References
- 1Alibali, M. W., Crooks, N. M., & McNeil, N. M. (2018). Perceptual support promotes strategy generation: Evidence from equation solving. British Journal of Developmental Psychology, 36(2), 153–168. 10.1111/bjdp.12203
- 2Ashcraft, M. H., & Kirk, E. P. (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology: General, 130(2),
224 . 10.1037/0096-3445.130.2.224 - 3Barbieri, C. A., Miller-Cotto, D., Clerjuste, S. N., & Chawla, K. (2023). A meta-analysis of the worked examples effect on mathematics performance. Educational Psychology Review, 35(1),
11 . 10.1007/s10648-023-09745-1 - 4Bjork, R. A. (1994).
Memory and metamemory considerations in the . Metacognition: Knowing about knowing (pp-185–205). Cambridge, MA: MIT Press. 10.7551/mitpress/4561.003.0011 - 5Bol, L., Hacker, D. J., Walck, C. C., & Nunnery, J. A. (2012). The effects of individual or group guidelines on the calibration accuracy and achievement of high school biology students. Contemporary Educational Psychology, 37, 280–287. 10.1016/j.cedpsych.2012.02.004
- 6Botelho, A., Chan, J. Y. C., Trac, C., Closser, A. H., Smith, H., Drzewiecki, K. C., & Ottmar, E. (July, 2021). State vs. trait: Examining gaming the system in the context of math perception task. Poster presented at the Annual Meeting of the Cognitive Science Society.
- 7Braithwaite, D. W., Goldstone, R. L., van der Maas, H. L. J., & Landy, D. H. (2016). Non-formal mechanisms in mathematical cognitive development: The case of arithmetic. Cognition, 149, 40–55. 10.1016/j.cognition.2016.01.004
- 8Bye, J. K., Chan, J. Y. C., Closser, A. H., Lee, J. E., Shaw, S. T., & Ottmar, E. R. (2024). Perceiving precedence: Order of operations errors are predicted by perception of equivalent expressions. Journal of Numerical Cognition, 10, 1–23. 10.5964/jnc.14103
- 9Carpenter, S. K., Cepeda, N. J., Rohrer, D., Kang, S. H., & Pashler, H. (2012). Using spacing to enhance diverse forms of learning: Review of recent research and implications for instruction. Educational Psychology Review, 24(3), 369–378. 10.1007/s10648-012-9205-z
- 10Chan, J. Y.-C., Lee, J.-E., Mason, C. A., Sawrey, K., & Ottmar, E. (2022). From Here to There! A dynamic algebraic notation system improves understanding of equivalence in middle-school students. Journal of Educational Psychology, 114(1), 56–71. 10.1037/edu0000596
- 11Closser, A. H., Botelho, A. F., & Chan, J. Y.-C. (2024). Exploring the impact of symbol spacing and problem sequencing on arithmetic performance: An educational data mining approach. Journal of Educational Data Mining, 16(1), 84–111. 10.5281/zenodo.11403249
- 12Closser, A. H., Bye, J. K., Zhang, P., Lee, J., Egorova, A., Colbert, M. A., Yu, S., Hornburg, C. B., & Ottmar, E. (under review). The unique and combined effects of perceptual cues on middle schoolers’ mathematical performance, learning, and retention.
- 13Closser, A. H., Chan, J. Y.-C., Smith, H., & Ottmar, E. R. (2022). Perceptual learning in math: Implications for educational research, practice, and technology. Rapid Community Report Series. Digital Promise and the International Society of the Learning Sciences.
https://repository.isls.org//handle/1/7668 - 14De Leeuw, J. R., Gilbert, R. A., & Luchterhandt, B. (2023). jsPsych: Enabling an open-source collaborative ecosystem of behavioral experiments. Journal of Open Source Software, 8(85),
5351 . 10.21105/joss.05351 - 15Ganley, C. M., & McGraw, A. L. (2016). The development and validation of a revised version of the math anxiety scale for young children. Frontiers in Psychology, 7,
1181 . 10.3389/fpsyg.2016.01181 - 16Gibson, E. J. (1969). Principles of perceptual learning and development. Englewood Cliffs, NJ: Prentice Hall.
