Volume 28, Number 3 
Table of Contents Genetic Epistemologist Homepage


President's update

Dear members,

I would like to share with you an update of events within JPS since our June meeting.

Memberships at 4 Year High

Current JPS membership is at 466. This is the highest level in four years, second highest in the past ten years, and more than 150 members above the number that we began the last decade. Our goal is to continue on this positive trajectory.

New JPS Journal

JPS has affiliated with the journal Cognitive Development which is now the official journal of the Society. Beginning 2001 JPS members will receive this quarterly journal at a special low subscription rate as part of their membership benefits. You will continue to receive the Genetic Epistemologist. You will also continue to receive the annual symposium series volume.

As we move forward with this new exciting venture, we will provide you with more information regarding the process of manuscript submissions and other information regarding the journal. One thing that we can share with you at this point is that the content of Cognitive Development will include articles dealing with social cognition and development. Peter Bryant, the current editor, is also open to theoretical articles that are brief, and interesting. We must all keep in mind that the process of accepting and publishing new material takes time and that our contributions to the journal will be a function of the quality of the work that we submit, the pre-existing manuscripts that have been accepted for publication, and the quality of manuscripts submitted in general. To insure a JPS contribution, the board in consultation with the editor will select a special issue editor each year to produce one volume.

We are very excited to be able to bring this new benefit to the JPS membership. The cost of the adding Cognitive Development as a membership benefit is reflected in a $35 increase to our yearly dues. The journal is optional for Student Members. Membership Dues for 2001:

President Elect

We are also delighted to announce that Elliot Turiel of the University of California at Berkeley has been unanimously elected to serve as the next President of JPS. Professor Turiel will assume office in October 2001.

It is a genuine pleasure to be able to share so much good news.


Larry Nucci, President
Jean Piaget Society


The development of linear measurement in elementary school children: An empirical study

Emily Bladen, Katey Wildish, Jonas A. Cox
George Fox University

Addresses for correspondence:

Jonas A. Cox
George Fox University
414 N. Meridian Street
Newberg OR 97132 USA

Emily Bladen: emilybladen@hotmail.com
Katey Wildish: katey_wildish@hotmail.com
Jonas Cox: jcox@georgefox.edu


There is a general need for persons in society to understand linear measurement. The need may be as simple as measuring the dimensions of a window for new drapes or as involved as understanding how the length of a girder in a bridge design affects the need for strength. Many aspects of our personal and professional lives require an understanding of standard and non-standard measures. The time we allot to subjects that require an understanding of measurement, e.g., geometry, geography, physics, trigonometry, cooking, astronomy, biology, and chemistry reflect the importance of these areas to our society. Virtually all math and science classes beyond primary grades require the use of measurement tools and the underlying reasoning involved. Unfortunately, many of our students do poorly in these classes.

Educators generally recognize that the concept of measurement is an essential component to success in the math and science curriculum. The American Association for the Advancement of Science (AAAS) and the National Council of Teachers of Mathematics (NCTM) have developed standards for curriculum and evaluation related to measurement. According to these standards, students should be able to use standard units of measure instead of simple qualitative comparisons in early primary grades and be using standard tools of measurement accurately to check their quantitative estimates. However, as these higher standards have been put in place in the past decade, most students are falling short of the expected levels.

Data reported by the National Assessment of Educational Progress (NAEP) show that only 14% of third grade students were able to answer a simple line-segment-measurement problem, and only 20% of fourth grade students demonstrate profiiency in mathematics. Students are clearly struggling to understand core concepts, such as measurement, that are foundational for success in math and science. Some argue that students and teachers are not working hard enough, others feel this is this due to mere test or math anxiety. Our own view is that the failure of most students to meet educational expectations is evidence of a deeper problem.

