Abstract Title Page
Title: Assessing a Linear Representation of the Counting Numbers
Authors: Erin E. Reid, Arthur J. Baroody, and David J. Purpura
(University of Illinois at Urbana-Champaign)
SREE 2011: Early Numeracy Assessment
Abstract Body
Background
By 4 years of age, children have begun to learn that the number words in the counting
sequence represent different specific quantities of increasingly larger size—the “increasing
magnitude” principle (Sarnecka & Carey, 2008). This principle, however, does not necessarily
entail understanding that the counting sequence is, in mathematical terms, a “growing pattern”
—that successive numbers increase by a constant amount. Such an understanding is embodied by
a linear representation of numbers with any appreciable slope more than 0. Instead, young
children seem to have a logarithmic mental representation of even the first 10 counting numbers.
That is, their representation seems to involve large magnitude differences between small
numbers and increasingly smaller ones between progressively larger numbers.
Constructing a linear representation of number is a critical foundational mathematical
competency. A linear representation of number provides children with a more accurate
understanding of the relations among numbers, and this permits them to make better estimates of
number and to better understand operations on numbers (e.g., Siegler & Ramani, 2009). Whyte
and Bull (2008) observed that the development of [a] linear representation … is believed to
facilitate improvements in numerical estimation, which has been found to significantly correlate
with improvements in mathematical abilities (Booth & Siegler, 2006; Siegler & Booth, 2004).
Past efforts to assess the linearity of at-risk preschoolers’ number representations may
have used an unfamiliar and misleading number-line task that prompted a response bias.
Commonly used to assess children’s linear representation is the number to a point on a linear
number line (N-P) task that involves presenting a child a sheet of paper with a horizontal line,
with ‘‘0’’ just below the left end and‘‘10’’ just below the right end, and a numeral (1 to 9) is
approximately 2 cm above the center of the line. An experimenter asks the child to identify the
numeral at the top (and, if needed, provides help) and then asks, “If this is where 0 goes
(pointing) and this is where 10 goes (pointing), where does N go?”
Unfortunately, the current N-P task may not accurately reflect a preschooler’s mental
representation of the counting sequence (Fuson, 2009), because they may not (immediately)
comprehend the task for two reasons. First, “zero” is not a counting number and may be
unfamiliar to young children, especially those at risk of academic difficulties. Thus, for some
children, the left-hand referent may have been meaningless and unhelpful. Second, the placement
of the number N in the upper middle of the testing form may confuse children and cause them to
respond by placing their mark directly under the presented number (i.e., marking where the
target number actually is on the page and not where it should be on the number line). Consistent
with this conjecture, prior research pretesting of at-risk preschoolers (Ramani & Siegler, 2008;
Siegler & Ramani, 2009) with a N-P task yielded results that were neither logarithmic nor linear
with a slope appreciably larger than 0 but a nearly flat line with an intercept of between 4 and 5.
This may indicate that regardless of the numeral shown (actual magnitude), children consistently
tended to indicate the middle of the number line as their estimated magnitude. On an immediate
posttest, the children who received training with a number list did exhibit a linear pattern of
results, whereas the control participants remained confused by the number line estimation task
and continued to exhibit the mid-point response bias.
Extant research (Ramani & Siegler, 2008; Siegler & Ramani, 2009; Whyte & Bull, 2008)
indicates that the training with a number list can help preschool children improve their
performance on the N-P task. In the past, this improvement has been attributed to replacing a
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logarithmic representation of the counting numbers with a linear one. However, a plausible
alternative explanation is that the training merely helped trained participants better understand
the task demands of the N-P task. That is, the conventional N-P task may not have accurately
reflected participants’ initial representation of number.
Purpose
The purpose of this study was to compare 5- and 6-year-olds responses to a conventional
N-P task and a N-P task modified to make it more immediately comprehensible. The primary
issue addressed was: Does preschool children’s accuracy vary as a function of the type of N-P
task? That is, do they perform significantly better on a N-P task that does not have the possible
confusing or misleading features of the conventional N-P task or not? Secondary issues were: (1)
How accurate are young children’s representations of representations of the numbers 1 to 9? (2)
Is best fit of their numerical estimations a linear, logarithmic, or exponential function? (3) Does
children’s performance on the tasks vary from 5 to 6 years of age?
