Do preschoolers see numbers on a log scale or a straight line?
Even 3- to 5-year-olds who could not yet count placed dot sets on a number line in a more evenly spaced (linear) than compressed (logarithmic) way, which challenges the idea that young children's number sense starts out logarithmic.
Source
More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children
Study at a glance
- Design
- Human experiment — Three computerised number-line experiments with non-symbolic (dot-set) stimuli: 1–20 lines in both directions (Exp 1), a 1–9 left-to-right line (Exp 2) and a forced-choice position-to-number version (Exp 3); children grouped by counting-principle knowledge using the Give-a-number task
- N
- Separate samples per experiment: Exp 1 had 35 subset-knowers and 29 CP-knowers (61 analysed after exclusions); Exp 2 had 64 children; Exp 3 had 95 children. The Methods section for Experiment 1 is missing from the available text.
- Population
- Polish preschool children aged roughly 3 to 6 years from preschools in Warsaw
- Outcome
- Fit of linear versus logarithmic (and cyclic power) models to children's placements, percent absolute error, variance of estimates, and effects of line direction and counting knowledge
Structured fields used in claim comparison tables when every cited study has a complete layer.
What they did
Preschoolers were first sorted into 'subset-knowers' (who only know the meaning of a few small number words) and 'CP-knowers' (who understand how counting works) using the Give-a-number task. They then played a touch-screen game placing plates of dots on a line anchored by 1 dot and 20 dots, run left-to-right and right-to-left (Experiment 1), or by 1 and 9 dots (Experiment 2). In Experiment 3 the task was reversed: a position on the line was marked and children chose which of three dot sets belonged there. The authors compared how well linear and logarithmic functions fitted children's estimates.
What they found
Across all three experiments the linear model fitted better than the logarithmic model, including in subset-knowers, although individual fits in Experiment 1 were low and estimates imprecise. The direction of the line made no difference, and Bayesian tests gave moderate evidence for no effect. In Experiment 2 the spread of estimates grew with the size of the number, the pattern expected from a linear representation with scalar variability. CP-knowers were more precise than subset-knowers in every experiment, and in Experiments 2 and 3 counting knowledge explained this better than age alone.
The limits
What it doesn't show
The study cannot tell whether the number-line task reveals the child's internal number representation or just a strategy for mapping numbers onto space on the spot; the authors themselves conclude the mapping may be built ad hoc for the task. Experiment 3 was dominated by an anchoring strategy of choosing the set most like the nearer end of the line, and Experiment 1's test of scalar variance came out null. Because the design is cross-sectional, counting knowledge and age are confounded, and the link between counting knowledge and precision could reflect other skills such as executive function. Dot sets could not be fully controlled for area or density cues given the small number of trials young children can tolerate.
Key terms
- Number-line estimation task
- A task in which people mark where a number (or a set of dots) belongs on a line with labelled ends; the pattern of placements is used to infer how numbers are mentally scaled.
- Log-to-linear shift
- The hypothesis that young children space numbers logarithmically (small numbers spread out, large numbers squashed together) and shift to even, linear spacing as they learn about numbers.
- Subset-knower vs CP-knower
- A subset-knower knows the exact meaning of only a few small number words; a cardinal-principle (CP) knower understands that the last number counted gives the size of the set.
- Scalar variability
- The idea that the noise in estimating a quantity grows in proportion to its size, so estimates of larger numbers are more variable.
- Percent absolute error (PAE)
- How far an estimate falls from the correct position, expressed as a percentage of the line's length; lower means more accurate.
- Non-symbolic number
- Quantity shown as a collection of items (like dots) rather than as a numeral or number word.
Flashcards
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Quiz yourself
What does the log-to-linear shift hypothesis claim?
Common questions
If the linear model won, does that mean children's number sense is perfectly accurate?
No. Estimates were quite imprecise, especially on the 1–20 line where mean error was above a quarter of the line, and individual model fits were modest. Linear just means the errors were not systematically compressed at the top end.
Why test the line in both directions?
Many cultures read left to right, and older children and adults often link small numbers to the left. Testing right-to-left lines checked whether preschoolers already rely on that direction. They did not: performance was the same either way.
Why do the results differ from earlier studies that found logarithmic patterns?
The authors suggest earlier studies often used zero as the left end of the line and very large numbers of dots. Young children may not treat an empty set as a number, which could squash their estimates and create an apparent logarithmic pattern.
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