A typical first grade measurement activity illustrates the problem. As Halloween draws near, students are often found engaged in an activity called "pumpkin math." They are asked to guess and then measure in inches the circumference of a pumpkin. They are given a ruler and some string to solve the problem. The assignment seems simple enough, yet in this activity teachers are often greeted with blank stares and anxious glances at the materials as the students try to decide what to do. Many wrap the string around the pumpkin as many times as possible while others pick up the ruler and try to roll it around the outside of the pumpkin. The teacher stands back, frustrated; regardless of the number of times she tries to explain measurement, her students don't seem to understand. Finally, she intervenes and essentially does the assignment for the students, giving them step by step procedures in order to arrive at a correct answer. Observing this activity and others like it leads the observer to ask, how much of this process is truly understood by students? What do students understand about measurement, and is there perhaps a pattern to how children acquire these concepts? If so, this knowledge would be very helpful to teachers trying to establish an experience base for mathematical instruction.

Piaget studied the development of the concept of measurement in children. Two of his best known works, The Child's Construction of Space, and The Child's Conception of Geometry include these notions. Piaget suggested that "the conservation of length is a necessary postulate for measurement" (1960, p.27); measurement is dependent on the fundamental concept that length is conserved (not altered by spatial arrangement). He also proposed that comparison between two objects does not involve measurement but is purely perceptual if the person does not possess a spatial coordinate system. His research involved students' construction of a map and then reconstruction of a 180-degree reversal of that map to assess the presence of a spatial coordinate system in the child's thinking. The combination of these two (conservation of length and a spatial coordinate system) is necessary or "there can be no operational transitivity" (1960, p.54).

According to Piaget then, one would expect to see two cognitive structures link to form an understanding of iterated units of measure. The first, conservation of length, concerns the child's ability to hold length constant regardless of the appearance. The second, the coordinate system, allows the unit (a conserved length) to be moved along the length of the larger object to be measured. The repeated comparison of the smaller unit to the larger object requires both the coordinate system (change of position as Piaget refers to it) and the constant length of the unit being moved. The coordinate system and conservation of length develop separately but according to Piaget are necessary for a child to perform iterated units of measure.

Piaget's work on measurement and the 3 stages in children's reasoning that he described, were based largely on two tasks: the change of position task, and the towers task. In the change of position task, the child viewed the city from a window and was asked to retrace their route to and from school. In the towers task, a block of wood -- "the tower" -- is placed on a table and is meant to serve as a model for the child who is told to construct another "tower" that is just the same size. The child is offered a series of sticks: one that is longer than the tower, one that is the same length as the tower and one that is much shorter than the tower. The interviewer watches to see which sticks, if any, the child is able to use effectively to compare the height of the towers. Piaget's results indicate that very young children are only capable of a visual comparison of the two towers. They visually compare the tops of the towers with no demonstration of transitivity, which is to say that they are unable to use the sticks offered as a means to refine their measurement of the towers. Later in development, children are able to use only the stick that is just the right length as a comparison of the two towers -- rejecting both the longer and the shorter stick. Later still, they reject the shortest stick but use the longest stick, comparing this stick to one of the towers, marking the height with their fingers, and then comparing it to the other tower. Finally, children are able to use even the smallest stick by marking the spot with their fingers and iterating the unit. Counting the number of lengths of the small stick it takes to run the entire length of one tower they then compare that number to the number of lengths of the other tower (Inhelder et al., 1960).

Researchers continue to use Piaget's work to identify the development of measurement abilities in children. Phillips (1996) used Piaget's towers task to chart the development of linear measurement in children of different ages. He found that only six percent of students in the second grade, and just 42% of eleventh graders were able to succeed in this task, which assessed the presence of conservation of length and transitivity. These findings suggest, to us at least, that something is wrong with either the way measurement skills are taught or the expectations educators have about students understanding of measurement.

Much of the current research, however, takes the concept of measurement and the structures included in its development out of the important practical context of the classroom. Kamii and Clark (1997) conducted research on the age at which children construct unit iteration out of transitive reasoning (proposed by Piaget) in order to discover implications this has for teaching linear measurement in the schools. These two cognitive abilities (iteration of unit and transitivity) are necessary for measurement. As an example, they proposed that rulers are useless for children who have not constructed transitive reasoning because they cannot compare two lengths that are not placed next to each other. The results of their research on almost 400 students found that unit iteration develops by the fourth grade, for most children, a year later than Piaget found iteration developing in children. (Piaget, 1960). Further Kamii's work supports Piaget's theory that children construct unit iteration out of transitive reasoning. No student was found to have demonstrated iterations who could not also demonstrate transitive reasoning. As a concluding comment, Kamii and Clark advise teachers to challenge students' thinking in order to modify their understanding instead of directly teaching empirical procedures, since direct teaching strategies do not tend to result in the infralogical understanding necessary for unit iteration.

