Annie Paladino

Dale Cohen

Emily Slusser

Hilary Barth

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COMMENTARY
A sense of proportion: commentary on Opfer, Siegler and Young
Hilary Barth,1 Emily Slusser,1 Dale Cohen2 and Annie Paladino1,3
1. Psychology Department, Wesleyan University, USA
2. Department of Psychology, University of North Carolina at Wilmington, USA
3. WestEd, San Francisco, USA
This is a commentary on Opfer, Siegler and Young (2011).
A growing body of evidence (Barth & Paladino, 2011;
Cohen & Blanc-Goldhammer, 2011; Sullivan, Juhasz,
Slattery & Barth, 2011) shows that ‘linear’ estimation
patterns in published number-line data actually follow
a cyclic power function, a signature of proportional
reasoning (Hollands & Dyre, 2000; Spence, 1990). Opfer,
Siegler and Young argue that fitting a cyclic power
function to number-line data is tantamount to capital-
izing on chance. We claim, in contrast, that the cyclic
power function is (1) theoretically motivated, (2) math-
ematically sensitive to important fluctuations in the data,
and (3) highly unlikely to be fitting noise in these data.
We caution OS&Y that linear functions are insensitive to
the systematic fluctuations in the data that describe the
observer’s estimation bias in the bounded number-line
task.
The bulk of OS&Y’s substantive critique rests on
analyses of previous datasets. But OS&Y’s compiled
studies were explicitly designed to distinguish between
logarithmic and linear models. The observations in their
datasets are sparsely distributed near the upper endpoint,
where proportion-judgment (PJ) models make specific
predictions. Therefore those datasets are biased against
PJ hypotheses and toward the logarithmic and linear
hypotheses they were designed to capture. We also note
that B&P found support for the PJ account in analyses of
individuals’ estimates, not just group analyses as implied
by OS&Y.
The PJ account explains changes in number-line esti-
mation in at least three ways. First, values of Stevens
exponents (b) for numerical magnitude might gradually
approach 1 with age, leading to more accurate estimates.
Second, older children might use more reference points,
an independent source of increased accuracy. Third,
accuracy requires evaluating the upper endpoint appro-
priately when deciding where to place a target number
(see also Cantlon, Cordes, Libertus & Brannon, 2009).
Implicit also in B&P is the possibility that very young
children might not recognize the bounded nature of the
task at all, instead treating it as an open-ended magni-
tude judgment; if so their estimates should be well
characterized by an unbounded power function.
OS&Y misinterpret the PJ account when they model
microgenetic changes in children’s estimates as changes
in b. They report abrupt changes in b after local feed-
back, claiming that this finding supports the log-to-
linear-shift hypothesis. But in our view, local feedback
probably improves accuracy by providing children with a
new reference point (predicting broad, not just local,
improvements in estimation accuracy; see Hollands &
Dyre, 2000, Figure 3). Adoption of a reference point can,
of course, occur abruptly. OS&Y unfortunately did not
test this hypothesis.
Finally, OS&Y raised the concern that while power
models may provide the best fits, they are not necessarily
the best predicting models. They applied leave-one-out
cross-validation (LOOCV) techniques to their compiled
dataset, finding support for the log-to-linear shift. We
conducted LOOCV analyses on B&P’s dataset (one
better suited to testing alternative hypotheses). We cal-
culated mean squared error (MSE) as the cross-valida-
tion error index, instead of using the mean absolute
percent error (MAPE) as OS&Y did (MAPE is at times
appropriate for evaluating time-dependent series data,
but it is not appropriate for datasets discussed here1).
We also did not separate children with ‘logarithmic’ vs.
‘linear’ estimation patterns, a serious problem with
OS&Y’s analysis (because preassigning these groups
Address for correspondence: Hilary Barth, Psychology Department, Wesleyan University, 207 High St, Middletown CT 06459, USA; e-mail:
hbarth@wesleyan.edu
1 By dividing the error (estimated value minus the actual value) by the
actual value, this calculation weighs errors from predictions on the
upper end of the scale less than those on the lower end. For example, if
an estimated value is 4 and the actual value is 3, the absolute percent
error is .33. However, if an estimated value is 91 and the actual value is
90 (again, a discrepancy of only one point), the [M]APE is .01. MSE, on
the other hand, weights each of these values equally.
MSE ¼ 1
n
X
ðE  AÞ2 MAPE ¼ 1
n
X E  A
A


where E refers to the estimated value and A refers to the actual value.
 2011 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
Developmental Science (2011), pp 1–2 DOI: 10.1111/j.1467-7687.2011.01081.x
essentially circumvents the very safeguards that cross-
validation techniques impart, inserting an additional
level of model complexity and bias that are not
accounted for by these analyses). Our LOOCV results
supported the PJ account. For 5-year-olds, an un-
bounded power model yielded the lowest MSE (26.356),
followed by the adapted one-cycle model (27.884) and
then the logarithmic model (34.397). The linear, one-
cycle, and two-cycle models yielded much larger MSEs
(51.278, 66.297, and 210.943, respectively). For 7-year-
olds, the two-cycle model yielded the lowest MSE
(13.645), followed by the linear model (15.475) and the
one-cycle model (19.484; 19.622 for the adapted version),
and unbounded power and logarithmic models produced
large MSEs (118.908 and 236.527, respectively). Unsur-
prisingly (see Stone, 1977), the LOOCV analysis describes
the models in much the same way as a comparison of
AICc scores (Akaike’s information criterion, used in
B&P’s original analyses).
‘The choice of a model may be influenced by measures
of fit but the final decision concerning which model to
use must involve human judgment’ (Browne, 2000, p.
110). We agree. Our intent is not to quibble about model
selection, but to present theoretical and empirical sup-
port for an alternative account of number-line estimation
and related paradigms. Given growing evidence that
bounded number-line tasks produce the signature ogival
functions that indicate proportion estimation, we urge
researchers to do the following. First, sample a sufficient
number of points along the number line to identify the
cyclic function if it is present. Second, model both the
cyclic function and linearity to determine relative fits
of each model.2 Limiting analyses to logarithmic and
linear models may be sufficient for those who simply
need a useful heuristic for classifying children. But for
researchers using these tasks to assess the nature of
numerical representation, failure to adequately assess
and fit proportion models in a bounded number-line task
would be a serious omission.
Acknowledgements
This work was supported in part by NSF DRL0950252
to HB and NIH RO1HD047796 to DC.
References
Barth, H., & Paladino, A.M. (2011). The development of
numerical estimation: evidence against a representational
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Browne, M. (2000). Cross-validation methods. Journal of
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Received: 18 May 2011
Accepted: 28 May 2011
2 A Microsoft Excel worksheet for performing simple versions of these
analyses (Slusser & Barth, n.d.) is freely available at http://hbarth.
faculty.wesleyan.edu/publications/.
2 Hilary Barth et al.
 2011 Blackwell Publishing Ltd.


COMMENTARY A sense of proportion: commentary on Opfer, Siegler and Young

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