Abstract
Rational numbers, such as fractions and decimals, are harder to understand than natural numbers. Moreover, individuals struggle with fractions more than with decimals. The present study sought to disentangle the extent to which two potential sources of difficulty affect secondary-school students’ numerical magnitude understanding: number type (natural vs. rational) and structure of the notation system (place-value-based vs. non-place-value-based). To do so, a 2 (number type) × 2 (structure of the notation system) within-subjects design was created in which 61 secondary-school students estimated the position of four notations on a number line: natural numbers (e.g., 214 on a 0–1000 number line), decimals (e.g., 0.214 on a 0–1 number line), fractions (e.g., 3/14 on a 0–1 number line), and separated fractions (3 on a 0–14 number line). In addition to response times and error rates, eye tracking captured students’ on-line solution process. Students had slower response times and higher error rates for fractions than the other notations. Eye tracking revealed that participants encoded fractions longer than the other notations. Also, the structure of the notation system influenced participants’ eye movement behavior in the endpoint of the number line more than number type. Overall, our findings suggest that when a notation contains both sources of difficulty (i.e., rational and non-place-value-based, like fractions), this contributes to a worse understanding of its numerical magnitude than when it contains only one (i.e., natural but non-place-value-based, like separated fractions, or place-value-based but rational, like decimals) or neither (i.e., natural and place-value-based, like natural numbers) of these sources of difficulty.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The raw data can be found at the following repository: https://osf.io/guk9x/?view_only=4f96a2db26ac4aca8e03e50c283391ad.
Code availability
Not applicable.
Notes
For terminology in the current paper, decimals refer to numbers in the form of W.XYZ and, fractions refer to numbers in the form X/Y, with WXYZ representing different Arabic numerals.
Numerical magnitude understanding refers to how well an individual grasps the quantity of a number and reflects how well our cognitive representation numerical quantity is calibrated, sometimes referred to as a “mental number line” (Siegler & Opfer, 2003). This definition of the term numerical magnitude understanding will be assumed throughout the manuscript.
It should be noted that there also exist decimal fractions, which are fractions in which the denominator is a power of 10 (i.e., 10, 100, 1000), allowing fractions to be easily converted to decimals making arithmetic with decimal fractions easier. The current study omits decimal fractions and focuses rather on fractions that are unrelated to multiples or powers of 10.
It should be noted that the distinction between the place-value-based and non-place-value-based notations for natural numbers was at the level of the number line’s endpoint, while for rational numbers, it was at the level of the target number. It is possible that changes at one level have more of an impact on numerical magnitude understanding than the other.
Each decimal was rounded to the thousandths.
A study using the same eye tracker and calibration settings compared two modes of responding in exactly the same NLE task. Responding with the mouse (i.e., estimation error but no calibration error) resulted in an average error rate of 2.13%, while responding with the eyes (i.e., estimation error and calibration error) yielded an average error rate of 3.47% (MacKay et al., 2020). Assuming that participants’ estimation error was the same in both conditions, it can be estimated that the additional average error due to calibration error is about (3.47–2.13% =) 1.34%, which is much less than the maximum of 8.82%.
Because the endpoint of the number line changed each trial in the separated fractions condition only, we opted for three AOIs instead of one large AOI encompassing the entire number line, to separate participants’ eye movement behavior around the endpoint from their eye movement behavior on the number line, which we were specifically interested in.
The cut-off of 120 ms was determined based on a visual inspection of the histogram of encoding gaze duration data across all trials for all participants, as the data below 120 ms were clearly separate from the remainder of the data.
These results should be interpreted considering the statistics used for the current manuscript (i.e., the mean of the medians). As pointed out earlier, if many participants did not fixate within an AOI on more than 50% of trials, the mean of medians was 0. As the values for some of the means in the positioning phase were 0 in the endpoint AOI, it is possible that smaller differences between each of notations might exist (i.e., some participants fixated on the endpoint AOI for some trials in one notation even though it had a mean of 0).
By means of parafoveal vision (covering a visual angle of 4–5° from the center of the visual field) or peripheral vision (i.e., covering a visual angle of > 5° from the center of the visual field; Larson & Loschky, 2009).