Participants
A total of 36 children were recruited to participate in the study. Participants were
recruited from 5 private preschools and 1 private kindergarten classrooms in a small Midwestern
city. Children ranged in age from 5 years 0 months to 7 years 0 months (M = 5.75, SD = .66).
Males composed 53 percent of the sample. The children’s race was reported as White (69%),
Asian (22%), Black (6%), and Hispanic (3%).
Data Collection and Analysis
Each participating children was assessed with two N-P tasks. The conventional task was
based on research conducted by Siegler and colleagues (Siegler & Opfer, 2003). Children were
presented with nine 4.24 in. x 11 in. sheets of white paper, one at a time. Printed on each sheet of
paper was a 25 cm horizontal line with hatch marks at each end of the line. Below the left-hand
hatch mark was printed “0” and below the right-hand hatch mark was printed “10.” A numeral (1
to 9) was printed 2 cm above the center of the horizontal line. On each trial, after asking the child
to identify the number at the top (and helping if needed), the examiner says, “If this is where 0
goes (pointing) and this is where 10 goes (pointing), where does n go? Use this pencil to mark on
the line where n goes. Take your best guess.”
The “modified” task differed from the conventional task in three important ways. First,
we removed the target numeral printed directly above the center of the line. The placement of the
numeral on the conventional task may lead to a response bias, where children exhibit an
increased tendency to mark their estimate near the center of the line. Second, we kept the length
of the horizontal line but moved the position of the left hand hatch mark to the position
corresponding to 1 on the number line and printed the numeral “1” below the hatch mark.
Finally, we attempted to make the task more meaningful to children by putting it into the context
of hopping bunnies and leaping frogs. A frog or bunny was printed above the left-hand edge of
the line (the position corresponding to “0” on the number line), and a lily pad or carrot was
printed above the hatch mark for “10.” For each trial, the examiner says, “It takes the bunny/frog
exactly 10 hops/leaps to get from here (point) to the carrot/lily pad. If this is where the
bunny/frog lands when he hops/leaps 1 time (point to the line above the “1”) and this is where
the bunny/frog lands he hops/leaps 10 times (point to the carrot/lily pad), where will the
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bunny/frog land when he hops/leaps n times? Use this pencil to mark on the line where the
bunny/frog would land. Take your best guess.”
Each task was presented during separate sessions, with approximately 2 to 7 days
between sessions. Trials were presented in random order. The children were randomly assigned
to one of two conditions. In the first condition, children (n=19) completed the conventional task
followed by the modified task. The children in the second condition (n=17) completed the tasks
in the reverse order.
Children’s performance on these tasks was evaluated in terms of accuracy and linearity.
Accuracy. Children’s accuracy was assessed using two methods. First, children’s
estimates were calculated as being correct if it fell within a 25% confidence interval around the
actual magnitude. For instance, for the target number 5, estimates falling between 4.75 and 5.25
were scored as correct and scores outside that range were scored as incorrect. These scores were
then summed to produce a total accuracy score. Accuracy was also assessed by calculating
absolute error values between the target number and the child’s estimate. The absolute error
values were then summed to produce a total absolute error score. In order to facilitate
comparisons between the two tasks, the total absolute error score for the conventional task
excluded children’s absolute error value for the number 1 (the number 1 was not estimated in the
modified task). A mixed ANOVA was conducted to determine if total absolute error scores
differed as a function of the type of task completed and the age of the child.
Linearity. Curve estimation regression analyses were conducted to determine if a linear,
logarithmic, or exponential function best fit estimates at the group and individual level. R2 and
slope values were examined. While previous literature suggests that children numerical
estimations tend to follow a linear or logarithmic function, preliminary analysis of scatterplots
indicated that an exponential function might also fit the data. Paired samples t-tests were
conducted to determine if R2 and slope values differed on the two tasks and one-way ANOVAs
were conducted to see if R2 and slope values differed for 5- and 6-year-olds.