Given that higher standards for performance are now in place, and that current educational practices fail to promote an understanding of measurement among students, some better understanding of the developmental structures of this important area of mathematics, and some better means of promoting understanding seems warranted. The present study was conducted to provide the first steps towards determining the sequence of development of structures that contribute to an understanding of measurement. Such efforts, we contend, would have both theoretical and practical significance.

The hypothesized sequence of these 20 points were derived from the existing literature and, therefore, the theory which supports this predicted sequence, using both existing measures and a new task constructed specifically for this research. More practically, providing teachers with knowledge about the sequence in which these notions are constructed will also help teachers organize their classrooms to provide the experiences necessary for students to master these concepts. If teachers know what mental structures are present in a child's thinking, they can provide resources and appropriate challenges to allow the students to construct the concept of measurement. Thus, the students will understand measurement, rather than merely memorizing a set of procedures offered by their teacher.

Choosing assessment tasks

The ultimate goal of this study is to establish a framework of cognitive structures leading to measurement that can then be used to guide teachers in the classroom. This preliminary study focuses broadly in this area rather than having the precision and detail of other Rasch modeled studies. For example, Bond and Bunting (1995) produced 18 data points for their pendulum task alone, while this entire study of measurement contains only 20 such points. A more refined list of criteria, some of which may be identified as a consequence of this study, will be needed to inform practicing teachers.

The cognitive tasks for this study were derived from Piaget and other constructivist sources. A total of 6 tasks were selected and are described below. Four of these tasks (seriation, topological ordering, conservation of length, and the towers task) are taken from Phillips (1996, 1999), who in turn, adapted procedures described by Piaget. The change of position task was adapted by the authors from Piaget's original task. The transitivity of length task is taken from Kamii and Clark (1997).

1. Seriation by length. In this task the student orders a series of 9 sticks (quarter inch dowels) by length. The sticks are cut in lengths from 8 to 16 centimeters in one centimeter increments. Once the child has successfully ordered the set they are asked to insert three additional sticks into the set. These sticks are cut in the following lengths 14.5, 11.5 and 9.5 centimeters

2. Topological ordering. In this task the child is asked to place 8 colored beads onto a pipe cleaner in the same order as the model provided. The child is then asked to place the beads so that if the model pipe cleaner is turned over their set would look just like the model set. In other words they are asked to put the beads on in reverse order to the model.

3. Conservation of length. This task is a standard conservation task where the child is asked to establish equality, in this case of length. The objects are then changed to give a visual miscue of perception to the child and the child is asked about the equality of the two items or sets. The items used to establish equivalence of lengths here were quarter inch dowels (sticks) cut to 25 cm. The sticks are placed parallel with both ends together for an easy comparison of length. The child is asked if these sticks are the same length. Once the equivalence is established, one stick is moved parallel to the other so that roughly 1/3 of one stick extends beyond the other. The child is asked if one stick is longer or are they the same and then probed for their reasoning. The stick is then moved perpendicular to the other stick and the child is again asked if one stick is longer or are they the same and again probed for their reasoning. Reasons considered acceptable were logical necessity, compensation and empirical reversibility (see Phillips, 1996 for a complete discussion).

4. Transitivity of length. In this task the student is asked to make an indirect comparison of length between two objects that are distanced from each other using an intermediary object which can be moved for direct comparison of the two other objects. Rod A (11cm) and rod C (10 cm) are both stationary and separated by a distance of three feet. A third rod B (10.5 cm) is used for comparison between the two stationary rods. The task seeks to test for the presence of the transitive logic. Namely, that if the length of rod A is greater that of rod B and rod B has a greater length than rod C then rod A must be longer than rod B.