References
Ashcraft, M. H., & Moore, A. M. (2012). Cognitive processes of numerical estimation in children. Journal of Experimental Child Psychology, 111(2), 246–267. https://doi.org/10.1016/j.jecp.2011.08.005
Carrasco, M. (2011). Visual attention: The past 25 years. Vision Research, 51, 1484–1525. https://doi.org/10.1016/j.visres.2011.04.012
Braithwaite, D. W., & Siegler, R. S. (2018). Developmental changes in the whole number bias. Developmental Science, 21(2), e12541. https://doi.org/10.1111/desc.12541
Dackermann, T., Kreomer, L., Nuerk, H.-C., Moeller, K., & Huber, S. (2018). Influences of presentation format and task instruction on children’s number line estimation. Cognitive Development, 47, 53–62. https://doi.org/10.1016/j.cogdev.2018.03.001
DeWolf, M., Grounds, M. A., Bassok, M., & Holyoak, K. J. (2014). Magnitude comparison with different types of rational numbers. Journal of Experimental Psychology: Human Perception and Performance, 40(1), 71–82. https://doi.org/10.1037/a0032916
DeWolf, M., & Vosniadou, S. (2015). The representation of fraction magnitudes and the whole number bias reconsidered. Learning and Instruction, 37, 39–49. https://doi.org/10.1016/j.learninstruc.2014.07.002
Di Lonardo, S. M., Huebner, M. G., Newman, K., & LeFevre, J.-A. (2019). Fixated in unfamiliar territory: Mapping estimates across typical and atypical number lines. Quarterly Journal of Experimental Psychology, 73(2), 279–294. https://doi.org/10.1177/1747021819881631
Duchowski, A. (2007). Eye tracking methodology: Theory and practice. Springer.
Fazio, L. K., Kennedy, C. A., & Siegler, R. S. (2016). Improving children’s knowledge of fraction magnitudes. PLoS ONE, 11(10), 1–14. https://doi.org/10.1371/journal.pone.0165243
Fuson, K. C. (1990). Issues in place-value and multidigit addition and subtraction learning and teaching. Journal for Research in Mathematics Education, 21(4), 273–280. https://doi.org/10.2307/749525
Ganor-Stern, D., & Weiss, N. (2016). Tracking practice effects in computation estimation. Psychological Research Psychologische Forschung, 80, 434–448. https://doi.org/10.1007/s00426-015-0720-7
Hamdan, N., & Gunderson, E. A. (2017). The number line is a critical spatial-numerical representation: Evidence from a fraction intervention. Developmental Psychology, 53(3), 587–596. https://doi.org/10.1037/dev0000252
Heine, A., Thaler, V., Tamm, S., Hawelka, S., Schneider, M., Torbeyns, J., De Smedt, B., Verschaffel, L., Stern, E., & Jacobs, A. M. (2010). What the eyes already ‘know’: Using eye movement measurement to tap into children’s implicit numerical magnitude representations. Infant and Child Development, 19, 175–186. https://doi.org/10.1002/icd.640
Holmqvist, K., Nyström, M., Andersson, R., Dewhurst, R., Jarodzka, H., & van de Weijer, J. (2011). Eye tracking: A comprehensive guide to methods and measures. Oxford University Press.
Holmqvist, K., Örbom, S.L., Hooge, I.T.C., Niehorster, D.C., Alexander, R. G., Andersson, R., Benjamins, J. S., Blignaut, P., Brouwer, A-M., Chuang, L. L. Dalrymple, K. A., Drieghe, D., Dunn, M. J., Ettinger, U., Fielder, S., Foulsham, T., van der Geest, J. N., Hansen, D. W., Hutton, S. B., Kasneci, E….Hessels, R. S. (2023) Eye tracking: Empirical foundations for a minimal reporting guideline. Behavior Research Methods, 55, 364–416. https://doi.org/10.3758/s13428-021-01762-8
Huber, S., Klein, E., Willmes, K., Nuerk, H. C., & Moeller, K. (2014a). Decimal fraction representations are not distinct from natural number representations—Evidence from a combined eye-tracking and computational modeling approach. Frontiers in Human Neuroscience, 8, 172. https://doi.org/10.3389/fnhum.2014.00172
Huber, S., Moeller, K., & Nuerk, H. C. (2014b). Adaptive processing of fractions—Evidence from eye-tracking. Acta Psychologica, 148, 37–48. https://doi.org/10.1016/j.actpsy.2013.12.010
Huebner, M. G., & LeFevre, J.-A. (2018). Selection of procedures in mental subtraction: Use of eye movements as a window on arithmetic processing. Canadian Journal of Experimental Psychology / Revue Canadienne De Psychologie Expérimentale, 72(3), 171–182. https://doi.org/10.1037/cep0000127
Hurst, M., & Cordes, S. (2016). Rational-number comparison across notation: Fractions, decimals, and whole numbers. Journal of Experimental Psychology: Human Perception and Performance, 42(2), 281–293. https://doi.org/10.1037/xhp0000140
