tree’s height to be a certain number of meters taller than their comparison to the school building. In addition, as similarly identified by Costillo (Segovia & Castro, 2009), it was observed that iteration as well as comparison strategies was employed that used either a mental or a physical referent.

The students’ use of these estimation processes also indicated that the girls possessed a high degree of measurement competence. Students demonstrated a familiarity with the measurable attribute of length, using both the standard unit of a meter and such non-standard units as the length of one’s palm or one’s height to quantify this attribute. They demonstrated an ability to select units that were appropriate to the size of the length to be measured. Similarly, students dealt successfully with area measurement. Many students possessed appropriate mental images of non-standard units of area. Other students found the area of the schoolyard from the product of its estimated length and width.

The students’ choice of estimating the schoolyard area either by using mental non-standard area units or by estimating linear units and using the area formula raise interesting questions for further investigation. Do these choices indicate underlying differences in the individuals’ conceptions of area? That is, is area conceived as a fundamental attribute of an object, or as a quantity found indirectly by the “coordination of two dimensions” (Outhred & Mitchelmore, 2000, p. 144) Also, as indicated in Table 2, most of the students were more accurate in their length than in their area estimates, suggesting that they are less familiar with the size of area referent units than those they employ for linear estimation.

4.4 Use of mental or physical Individual Frames of Reference as a function of task and of student

A second aspect of our analysis was to investigate how the uses of either mental or physical referents were distributed across tasks and across students. The data were reexamined to see if patterns could be identified in the types of referents particular students tended to use, and also to search for patterns in the selection of particular frames of reference for each separate task.

Information about each student’s use of either a mental or a physical frame of reference to estimate the chalkboard dimensions was shown above in Table 1. Note that, even with the close proximity of the floor and wall tiles to the chalkboard, students did not universally avail themselves of this physical referent. Of the eight instances in which a student used a physical referent to estimate the length, six girls used tiles, one used her palm, and one compared the length to the previously estimated width. It is also interesting to note that of the six students who used tiles, only three of them used the floor tiles. Six out of ten times, students used a physical referent for estimating the width; three girls used tiles (but only one used those on the wall), one used her palm, another used her height, and the last compared the width to the previously estimated length. From the other perspective, while the use of a mental referent was low, two students used a mental meter to estimate the length and four used it to estimate the width.

To extend the analysis, frequency counts of the uses of the different reference types were generated across all five tasks and across all ten students. Table 3 indicates the frequency of type of IFR employed by each student as a function of task. The category of mental referent is divided into two sub-types: (M) the use of a standard unit (meter) or (Mn) some kind of non-standard unit, such as the familiar area of a room in a house. Physical referents are also split into groups: Students use of either wall or floor tiles are designated by P(w) and P(f), respectively, other referents such as the height of the building or the school wall are indicated by P, and those that were present, but not directly visible, such as the girl’s own height are shown as P*. Instances are noted where a student employed a mixture (Mix) of either a mental and a physical referent or a mental referent and prior knowledge to accomplish the task. The nature of each referent is indicated in parentheses for that task. The use of prior knowledge or fact is indicated by F.
Table 3

Frequency of use of mental or physical referent or known fact across students and tasks

 

Board l

Board W

Build H

Tree H

Yard area

Sahel

P(w)

M

M

Mix (P, M)

M

Houra

P

P

F

P

Mix (Mn, P)

Shafagh

P(f)

P(f)

M

P

Mn

Arefe

M

M

M

P

M

Farnaz

P(w)

P(w)

Mix (M, F)

M

M

Mahtab

P(f)

P(f)

M

Mix (P, M)

Mix (Mn, P)

Rahele

P(f)

P

F

Mix (P, M)

Mn

Saede

P(w)

M

P*

Mix (P, M)

M

Mitra

P

P*

P*

Mix (P, M)

Mn

Minou

M

M

Mix (M, F)