5. Change of position. Piaget's work includes a description of the coordinate system or change of position task, but his interviews involve the child viewing the city through a window and retracing their path to and from school, an uncommon experience for most American school children. Many students are bussed to and from school and few schools set on a substantial enough hill such that the child can clearly see their path. Additionally, the task needs some standardization such that it represents the same level of difficulty for each interviewee. It was decided that the child would be asked to draw a map of their path to a familiar landmark in the school either their homeroom or the office depending on location.

In our version of this task, the students are asked to draw a map of how to get from the room they are being interviewed in to another distant, yet well known location in the school (i.e. the office or a teacher's classroom). The task begins by placing a small model of the desk the student is sitting at for the assessment on a large sheet of butcher paper. The student is asked to draw a map of the route to a familiar place within the building. They are asked to place other significant landmarks such as the playground, cafeteria, and gymnasium. Once the child has drawn the map, placed the landmarks, and labeled them, the model of the desk is turned 180 degrees and they are asked to draw another map with the same destination and landmarks. It is the child's ability to reverse this map, while mentally maintaining the relationships present that tests for the presence of a coordinate system.

6. Towers. As described above, in the towers task, a block of wood -- "the tower" -- is placed on a table and the child is told to construct another "tower" that is just the same size. After the child constructs the tower, he/she is asked to compare the height to be sure it is the same size. The child is offered a series of sticks -- one that is longer than the tower, one that is the same length as the tower, and one that is much shorter than the tower. The interviewer watches to see which sticks, if any, the child is able to use effectively to compare the height of the towers.

Predicted order of development

As noted above, Piaget claimed that an understanding of iterated units of measurement depends on two structures: conservation of length and a spatial coordinate system. In addition to the tasks for change of position, unit iteration and conservation of length, this study also involves tests for a few more primitive structures including seriation of length, topological ordering and transitivity of length. Transitivity (Kamii & Clark,1997) and seriation of length (Phillips, 1996) were chosen because they both related to length and it was thought that they would be primitive to conservation of length. Topological ordering (Phillips, 1996) was chosen because, similar to change of position, it involves mentally reversing the order of objects by the child. It was thought that this would be primitive to change of position.

The predicted order of development for these tasks was as follows: seriation by length, conservation of length, and transitivity of length. Then in a separate developmental sequence it would be expected that topological ordering would preceed change of position. Both change of position and transitivity would need to be in place before the child would iterate a unit in the measurement task. As indicated in Figure 1, current theory would predict that there are two lines of development (here depicted in the left and right hand boxes), and that these two lines merge to produce an understanding of iterated units of measure (the center box).

Figure 1


A total of 99 children, age 9-11, consisting of 45 males and 54 females were tested in individual interviews. All six tasks were administered at the child's school during a single testing session lasting 20 minutes. All of the tasks were administered by two of the authors (Bladen & Wildish) during their student teacher placements at two elementary schools in the greater Portland area. Both researchers interviewed students at a third school. All three of the schools involved in the study serve middle and upper-middle class areas and are populated by predominantly Caucasian students.


Pass or Fail assignments were given for performance on multiple aspects of each of the tasks. For the seriation task, only one Pass/Fail (1,0) assignment was made (the child either did or did not insert the three dowels correctly). For the other tasks, multiple assignments could be made -- for example, on the transitivity task, the child's response could be judged as Pass/Fail. Providing a correct response and a sufficient reason for that response could also be judged as Pass/Fail. Within the six tasks used, there were a total of 20 such data points (see Table 1, below). Most tasks averaged two or three points, with the exception of the towers and change of position tasks with five and six points respectively.

Table 1

Data Scoring Criteria

1 is able to insert the three provided dowels correctly

Topological Ordering
2 is able to match a linear arrangement
3 is able to strings beads in reverse order from model

Conservation of Length
4 answers same with parallel sticks
5 answers same with parallel sticks and provides a sufficient reason
6 answers same with perpendicular sticks
7 answers same with perpendicular sticks and provides a sufficient reason

8 answers with the correct response
9 answers with the correct response and a sufficient reason

Change of Position Task
10 points in the correct target location for the office
11 correctly places the first location on the 180 degree turn map (office)
12 correctly places the second location on the 180 degree turn map (room)
13 correctly places the third location on the 180 degree turn map (playground)
14 left/right reversals