Iuculano, T., & Butterworth, B. (2011). Understanding the real value of fractions and decimals. The Quarterly Journal of Experimental Psychology, 64(11), 2088–2098. https://doi.org/10.1080/17470218.2011.604785
Johnson, J. T. (1956). Decimal versus common fractions. The Arithmetic Teacher, 3(5), 201–206. https://doi.org/10.5951/at.3.5.0201
Just, M. A., & Carpenter, P. A. (1980). A theory of reading: From eye fixations to comprehension. Psychological Review, 87(4), 329–354. https://doi.apa.org/doi/10.1037/0033-295X.87.4.329
Kloosterman, P. (2010). Mathematics skills of 17-year-olds in the United States: 1978 to 2004. Journal for Research in Mathematics Education, 41(1), 20–51. https://doi.org/10.5951/jresematheduc.41.1.0020
Larson, A. M., & Loschky, L. C. (2009). The contributions of central versus peripheral vision to scene gist recognition. Journal of Vision, 9(10), 6–6. https://doi.org/10.1167/9.10.6
LeFevre, J.-A., Jimenez Lira, C., Sowinski, C., Cankaya, O., Kamawar, D., & Skwarchuk, S.-L. (2013). Charting the role of the number line in mathematical development. Frontiers in Psychology, 4, Article 64. https://doi.org/10.3389/fpsyg.2013.00641
Lilienthal, A., & Schindler, M. (2019). Current trends in the use of eye tracking in mathematics education research: A PME survey. In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of 43rd Annual Meeting of the International Group for the Psychology of Mathematics Education. PME.
MacKay, K. J., Germeys, F., Van Dooren, W., Verschaffel, L., & Luwel, K. (2020). Comparing eye fixation and mouse cursor response modes in number line estimation. Journal of Cognitive Psychology, 32(8), 827–840. https://doi.org/10.1080/20445911.2020.1817039
MacKay, K. J., Germeys, F., Van Dooren, W., Verschaffel, L., & Luwel, K. (2022). The structure of the notation system in adults’ number line estimation: An eye-tracking study. Quarterly Journal of Experimental Psychology, 76(3), 538–553. https://doi.org/10.1177/17470218221094577
Meert, G., Grégoire, J., & Noël, M. P. (2009). Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task. Quarterly Journal of Experimental Psychology, 62(8), 1598–1616. https://doi.org/10.1080/17470210802511162
Meert, G., Grégoire, J., & Noël, M. P. (2010). Comparing the magnitude of two fractions with common components: Which representations are used by 10- and 12-year-olds? Journal of Experimental Child Psychology, 107(3), 244–259. https://doi.org/10.1016/j.jecp.2010.04.008
Miura, I. T., Okamoto, Y., Kim, C. C., Chang, C. M., Steere, M., & Fayol, M. (1994). Comparisons of children’s cognitive representation of number: China, France, Japan, Korea, Sweden, and the United States. International Journal of Behavioral Development, 17(3), 401–411.
Mock, J., Huber, S., Klein, E., & Moeller, K. (2016). Insights into numerical cognition: Considering eye fixations in number processing and arithmetic. Psychological Research Psychologische Forschung, 80, 334–359. https://doi.org/10.1007/s00426-015-0739-9
Moseley, B. J., Okamoto, Y., & Ishida, J. (2007). Comparing U.S. and Japanese elementary school teachers’ facility for linking rational number representations. International Journal of Science and Mathematics Education, 5, 165–185. https://doi.org/10.1007/s10763-006-9040-0
Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52. https://doi.org/10.1207/s15326985ep4001_3
Obersteiner, A., & Staudinger, I. (2018). How the eyes add fractions: Adult eye movement patterns during fraction addition problems. Journal of Numerical Cognition, 4(2), 317–336. https://doi.org/10.5964/jnc.v4i2.130
Reinert, R. M., Huber, S., Nuerk, H. C., & Moeller, K. (2015). Strategies in unbounded number line estimation? Evidence from eye-tracking. Cognitive Processing, 16, 359–363. https://doi.org/10.1007/s10339-015-0675-z
Schindler, M., & Lilienthal, A. J. (2019). Domain-specific interpretation of eye tracking data: Towards a refined use of the eye-mind hypothesis for the field of geometry. Educational Studies in Mathematics, 101, 123–139. https://doi.org/10.1007/s10649-019-9878-z
Schneider, M., Heine, A., Thaler, V., Torbeyns, J., De Smedt, B., Verschaffel, L., Jacobs, A. M., & Stern, E. (2008). A validation of eye movements as a measure of elementary school children’s developing number sense. Cognitive Development, 23(3), 424–437. https://doi.org/10.1016/j.cogdev.2008.07.002
