Where is 150 on the ruler?
Write 0 at one end of a line and 1000 at the other, and ask a seven-year-old 'where is 150?', and the mark often lands much further right than it should; young children's estimates give small numbers a large part of the line. An adult puts the mark at about the 15 per cent point. The difference opens an interesting window onto how numbers are arranged in our minds.
In this experiment you place twenty numbers on two different lines: eight on the 0–100 line and twelve on the 0–1000 line. At the end we plot your estimates against the numbers' true values and compare them with two models: a straight line, like an evenly spaced ruler, or a logarithmic curve that gives small numbers too much room?
Siegler and Opfer's experiment
In a study published in Psychological Science in 2003, Robert Siegler and John Opfer gave the same task to second, fourth and sixth graders and to university students. On the 0–1000 line, second graders' median estimates fitted a logarithmic curve very well: the curve explained 95 per cent of the variance, while a straight line explained only 63 per cent. Fourth graders were in between; for sixth graders and adults the linear model explained almost all of the variance.
The study's most interesting finding was about context. The same second graders made much more linear estimates on the 0–100 line. According to Siegler and Opfer, children did not have a single representation of number; they chose between several representations depending on the context, and with age they chose the right one more often.
One piece of feedback may be enough
How does this change happen? In 2007 Opfer and Siegler showed second graders who made logarithmic estimates the correct positions of some numbers. Feedback on numbers where the linear and logarithmic representations differ most had the biggest effect. The change was strikingly abrupt and broad: often after a single piece of feedback, all estimates between 0 and 1000 straightened out at once.
In 2006 Booth and Siegler found the same development across four different estimation tasks, including the number line, in children from kindergarten to fourth grade: increasing reliance on linear representations and decreasing reliance on logarithmic ones. All estimation skills were positively related to scores on math achievement tests.
A number line without school
Is logarithmic intuition found only in children? In 2008 Dehaene, Izard, Spelke and Pica worked in Science with the Mundurucu, an Amazonian indigenous group with very little schooling and a limited vocabulary of number words. At every age the Mundurucu placed both sets of dots and numbers on a logarithmic scale. Western adults placed small or written numbers linearly, and clouds of dots that discouraged counting logarithmically.
According to the authors, the intuition of mapping numbers onto space is universal and this initial intuition is logarithmic; the evenly spaced linear number line is a cultural invention that does not develop without formal education. This strong interpretation was debated in the following years.
Representation, or proportion?
In 2011 Barth and Paladino proposed a different explanation: the number-line task is really a proportion judgment. When answering 'how much of 1000 is 150?', people use the end points and the middle of the line as references; models of perceptual proportion judgment can also explain curves that look logarithmic. On this view there may be no need to assume a 'shift from logarithmic to linear'; what changes is the precision of proportion estimation.
In 2012 Anobile, Cicchini and Burr asked adults to place clouds of dots on a line. Under normal conditions the adults placed them linearly; but when they were given another attention-demanding task at the same time, their placements turned into a compressed, logarithmic-like curve. The linear number line seems to be not a spontaneous intuition but a tool that requires attention.
Your score and your chart
Each number's score is calculated from the distance between where you put it and where it belongs, as a share of the line's length: 10 / (1 + (error / 5%)^1.6). An error of 2 per cent of the line earns about 8 points, 5 per cent earns 5 points and 10 per cent about 2.5 points. A round's score is the average of its four numbers; the five rounds add up to at most 50 points.
In the chart on the results screen each dot is a number: horizontally the number itself, vertically where you placed it. The dashed line is the perfect answer. The orange line is the straight line that best fits your dots, and the blue curve the best-fitting logarithmic curve; the closer R² is to 1, the better the model fits. With just a few numbers the two models can easily come close to each other; if the R² values are close, don't make too much of the difference.