0 = no understanding
1 = slight or beginning understanding (few correct reversals)
2 = moderate understanding (most correct reversals)
3 = complete understanding
15 front/back reversals 0 = no understanding
1 = slight or beginning understanding (few correct reversals)
2 = moderate understanding (most correct reversals)
3 = complete understanding

Towers Task
16 does not use or believe visual measurement is accurate
17 does not use or believe body measurement is accurate
18 is able to correctly use dowel A
19 is able to correctly use dowel B
20 is able to correctly use dowel C

All but two of the individual data points were dichotomous with a student being judged as either passing or not passing a particular aspect of the task. The two exceptions were contained in the change of position task, which were scored as polytomous items. On the two polytomous items the students were scored based on the level of development demonstrated in performance. Two points on the change of position task were not easily judged "Pass" or "Fail". When the students were asked to draw their second map with a 180-degree turn there were several different levels of understanding shown by students in both left/right reversals and front/back reversals. Four different levels of understanding were defined for both right/left and front/back reversals. If a student showed no concept of the reversal on their map they received a 0. If the student's map showed a beginning of the reversal concepts (a few correct reversals) the performance received a score of 1. When a student had a grasp of the concept but did not apply it consistently (most reversals correct but a few minor errors), a score of 2 was given. To earn a score of 3, the 180-degree map had to be completely correct. The students received separate scores for right/left and front/back reversals. ).

Table 1 shows the predicted order of difficulty for the items.


There were a total of 20 items, 18 of which were dichotomous and two of which were polytomous (4 levels). The first analysis presented is a Rasch partial credit analysis, conducted with Quest (Adams et.al,1993). Rasch models permit the analysis of data with a mixture of dichotomous and polytomous items (Bond & Fox, 2001).

Overall Developmental Sequence

Rasch models test for a hierarchical, unidimensional sequence within certain probabilistic constraints. Consider typical data from the informal classroom assessments. Little can be said of these data in their raw form but that one item seemed more difficult to these students than another. When participants are ordered by score the persons whose raw scores are high will likely be toward the top of the developmental continuum, and the persons whose raw scores are lower will likely be toward the bottom of the continuum. Rasch modeling tests for a developmental sequence by seeing how well data, persons and items, fits the this theoretical idealization or unidimential developmental sequence. How well the data fits the model is assessed with infit and outfit statistics.

Both infit and outfit statistics are based on the difference between expected and observed performance. They are used to assess whether a given person's results or the perfomance of an item on the test is consistent with other persons' results or other items on the test. While outfit statistics are based solely on the difference between observed and expected scores, infit statistics are downweighted by extreme persons or items. Usually, the weighted infit statistics are more useful for assessing fit with other persons or items, because they are not affected by outlying bits of data, the argument being that persons whose ability is close to that of the item will give us better information about that item's performance. Infits (or outfits) near 1 are desirable. T-scores are calculated to assess the significance of both positive and negative divergences from 1. Interpretation of fit statistics is demonstrated below, in the results of the analysis.

Change of Position Task Performance

In a second analysis we qualitatively scrutinized the results of the newly derived change of position task. In this task, the students are asked to draw a map (and then a reversed map) of how to get from the room they are being interviewed in to another distant, yet well known location in the school. It is the child's ability to reverse this map, while mentally maintaining the relationships present that tests for a coordinate system. As noted above, the task is scored based on the number of errors in left/right reversals and front/back reversal's. According to Piaget (1963), students passing this task are able to perform the task with no inconsistencies or incorrect placement of locations between the original and the 180-degree turn map. It is predicted that the students' performance on this task based on Piaget's definition of "passing" will fit with the data from the other tasks used in the study. That is to say that students who pass the task for iterated unit should also pass the change of position task and that not all students who pass the task for change of position should pass the task for iterated unit. This pattern would then indicate that the change of position task fit within the anticipated sequence.


The data listed in Table 1 are plotted on the Rasch map of person and item estimates shown in Figure 2. At the center of the Figure is the logit scale, which spans 8 logits (-4 to +4). To the left are the student performance estimates, and to the right are the item and item step difficulties. The items students' found easiest are at the bottom of the map and those they found hardest are at the top of the map. Similarly, the students who performed correctly on more items are at the top of the map while those who performed correctly on fewer items are at the bottom of the map.

