Number-Line Estimation and Math Difficulties

 Should Number-Line Estimation Be Used as a Measure of the Perception and Mental Representation of Quantities?

In the field of reading, there are standardized tests with established norms, as well as considerable knowledge about the cognitive functions that should be assessed in a child who is experiencing difficulties. This is not the case in arithmetic. In this field, there is a shortage of norm-referenced tests, and the body of knowledge concerning the cognitive functions that should be examined is still developing. In this post, I will trace some of the obstacles encountered in the development of the number-line estimation task as an assessment tool. This developmental process reveals how children of different ages, as well as adults, perceive and process quantities.

One of the abilities commonly assessed in a child who has difficulty with arithmetic is the accuracy of quantity perception. This refers to the extent to which the child can distinguish subtle differences between quantities, for example, the difference between a group of 11 objects and a group of 12 objects of the same kind. Quantity perception probably underlies many arithmetic functions, and researchers therefore hypothesize that it is related to arithmetic difficulties in both children and adults.

One common way of assessing the perception and representation of quantities in children of elementary-school age and older, as well as in adults, is the number-line estimation task. In this task, the child sees an empty number line on which only the two endpoints are marked. The left endpoint is labeled 0, and the right endpoint is labeled 10, 100, 1,000, or another number. The child is asked to estimate the position of a particular number, for example 42, on the line and to mark that position with a vertical line. After the child has completed many such estimates, the examiner evaluates their accuracy by measuring the differences between the child’s marks and the correct locations of the estimated numbers.

Studies using this task found that young children represent numbers on the number line according to a logarithmic scale. On such a scale, the distances between successive numbers become progressively smaller as the numerical values increase. For example, the distance between 1 and 2 is greater than the distance between 8 and 9. A child in kindergarten or first grade may place the number 9 at the location corresponding to 40 on a number line ranging from 0 to 100. The number 9 is shifted to the right because the child perceives the distances between the numbers 1 and 9 as very large relative to the distances between the other numbers. From the child’s perspective, the distances between the numbers 1 and 9 occupy 40% of the space on a number line ranging from 0 to 100.

As children grow older and gain experience with arithmetic exercises and with number lines, their representation of quantities becomes more accurate and their estimates become increasingly linear. On a linear scale, the distances between successive numbers are constant. For example, the distance between 1 and 2 is the same as the distance between 8 and 9. A lower degree of linearity than expected for the child’s age may be one indication of arithmetic difficulties.

Studies have found correlations between performance on number-line estimation tasks and performance on arithmetic tasks such as counting, basic arithmetic, particularly subtraction, and algebra. Children with mathematics learning disabilities have difficulty with number-line estimation. Number-line estimation ability is also related to later success in learning addition, and training in estimation has been found to improve performance on addition tests.

All of this appeared quite promising until some researchers recently began to argue that the number-line estimation task does not actually measure what it is intended to measure. These researchers observed that both children and adults perform the estimation task by using strategies. For example, they mentally divide the number line into halves or quarters (they are not allowed to mark these reference points on the line while performing the task). They then use these reference points to locate the number whose position they are estimating.

For example, when a child is asked to place the number 42 on a number line ranging from 0 to 100, the child identifies the midpoint, where the number 50 would be located. Because 42 is smaller than 50, the child moves leftward from 50 to estimate its position. The estimation process is therefore influenced by knowledge of the decimal structure of numbers and by the ability to develop a strategy. It is consequently not surprising that performance on this task correlates with performance on arithmetic tasks. Because the task relies on knowledge of the decimal structure of numbers, these researchers argue that number-line estimation does not provide a “pure” measure of quantity perception.

To overcome this problem, researchers developed a new version of the task: the unbounded number line. On such a line, the number 0 is marked at the left endpoint, while the right endpoint remains unlabeled. To the left of 0, and very close to it, a segment representing a particular value is shown, for example 1, 4, 10, or even 100. The child or adult uses this segment to estimate the location of the target number. This task is intended to provide a “purer” measure of quantity perception because the participant cannot use two labeled endpoints to divide the line into halves or quarters.

The image below shows two bounded number lines at the top and two unbounded number lines at the bottom.

Performance on an unbounded number-line task appears to be less influenced by age and development and may therefore reflect the quality of an innate sense of quantity. On the other hand, it is also less strongly related to arithmetic performance in children from elementary school through seventh grade. Does this mean that quantity perception does not influence arithmetic performance?

In any event, if performance on this task is not related to arithmetic performance, it is difficult to justify using it to identify the source of difficulties in children who struggle with arithmetic, unless research demonstrates that such children also perform more poorly on the unbounded number-line task. To the best of my knowledge, no such study involving children has yet been published. In an unpublished master’s thesis, van Wijk found no difference in unbounded number-line estimation ability between children with and without mathematics learning disabilities. Similarly, no differences in unbounded number-line estimation were found between adults with and without dyscalculia (van der Weijden et al., 2018).

It therefore appears necessary to develop a number-line estimation task that, on the one hand, cannot be solved through the use of strategies and, on the other hand, is correlated with performance on arithmetic tests.

van der Weijden, F. A., Kamphorst, E., Willemsen, R. H., Kroesbergen, E. H., & van Hoogmoed, A. H. (2018). Strategy use on bounded and unbounded number lines in typically developing adults and adults with dyscalculia: An eye-tracking study. Journal of Numerical Cognition, 4, 337–359.

Jung, S., Roesch, S., Klein, E., Dackermann, T., Heller, J., & Moeller, K. (2020). The strategy matters: Bounded and unbounded number line estimation in secondary school children. Cognitive Development, 53, 100839.

Link, T., Nuerk, H. C., & Moeller, K. (2014). On the relation between the mental number line and arithmetic competencies. Quarterly Journal of Experimental Psychology, 67(8), 1597–1613.