P

Mn

The charts in Figs. 1 and 2 summarize the distribution of the use of either mental or physical referents by student and across tasks. In the charts, the detail regarding the nature of the reference unit has been collapsed for both mental and physical frames in order to investigate general patterns. Frequency counts display (M) mental, (P) physical and mixtures of mental and physical, or factual (M + P, M + F) referents.
https://static-content.springer.com/image/art%3A10.1007%2Fs11858-011-0338-1/MediaObjects/11858_2011_338_Fig1_HTML.gif
Fig. 1

Distribution of type of IFR by student

https://static-content.springer.com/image/art%3A10.1007%2Fs11858-011-0338-1/MediaObjects/11858_2011_338_Fig2_HTML.gif
Fig. 2

Distribution of IFR within task

Figure 1 displays, for each student, the frequency count for each of the different types of IFR that they employed across all five tasks. Student preferences ranged over a continuum from using mostly mental referents (including mental mixes) to mostly physical ones (including physical mixes). At one extreme, the two students Arefe and Minou each used four mental referents and only one physical referent. At the other extreme, Houra used a physical referent in four of her tasks, employing a mental referent only once in an M + P mix. Two other students (Mahtab and Mitra) also used physical referents in four of the five tasks, but included mental referents as well in several of the tasks either alone or as mixes.

The variation by student, in terms of the relative frequency of their use of either mental or physical referents across all five tasks, can be interpreted as an indication that, for some students, their individual preferences for a particular referent type may play a strong role in how they approach contextually situated measurement estimation tasks. In other words, particular measurement cues that are present within a task environment (such as the close proximity of wall tiles to the sides of the blackboard) may have been ignored by particular students as they selected a preferred estimation strategy. On the other hand, the similarity in choice of referent across students for specific tasks indicates that many students appear to be influenced by the physical features of the estimation situation.

Figure 2 displays the frequency count of mental and physical referents used by all ten students within each of the five tasks. While none of the tasks elicited the use of only one kind of referent, on two tasks the students showed a marked preference for either mental or physical frames. For estimating the area of the schoolyard, all students used a mental referent (either alone or in an M + P mix). At the other extreme, for the tree height task, all but one student (Farnaz) used a physical referent (either alone or in an M + P mix). Estimating the length of the chalkboard also primarily elicited the use of a physical referent, with only two students employing a mental IFR. Interestingly, for estimating the chalkboard width, the distribution between physical and mental referents was more equal (6 M and 4 P). As can be seen from Fig. 2, the estimation activities performed in the schoolyard environment elicited the only instances in which students used mixes of two different types of referents.

The patterns noted in Fig. 2 suggest that, in certain situations, the specific nature of the measurement estimation task and the physical environment in which it was situated appeared to influence a student’s choice of IFR. We present this statement as a very qualified conjecture. While extremes between a choice of physical or mental referent were observed for two of the tasks, the distribution between these two types in the other three tasks suggests that the influence, on a student’s choice of IFR, of particular physical features within the estimation context may be a complex issue.

5 Conclusions

5.1 Findings from the study

While this study was exploratory in nature and involved only a small group of ten participants, several findings are of general interest. These findings are a result of the richness of the real-world task environments and the fact that the participants were proficient in the use of measurement processes. In this respect, the study does not address how measurement estimation develops but, rather, probes aspects of the variation and complexity of competent estimation performance. The analytical emphasis on the use of Individual Frames of Reference provides a useful classification scheme to investigate individuals’ use of non-tool units in measurement estimation activities. Thus, this research provides a different investigative lens than that used in other research that focuses on the processes, or strategies of estimation.

First, the notion of an IFR was found to be a functional construct for unpacking an individual’s performance across a range of measurement estimation tasks. Such referents provide an important analytic tool for understanding how individuals actualize the basic measurement principle of quantifying an object’s attribute by the application of a relevant unit of measure during an estimation process.