Results
Accuracy. A summary of children’s number line estimates and absolute errors is provided
in Table 1. A summary of total accuracy scores is provided in Table 2. Analysis of children’s
total correct scores indicates that children’s number estimations were not very accurate on the
conventional task (M = .72, SD = .94) or on the modified version (M = 1.58, SD = 1.73). In fact,
53 percent of children received a total score of 0 on the conventional task and 33 percent on the
modified task. Although 6-year-olds generally obtained larger total scores, this difference did not
reach statistical significance for the conventional task (F (1, 34) = .15, p = .69), but was
marginally significant for the modified task (FBF [1, 12.74] = 4.42, p = .06). Expanding the
confidence interval to 50% resulted in increases in the mean total scores of .81 and 1.09 for the
conventional and modified task, respectively. Children were least accurate in estimating the
positions of 3 (0%), 4 (2.8%), and 7 (0%) on the conventional task and positions of 5 (13.9%), 6
(11.1%), and 7 (13.9%) on the modified task. Despite the general inaccuracies observed on both
tasks, results of a mixed ANOVA indicated a significant main effect of task type on total
absolute error scores (F [1, 34] = 38.15, p < .001, r = .73). A significant main effect of age on
total absolute error scores was not found (F [1, 34] = 1.19, p = .283, r = .18). However, a
significant interaction between task type and age was observed (F [1, 34] = 8.78, p = .006, r =
.45). Six-year-olds were less accurate on the conventional task, but substantially more accurate
on the modified task (see Figure 1).
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Linearity. Curve estimation regression analyses were run to determine the best fitting
model for the children’s number estimations (see Figures 2 and 3). Mean estimates of the sample
were closely fit by both a linear (R2 = .95) and exponential (R2 = .98) function for the
conventional task and for the modified task (linear: R2 = .98; exponential: R2 = .99). Curve
estimation regression analyses also were run on individual number line estimations. A summary
of fit statistics are provided in Table 3. For the conventional task, the linear function fit 47.2
percent of cases, the logarithmic function fit 11.2 percent of cases, and the exponential function
fit 58.3 percent of cases. Several cases were not fit by any function (13.9%), while 63.9 percent
of cases were fit by more than one function. For the modified task, the linear function fit 63.9
percent of cases, the logarithmic function fit 27.8 percent of cases, and the exponential function
fit 80.6 percent of cases. Fewer cases were not fit by any function (8.3%) and 55.6 percent of
cases were fit by more than one function. Paired samples t-tests indicated that fit of the linear (t
[35] = -.26, p = .797), logarithmic (t [35] = -1.42, p = .164), and exponential (t [34] = -.316, p =
.754) functions did not differ significantly on individual children’s estimates on the two tasks.
The individual slopes associated with the linear function on each task did not significantly differ
(t [35] = -.29, p = .776). A series of ANOVAs were conducted to determine if fit or slope values
varied with age. No significant differences were observed.
Conclusions
Numerical magnitude estimation has been identified as an essential element of early
mathematics education (National Council of Teachers of Mathematics, 2006) and has been
associated with mathematics achievement outcomes (Siegler & Booth, 2004). This study sought
to investigate one means of assessing numerical magnitude estimation, namely number line
estimation. This study differs from previous studies in that it compares a well-researched number
line estimation task with a modified version that controls for possible confounds of the
conventional task. Results of this study indicated that children’s estimations on the modified
number line task were more accurate than those made on the conventional task, particularly for
6-year-olds. It appears that providing a context for the task (e.g., a bunny hopping to a carrot)
makes the task meaningful to children and facilitates their ability to visualize where a number
falls on a number line. Provided a context, 6-year-olds may be more capable of drawing on their
experiences and existing number skills to complete the task. Even though children were more
accurate on the modified task, children’s estimates were rather inaccurate overall. Further
analyses need to be conducted to determine if these children were deficient in other early
mathematics skills and if those deficiencies were related to their number line estimates.
