Are books like number lines?

author-approved-badge Reference: Thompson, C. A., Morris, B. J., & Sidney, P. G. (2018). Are books like number lines? Children spontaneously encode spatial-numeric relationships in a novel spatial estimation task. Frontiers in Psychology. 8(2242).

Nearly everyone knows the value and importance of reading for developing literacy, but school teachers are often questioned about the importance of mathematics. For example, my niece innocently asked me why she had to learn math when her phone has a calculator. But we use mathematical reasoning every day — and often in surprising ways! For example, have you ever considered that finding the right page in a book relies on your mathematical reasoning? This reasoning is not limited to your ability to identify the page number, but also relies on numerical magnitude representations (i.e., how big a number is).


Numerical magnitudes are represented along a mental number line where, in cultures that read from left-to-right, smaller numbers are associated with space to the left and larger numbers to the right. When activities support these left-to-right spatial-numeric associations, people tend to perform better. For example, American preschoolers are faster and more accurate at playing a matching game when verbally labeled boxes increase in numeric value from left-to-right, compared to when the boxes are labeled from right-to-left (1). Scientists often use the number line estimation task to measure numerical magnitude representations because numbers are thought to be represented along a mental number line.

In one version of the number line estimation task, participants are presented a line with two endpoints, typically labeled zero on the left and a larger number, such as 100, on the right (see figure below). Participants’ task is to estimate the location of a third number that is presented above the line (e.g., 15). Participants who estimate numbers closer to their true location on the line tend to have higher math achievement. Improving the accuracy of their estimates can improve performance on other math tasks. The relationship between performance on number line tasks and math is, in part, thought to be due to the task aligning with how numbers are represented in the brain: along a mental number line. More precise internal representations of numbers are thought to be foundational for higher-level mathematics.

Cobites figure1
(2) Example trial from the number line estimation task.

Other activities capitalize on left-to-right spatial orientations of numbers. For example, playing linear board games that are oriented with smaller numbers on the left and larger numbers on the right side, improves children’s precision in representing magnitudes on number lines (2). However, circular numeric board games do not afford these same learning advantages. These games are thought to improve numerical representations because they align with the mental number line and support spatial-numeric associations. This suggests that everyday activities, like games, may promote spatial-numeric associations.

Spatial-numeric associations of books

Just like linear board games and number lines, page numbers in books increase from left-to-right, have equal spacing between pages, and provide a way to compare and contrast numbers relative to others. Using this knowledge, researchers Thompson, Morris, and Sidney developed a new page-finder task in which participants were asked to estimate the location of a page, without counting, in a non-numbered book. Because the task shares similar spatial-numeric associations to number-lines, the authors predicted that performance on this task would be related to performance on the number-line estimation task.

Books gif

Method and Results

Sixty first graders, around age seven, completed tasks assessing two aspects of their numerical understanding: numerical magnitude knowledge and knowledge of symbolic numbers (i.e., recognize what 3 is). To assess their numerical magnitude knowledge, the children completed the number line estimation task (described above) and a magnitude comparison task, which involved selecting the larger of two presented numbers. To assess knowledge of symbolic numbers, children identified numbers (e.g., verbally name a symbolically presented number “17”), and counted on from provided numbers (e.g., count up three from 18).

Finally, children completed the page finder task developed by the authors. In this task, children were presented with a non-numbered, 100-page, spiral bound book and asked to quickly flip to a page (e.g., flip to page 21) without counting. After being oriented to the task and completing three practice trials, they were asked to find twelve different pages ranging from four to 90 without feedback.

In line with the authors’ predictions, performance on the number line estimation and page finder tasks were related, suggesting that the two tasks were tapping a common underlying magnitude representation. Critically, performance on the page finder task did not relate to the tasks measuring symbolic number knowledge, such as identifying numerals, which are thought to rely on different cognitive processes than magnitude understanding.

Books are like number lines

Children’s similar performance on the novel page finder task and the number line estimation task seems to suggest that children used similar representations to find pages in a non-numbered book as they do when they estimate the magnitude of numbers. To find a page in a book, children (and adults) need to consider how far that page is from the front of the book and the end of the book — similar to when they estimate where a number belongs on a number line labeled with two endpoints.

Note, however, that these results are correlational. It is exciting to consider that parents and teachers may be able to develop literacy and numeracy skills during reading tasks. However, more research needs to be done to evaluate whether interventions using the page-finder task can improve magnitude knowledge. But, the next time you open a book to find a specific page, know that this likely relies in part of your mathematical reasoning ability. Who knew interacting with books involved mathematical reasoning?

Images: (1) (2) Image generated by C.Fitzsimmons 12/2018. (3) Pixabay. Accessed 12/16/2018

Additional References:

(1) Opfer, J. E., Thompson, C. A., & Furlong, E. E. (2010). Early development of spatial-numeric associations: evidence from spatial and quantitative performance of preschoolers. Dev. Sci. 13, 761-771. doi: 10.1111/j.1467-7687.2009.00934.x

(2) Siegler, R. S. & Ramani, G. B. (2009). Playing linear board games – but not circular ones – improves low-income preschoolers’ numerical understanding. J. Educ. Psychol. 101, 545-560. doi: 10.1037/a0014239