Person performance analysis

The overall person separation reliability of the person performance estimates is 0.77. The infit and outfit statistics for all person performance estimates were considered to fit the model if t-scores were not greater than 2.0. Three (3%) had infit t-scores over 2.0. These are said to underfit the model. One or more of their items were scored higher or lower than expected at their overall performance level. This is an acceptable rate of underfit, which can reasonably be considered measurement error (Bond & Fox, 2001). The mean of the case esitmates was 1.83 with an adjusted standard deviation of 1.22. The reliability of the estimate was 0.77.


Item analysis

The infit and outfit statistics for the item difficulty estimates were considered to fit the model if their t-scores were smaller than 2.0. Three items misfit the model. These were 17, 19 and 20. These items are all contained within the task for iterated unit. This is more than 5% underfit and usually this is considered to be too much. The mean of the item estimates was 0.00 with an adjusted standard deviation of 1.62. The reliability of the estimate was 0.93.

For these six tasks the observed developmental sequence matched, with some exceptions, our predicted sequence. Seriation was found to be the easiest task. Topological ordering was next. It was predicted that conservation of length would come before the understanding of transitivity of length, but from analysis it appears they develop simultaneously. According to past research, an understanding of linear measurement should only be present after the development of all other structures tested for here, however this conclusion was not supported by these data. Several students were able to perform measurement, but were unable to perform change of position. This violation of the expected order raises both theoretical and methodological questions we take up in the discussion. This also raises the possibility that these are separate dimensions of performance. For example, multidimensional analyses of these two task types may reveal two latent dimensions -- at least up to this point in development.

It was predicted and theoretically supported that an understanding of iterated units of measure should develop out of transitivity. This point is supported by these data. All persons who passed iterated unit (item twenty) also passed transitivity (item 8) though 7 out of 15 were unable to give sufficient reasons for their answers on transitivity (item 9).

The only items that underfit the model (t-scores in excess of 2.0) were contained within the task testing for iterated unit. The protocol adopted from Phillips represents a significant shift away from Piaget's method clinique toward a more highly structured interview. The students were offered both the longer (item 19) and shorter (items 20) sticks but were not challenged to use them as Piaget might have done. If the student claimed that the stick was not of use to them in measuring the towers then a score was recorded and the interviewer moved on. This limited interaction, like most shifts toward more quantifiable data has benefits and costs. Here the cost may have been an underestimation of some subjects' development. This might account for the underfit of these items.

Looking at the map in Figure 2, there is a large gap between seriation and topological ordering and an even larger one between topological ordering and conservation of length. Conservation of length and transitivity of length appear at the same level, which indicates that children who can do one can most often do the other.

Some of the items are ordered in a way consistent with Piaget's notion of horizontal decalage. Decalage, as defined by Piaget in Gruber and Voneche (1997), is a lag or unevenness in development among structurally similar concepts. For example, the ability to understand right/left reversals precedes the understanding of front/back reversals. If a student is at level 1 in right/left reversal understanding, then, before this student is expected to reach level 2 in right/left reversal, he or she will likely reach level 1 in front/back reversal understanding. No hypothesis about what type of reversal precedes another was made here, but the question of why right/left precedes front/back is very interesting. Are right/left reversals somehow structurally easier than front/back, or do students simply have more left-right experiences, such as the way we read or turn pages in a book?

An unexpected finding concerns the change of position task. From the item person map, this task does not appear to fit into the predicted developmental sequence, though from this sample we can not support statistical significance. Given that measurement error is closely related to sample size, a readministration of these assessments with a larger number of subjects should be undertaken.


It seems important to examine math and science curriculum standards in light of developmental theory. The type of analysis conducted here not only helps identify learning sequences, it tells us when we can reasonably expect specific learning to take place. These findings may serve as a warning to organizations that develop content standards to reconsider expectations of students in the lower grades as they relate to measurement. These data reveal a wide discrepancy between educational expectations and development of the concept of measurement. For example, only 53% of the 10 year olds in this study demonstrated an understanding of iterated unit, yet the national standards require this logic in early primary grades.

