# Active Learning for a Number-Line Task with Two Design Variables

- Sang Ho Lee, The Ohio State University, Columbus, Ohio, United States
- Dan Kim, The Ohio State University, Columbus, Ohio, United States
- John Opfer, The Ohio State University, Columbus, Ohio, United States
- Mark Pitt, Psychology, Ohio State University, Columbus, Ohio, United States
- Jay Myung, Department of Psychology, The Ohio State University, Columbus, Ohio, United States

**Abstract**The number-line task is a widely used task in diverse fields
of study. In the task, a given number that varies every trial is
estimated on a continuum flanked with 0 and an upper-bound
number. An upper-bound of a number-line is often arbitrarily
selected by researchers, although this design variable has
been shown to affect the non-linearity in estimates. Examining
estimates of varying given numbers (design variable 1) with
varying upper-bound numbers (design variable 2) can be costly
because adding a new design dimension into a number-line
task could drastically increase the number of trials required
for examining the underlying representation of number. The
present study aims to conduct a number-line task with
the given number and the upper-bound being the design
variables. A design optimization algorithm, Gaussian Process
Active Learning (GPAL), made this new paradigm feasible
without increasing the number of trials, by presenting only
the most informative combinations of the design variables every
trial. Our experimental data showed that the non-linearity
of the number-line estimates increases with the upper-bound
of the number line. The degree of non-linearity could predict a
math skill (i.e., addition proficiency), but only when the upper-bound
was relatively large. The observed range-dependency of
the number-line estimates would not be fully explored without
systematically manipulating the upper-bound as an additional
design variable. As in the present number-line task, GPAL
would be a useful tool for the research problems that require
multidimensional design experiments to be solved.