Previous research has suggested that children’s representation of number starts out
logarithmically and then proceeds to a linear representation. Results from this study are
somewhat inconsistent with that previous research. An exponential function was more often a
best fit for individual children’s number line estimates. That is, children’s estimates of the first 5
or 6 numbers tended to bunch up on the lower limit of the number line while the last 3 or 4
numbers tended to be spaced further apart and closer to the upper limit of the number line. There
were no significant differences in model fit between the two tasks. Additionally, there were no
differences in fit or slope between 5- and 6-year-olds. Examination of scatter plots indicated that
some individual children’s estimates may be better fit by two separate regression lines, one for
numbers 1(or 2) to 6 and one for numbers 7 to 9.
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Appendix A: References
Booth, J. L., & Siegler, R. S. (2006). Developmental and individual differences in pure
numerical estimation. Developmental Psychology, 41, 189-201.
Fuson, K. C. (2009). Avoiding misinterpretations of Piaget and Vygotsky: Mathematical
teaching without learning, learning without teaching, or helpful learning-path teaching?
Cognitive Development, 24, 343-361.
National Council of Teachers of Mathematics. (2006). Curriculum focal points for
prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA:
Author.
Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in lowincome children’s numerical knowledge through playing number board games. Child
Development, 79, 375-394.
Sarnecka, B. W., & Carey, S. (2008). How counting represents number: What children must
learn and when they learn it. Cognition, 108, 662-674.
Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children.
Child Development, 75, 428-444.
Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games – but not circular
ones – improves low-income preschoolers’ numerical understanding. Journal of
Educational Psychology, 101, 545-560.
Whyte, J. C., & Bull, R. (2008). Number games, magnitude representation, and basic number
skills in preschoolers. Developmental Psychology, 44, 588-596.
SREE 2011: Early Numeracy Assessment
A-1
2.39
2.93
3.59
5.49
6.21
7.99
4
5
6
7
8
9
SREE l 2011: Early Numeracy Assessment
1.47
M
.58
1.42
3
Number
1
2
Conventional Task
Estimates
Absolute Errors
SD
M Range
M
SD
M Range
.79
.1 - 3.5
.71 .52
.1 - 2.5
1.40
.2 - 4.8
1.3 .68
.1 - 2.8
2
.81
.5 - 3.5
1.5 .68
.5 - 2.5
9
1.38
.6 - 5.9
1.8 .95
.4 - 3.4
8
1.63 1.0 - 6.5
2.3 1.1
.0 - 4.0
6
2
2.03
.6 - 8.0
2.7 1.4
.0 - 5.4
6
8
2.09 1.8 - 8.8
2.1 1.3
.5 - 5.2
8
2
2.55 2.2 - 9.2
2.2 2.1
.0 - 5.8
5
3
2.07 3.8 - 9.9
1.5 1.6
.0 - 5.2
9
4
M
--1.9
8
2.5
2
3.1
8
3.9
4
4.4
8
6.5
5
6.9
2
7.8
9
Condition 1
1.0
6
1.5
7
1.8
7
1.7
3
2.0
6
.90
.83
B-1
2.3 - 9.7
3.9 - 8.9
1.7 - 9.7
2.0 - 6.9
2.1 - 5.5
1.7 - 4.6
1.3 - 4.8
1.2
1
1.6
7
1.3
7
1.4
3
1.4
2
.98
.79
1.3
9
1.3
1
1.4
3
1.8
5
.89
.72
.53
.0 - 6.7
.0 - 4.1
.0 - 5.3
.0 - 4.0
.2 - 2.9
.0 - 2.3
.2 - 1.8
Modified Task
Estimates
Absolute Errors
SD
M Range
M SD
M Range
--------------.59
1.3 - 3.6
.44 .38
.0 - 1.6
Appendix B: Tables & Figures
Table 1
Summary of Estimates and Absolute Errors for the Number Line Estimation Task by Condition
3.09 1.67
3.62 1.69
4.52 1.91
5.02 1.99
6.17 1.67
6.84 1.88
8.78 1.47
3
4
5
6
7
8
9
7.5
9.8
4.0 - 10.
0
8.6
2.2 - 10.
0
2.0 - 10.