The students demonstrated important principles of linear measurement in the ways that they employed a range of Individual Frames of Referents as their length estimation units. Besides familiarity with a “mental meter,” benchmarks, such as the height of one story of a building or the width of one’s palm, were used in comparison and iteration strategies. In accordance with Joram (2003), we interpret this familiarity with both standard units and the possession of a set of personal reference points as indicators of a high degree of measurement sense.

Second, the distinction between the use of a frame of reference that is purely mental and one that is physically present in the environment highlights the variation in how different individuals interact with the context of a situated estimation task. This variation was also described by Castillo (reported in Segovia & Castro, 2009), who found that individuals employed either “present” or “absent” referents with iteration and comparison strategies for linear and area measurement estimations.

Our findings, however, go beyond simply a list of reported strategies as, e.g., given by Castillo in Segovia & Castro (2009). Our analysis of patterns across tasks and students suggests that individual estimation preferences may, in some cases, override measurement cues in particular task environments. Additional research is needed to probe more deeply the ways in which individuals personally interact with measurement situations as they make sense of and quantify physical attributes of objects in the world around them.

Third, research should be designed to provide students with a wide variety of measurement estimation situations. Our analysis revealed that the environment in which a task is presented may influence the choice of referent that is selected. The variation in the physical referents that was used by the students while carrying out the schoolyard tasks indicates that many different aspects of the environment may serve as a non-standard unit of measure. Tasks situated in the world outside a school environment would provide another area for significant investigation.

Joram et al. (1998, p. 436) call for “systematic research that examines how measurement estimation is used in daily life.” We recommend that task selection be a carefully considered aspect of any such research. As we observed in our study, tasks that elicit a range of choices of IFR across participants provide a rich environment for in-depth studies of important issues related to measurement estimation and the characterization of an individual’s measurement competence.

Finally, the limited nature of our findings raises more questions than answers. Our study has relevance less in what can be claimed than in suggestions for what more can be uncovered through further research focused on the use of Individual Frames of Reference in measurement estimation tasks—studies that can complement the already developed corpus of research related to measurement estimation strategies.

5.2 Directions for further research

More in-depth interviews of the students at the completion of each estimation task are needed to probe further into the reasons why students select a particular IFR for a given task. For example, are students more comfortable with, confident in, and/or able to carry out particular types of measurement estimations (i.e., length vs. area or volume), and how do such preferences influence their choice of IFR? Are choices of IFR a matter only of past measurement experience, or are they influenced by a persons’ tendency to be, e.g., a verbal, spatial, or linear thinker? And finally, where and how do students gain their ability to estimate, besides that offered in formal school instruction?

Systematic research is needed in the area of task design to study relationships among the physical setting, the features of the estimation situation and an individual’s choice of Individual Frame of Reference. For example, how does the physical context in which a task is placed influence students’ choices of IFR? Does the choice of IFR depend on the magnitude of the quantity to be estimated? In other words, do students select different kinds of units for estimating small lengths, long distances, or vertically or horizontally displayed lengths? How does keeping the task constant and varying the environment in which the task is situated affect the selection of an estimated measurement unit? What are the interactions between context and choices of estimation strategy and IFR, and which comes first—the choice of strategy or IFR?

As we noted in the beginning of the paper, an individual’s ability to estimate relies on an underlying measurement competence. Thus, investigations of older students’ performance in measurement estimation may provide insights into the particular aspects of measurement that students have developed and are able to utilize. The results of measurement estimation studies may also provide insights into how important aspects of this skill might be taught in schools.

Footnotes
1

In Iran, there is one national textbook series for each school subject. See reference list for details about the textbook series for mathematics for grades 1 through 8.

 
2

All schools in Iran are segregated and our study was conducted at a girls’ high school.

 

Acknowledgments

The authors wish to thank Marja van den Heuvel-Panhuizen, John Smith III, and an anonymous reviewer for their extensive help and encouragement in writing this article.

Copyright information

© FIZ Karlsruhe 2011