The finding that several students were able to pass the towers task before the change of position task has several possible explanations (other than challenging Piaget's position). These include problems with tasks, interviewer error, or school contamination. Developmental assessment can be contaminated when children have memorized a procedure rather than constructing an understanding. Measurement is very often part of math curriculum and students may have been taught to iterate units when measuring. If this is true, these tasks may not have tested for understanding, but might actually have tested students' memories for previously learned procedures.

Another interesting question to emerge from this study is why children seem to develop understanding for left/right reversal before front/back reversals. The task of topological ordering directly looks at the ability to right/left reverse, but is there a task for front/back ordering and how would this fit into the sequence of concepts?

After looking at these results, an obvious question surfaces: how can these results be used to help teachers and students meet the state and national standards in math and science? Many of the high stakes assessments used in standards based systems require a much deeper infralogical awareness than is present in children at these grade levels. Studies like this one can serve to improve instruction in two distinct ways. First, the study highlights the complexity of the infralogical thought required to understand iterated units of measure. Second, it provides a map for the development of these ideas in children. This map can serve to orient practicing teachers so that they ask the right questions of their students and have a deeper understanding of the responses they hear. Further, those involved in curriculum development should use findings like these to build experiences for children -- experiences that assist the child in constructing the infralogical structures necessary for understanding measurement rather than teaching them to blindly apply a step-by-step procedure resulting in an answer that is not understood by the child who produced it.

Further work should be done to enlarge the sample and to build a more detailed set of data points, thereby allowing for a more complete analysis. Possible benefits might be a better defined sequence with more data points, finding the proper fit for the change of position protocol and developing an interval scale on which to plot each of the items. Beyond these, the real work that has yet to be done involves finding ways to link the results of studies like this one to classroom practices. It should be the goal of developmental researchers to enrich the educational experiences of children by developing teacher friendly protocols that can be used to assess the development of important concepts in the classroom. Further work needs to be done by curriculum developers that take a well-defined developmental sequence and design activities and questions that focus the child's thinking on the questions pertinent to these concepts. Though some may be led by these arguments to develop a means of directly teaching these tasks to children, that is not what is being suggested here. Rather it is our position that certain experiences are necessary for the child as input for self-organization. We as educators should provide the experiences necessary for the child's continued development, such that the lagging factor in development is physiological rather than experiential.


Adams, R.J., & Khoo, S.T. (1993). Quest: The interactive test analysissystem [computer software]. Camberwell, Victoria: Australian Council for Educational Research.

Bond, T. G. (1995). Piaget and Measurement I: The twain really do meet. Archives de Psychologie, 63, 71-87.

Bond, T. G., & Fox, C. M. (2001). Applying the Rasch Model: Fundamental Measurement in the Human Sciences. Mahwah, NJ: Erlbaum.

Gruber, H. E., & Voneche, J. (1997). The Essential Piaget, An Interpretive Reference and Guide New York Basic Books.

Inhelder, B., Piaget, J., & Szeminska, A. (1960). The child's conception of geometry (E.A. Lunzer, Trans.). New York: W.W. Norton & Company.

Kamii, C., & Clark, F.B. (1997). Measurement of length: the need for a better approach to teaching. School Science and Mathematics, 97, 116-121.

Phillips, D. (1996). Structures of thinking: Concrete operations. Dubuque, IA: Kendall/Hunt Publishing Company.

Phillips, D. (1999). Developing logical thinkers. Dubuque, IA: Kendall/Hunt Publishing Company.

Piaget, J., & Inhelder, B. (1967). The Child's Conception of Space. New York: WW Norton and Company, Inc.


SRCD Biennial Meeting

The 2001 SRCD Biennial Meeting will be held April 19-22 in Minneapolis, Minnesota at the Minneapolis Convention Center. Professionals in the all fields of child development are invited to attend. Please visit SRCD's website (www.srcd.org) or contact SRCD by e-mail (srcd@umich.edu) for more information.

Table of Contents | Genetic Epistemologist Homepage

JPS icon© 1995-2001 The Jean Piaget Society
Last Update: 4 March 2001
Address Comments to: webmaster@piaget.org
SearchTop of page