0
3.2 9.4
1.8 8.7
.7 -
1.4 -
1.2 -
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2.35 1.88
M
---
2
Number
1
.83
1.1
4
1.2
2
1.4
8
1.7
3
1.4
5
1.4
5
.83
1.2
1
1.7
1
1.2
0
1.1
9
1.2
6
1.3
5
1.1
4
1.6
5
.0 - 5.0
.0 - 6.2
.1 - 3.8
.0 - 4.0
.0 - 5.0
.0 - 4.6
.1 - 4.5
.0 - 7.8
Modified Task
Estimates
Absolute Errors
SD
M Range
M
SD
M Range
---------------
7.9
3
M
1.0
2
1.5
1
2.1
1
2.7
8
3.2
8
3.7
1
5.1
4
6.4
6
Condition 2
2.3
6
B-2
2.7 - 10.
0
1.6
1
2.0
1
.0 - 6.3
Conventional Task
Estimates
Absolute Errors
SD
M Range
M SD
M Range
2.0
.0 1.1 1.6
.0 - 7.0
2
8.0
5
4
1.7
.2 1.4 1.1
.2 - 5.5
8
7.5
1
6
1.6
.4 1.7 .67
.9 - 3.2
9
6.2
5
1.3 1.0
1.6 .86
.2
3.0
8
5.5
1
1.5
.8 1.9 1.3
.0 - 4.2
9
6.7
2
2
1.5
1.7 2.3 1.4
.2 - 4.3
3
6.3
2
8
2.3
1.0 2.5 1.5
.4 - 6.0
3
9.6
0
7
2.2 2.2
2.0 1.7
.0
5.8
8
- 9.7
7
7
-
Table 2
Mean Total Scores for the Conventional and Modified Tasks and the Percent Correct for the
Numerals 1 to 9 by Task
Conventional Task
Modified Task
25%
50%
25%
50%
a
b
c
Mean Total Score (SD)
.72 (.94)
1.53 (1.28)
1.58 (1.73)
2.97d (1.80)
Estimates
%
%
%
1
11.1
27.8
--2
8.3
11.1
27.8
3
0
2.8
22.2
4
2.8
11.1
19.4
5
13.9
16.7
13.9
6
8.3
11.1
11.1
7
0
5.6
13.9
8
16.7
25
22.2
9
11.1
41.7
27.8
a
b
c
d
Note. Range = 0-4 / 9. Range = 0-5 / 9. Range = 0-7 / 8. Range = 0-8 / 8.
SREE l 2011: Early Numeracy Assessment
%
--69.4
30.6
33.3
27.8
25
25
30.6
55.6
B-3
Table 3
Summary of Individual Fit Statistics of Linear, Logarithmic, and Exponential Models for the
Number Line Estimation Task by Condition
Model
Linear
R2
Slope
Logarithmic
R2
Slope
Exponential
R2
Slope
M
Conventional Task
Time 1 (n=17)
Conventional Task
SD
M Range
M
Time 2 (n=17)
Modified Task
SD
M Range
.76
.89
.19
.34
.24
.32
- .97
- 1.32
.79
.88
.19
.26
.36
.25
- 1.00
- 1.24
.64
3.04
.20
1.11
.17
1.10
- .88
- 4.64
.72
4.00
.20
1.17
.32
1.13
- .95
- 5.60
.79
.35
.22
.14
.18
.07
-
.82
.20
.15
.05
.46
.11
M
Time 1 (n=19)
Modified Task
SD
M Range
M
.70
.86
.28
.26
.01
.07
- .98
- 1.16
.70
.82
.24
.41
.11 - .98
-.36 - 1.42
.63
3.87
.25
1.24
.02
.83
- .92
- 5.59
.61
2.84
.19
1.51
.24 - .84
-2.08 - 4.61
.73
.27
.08
-
.73
.22
.19
.05
.07
- .27
.31
.14
.98
.59
-
.98
.27
Modified Task
Model
Linear
R2
Slope
Logarithmic
R2
Slope
Exponential
R2
Slope
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.99
Time 2 (n=19)
Conventional Task
SD
M Range
.08
-.09 -
.98
.49
B-4
Figure 1
Graph of Task by Age Interaction
SREE l 2011: Early Numeracy Assessment
B-5
Figure 2
Linear, Logarithmic, and Exponential Pattern of Mean Estimates for Conventional Task
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B-6
Figure 3
Linear, Logarithmic, and Exponential Pattern of Mean Estimates for Modified Task
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B-7