Schneider, M., Merz, S., Stricker, J., De Smedt, B., Torbeyns, J., Verschaffel, L., & Luwel, K. (2018). Associations of number line estimation with mathematical competence: A meta-analysis. Child Development, 89(5), 1467–1484. https://doi.org/10.1111/cdev.13068
Schneider, M., Thompson, C. A., & Rittle-Johnson, B. (2017). Associations of magnitude comparison and number line estimation with mathematical competence: A comparative review. Cognitive development from a strategy perspective, 100–119. https://doi.org/10.4324/9781315200446-7
Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14(3), 237–243. https://doi.org/10.1111/1467-9280.02438
Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. https://doi.org/10.1016/j.cogpsych.2011.03.001
Strohmaier, A. R., MacKay, K. J., Obersteiner, A., & Reiss, K. M. (2020). Eye-tracking methodology in mathematics education research: A systematic literature review. Educational Studies in Mathematics, 104(2), 147–200. https://doi.org/10.1007/s10649-020-09948-1
Sullivan, J. L., Juhasz, B. J., Slattery, T. J., & Barth, H. C. (2011). Adults’ number-line estimation strategies: Evidence from eye movements. Psychonomic Bulletin & Review, 18(3), 557–563. https://doi.org/10.3758/s13423-011-0081-1
Thomaneck, A., Vollstedt, M., & Schindler, M. (2022). What can eye movements tell about students’ interpretations of contextual graphs? A methodological study on the use of the eye-mind hypothesis in the domain of functions. Frontiers in Education, 7. https://doi.org/10.3389/feduc.2022.1003740
Tian, J., & Siegler, R. S. (2017). Which type of rational numbers should students learn first? Educational Psychology Review, 1–22. https://doi.org/10.1007/s10648-017-9417-3
Vamvakoussi, X., Christou, K. P., Mertens, L., & Van Dooren, W. (2011). What fills the gap between discrete and dense? Greek and Flemish students’ understanding of density. Learning and Instruction, 21(5), 676–685. https://doi.org/10.1016/j.learninstruc.2011.03.005
Vamvakoussi, X., Christou, K. P., & Vosniadou, S. (2018). Bridging psychological and educational research on rational number knowledge. Journal of Numerical Cognition, 4(1), 84–106. https://doi.org/10.5964/jnc.v4i1.82
Van Dooren, W., Lehtinen, E., & Verschaffel, L. (2015). Unraveling the gap between natural and rational numbers. Learning and Instruction, 37, 1–4. https://doi.org/10.1016/j.learninstruc.2015.01.001
Van Hoof, J., Lijnen, T., Verschaffel, L., & Van Dooren, W. (2013). Are secondary school students still hampered by the natural number bias? A reaction time study on fraction comparison tasks. Research in Mathematics Education, 15(2), 154–164. https://doi.org/10.1080/14794802.2013.797747
Van Hoof, J., Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2017). The transition from natural to rational number knowledge. In D. Geary, D. B. Berch, R. Ochsendorf, & K. Mann Koepke (Eds.), Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts (pp. 101–123). Elsevier. https://doi.org/10.1016/B978-0-12-805086-6.00005-9
van’t Noordende, J. E., van Hoogmoed, A. H., Schot, W. D., & Kroesbergen, E. H. (2016). Number line estimation strategies in children with mathematical learning difficulties measured by eye tracking. Psychological Research, 80(3), 368–378. https://doi.org/10.1007/s00426-015-0736-z
Wang, Y. Q., & Siegler, R. S. (2013). Representations of and translation between common fractions and decimal fractions. Chinese Science Bulletin, 58(36), 4630–4640. https://doi.org/10.1007/s11434-013-6035-4
Acknowledgements
The authors would like to thank all teachers, parents, and principals for the help in conducting the current study as well as the students who participated in it.
Funding
Partial financial support was received from the Internal Funds of KU Leuven (grant number: KA/16/009).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Ethics approval
The current study was approved according to the Social and Societal Ethics Committee of KU Leuven under approval number G-2018 11 1422.
Conflict of interest
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
MacKay, K.J., Germeys, F., Van Dooren, W. et al. Numerical magnitude understanding of natural and rational numbers in secondary-school students: a number line estimation study. Educ Stud Math (2024). https://doi.org/10.1007/s10649-023-10291-4
Accepted:
Published:
DOI: https://doi.org/10.1007/s10649-023-10291-4