- 17Goldstone, R. L., Marghetis, T., Weitnauer, E., Ottmar, E. R., & Landy, D. (2017). Adapting perception, action, and technology for mathematical reasoning. Current Directions in Psychological Science, 26(5), 434–441. 10.1177/0963721417704888
- 18Harrison, A., Smith, H., Hulse, T., & Ottmar, E. R. (2020). Spacing out! manipulating spatial features in mathematical expressions affects performance. Journal of Numerical Cognition, 6(2), 186–203. 10.5964/jnc.v6i2.243
- 19Hawes, Z., & Ansari, D. (2020). What explains the relationship between spatial and mathematical skills? A review of evidence from brain and behavior. Psychonomic Bulletin & Review, 27, 465–482. 10.3758/s13423-019-01694-7
- 20Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21(1), 33–46. 10.5951/jresematheduc.21.1.0033
- 21Hornburg, C. B., Lee, J.-E., Closser, A. H., Bye, J. K., Egorova, A., Colbert, M. A., Yu, S., Zhang, P., Valdivia, I., & Ottmar, E. (2025). Effects of spacing and color perceptual cues on middle schoolers’ order-of-operations performance. Journal of Experimental Education, 1–22. 10.1080/00220973.2025.2528691
- 22Hornburg, C. B., Zhang, P., Closser, A. H., Bye, J. K., Lee, J.-E., Egorova, A., Colbert, M. A., Yu, S., & Ottmar, E. (2025, June). Understanding the effects of perceptual cues on middle schoolers’ mathematical performance, learning, and retention. A. H. Closser (Chair), Leveraging cognitive principles in low-cost interventions to improve mathematics problem solving and learning. Paper to be presented at the 8th Annual Meeting of the Mathematical Cognition and Learning Society (MCLS), Hong Kong, China.
- 23Jiang, M. J., Cooper, J. L., & Alibali, M. W. (2014). Spatial factors influence arithmetic performance: The case of the minus sign. Quarterly Journal of Experimental Psychology, 67(8), 1626–1642. 10.1080/17470218.2014.898669
- 24Kellman, P. J., Massey, C. M., & Son, J. Y. (2010). Perceptual learning modules in mathematics: Enhancing students’ pattern recognition, structure extraction, and fluency. Topics in Cognitive Science, 2(2), 285–305. 10.1111/j.1756-8765.2009.01053.x
- 25Kirshner, D., & Awtry, T. (2004). Visual salience of algebraic transformations. Journal for Research in Mathematics Education, 35(4), 224–257. 10.2307/30034809
- 26Labuhn, A. S., Zimmerman, B. J., & Hasselhorn, M. (2010). Enhancing students’ self-regulation and mathematics performance: The influence of feedback and self-evaluative standards. Metacognition and learning, 5, 173–194.10.1007/s11409-010-9056-2
- 27Landy, D., & Goldstone, R. L. (2007). Formal notations are diagrams: Evidence from a production task. Memory and Cognition, 35(8), 2033–2040. 10.3758/BF03192935
- 28Landy, D., & Goldstone, R. L. (2010). Proximity and precedence in arithmetic. Quarterly Journal of Experimental Psychology, 63(10), 1–16. 10.1080/17470211003787619
- 29Landy, D. H., Jones, E. L., & Hummel, J. E. (2008). Why spatial-numeric associations aren’t evidence for a mental number line. In Proceedings of the Annual Meeting of the Cognitive Science Society (Vol. 30, No. 30).
https://escholarship.org/uc/item/56f5w2k1 - 30Marghetis, T., Landy, D., & Goldstone, R. L. (2016). Mastering algebra retrains the visual system to perceive hierarchical structure in equations. Cognitive Research: Principles and Implications, 2(25). 10.1186/s41235-016-0020-9
- 31Mielicki, M. K., & Wiley, J. (2022). Exploring the necessary conditions for observing interleaved practice benefits in math learning. Learning and Instruction, 80,
101583 . 10.1016/j.learninstruc.2022.101583 - 32Ngo, V., Lacera, L. P., Closser, A. H., & Ottmar, E. (2023). The effects of operator position and superfluous brackets on student performance in simple arithmetic. Journal of Numerical Cognition, 9(1), 107–128. 10.5964/jnc.9535
- 33Norton, S. J., & Cooper, T. J. (2001). Students’ perceptions of the importance of closure in arithmetic: implications for algebra. In Proceedings of the International Conference of the Mathematics Education into the 21st Century Project (pp. 198–202).
https://academic.sun.ac.za/mathed/174/ANortonCooper.pdf - 34Roediger,
III , H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3), 249–255. 10.1111/j.1467-9280.2006.01693.x - 35Rohrer, D., Dedrick, R. F., Hartwig, M. K., & Cheung, C. N. (2020). A randomized controlled trial of interleaved mathematics practice. Journal of Educational Psychology, 112(1),
40 . 10.1037/edu0000367 - 36Wigfield, A., & Eccles, J. S. (2000). Expectancy–value theory of achievement motivation. Contemporary Educational Psychology, 25(1), 68–8. 10.1006/ceps.1999.1015
- 37Wilkinson, M. D., Dumontier, M., Aalbersberg, I. J., Appleton, G., Axton, M., Baak, A., … & Mons, B. (2016).
The FAIR Guiding Principles for scientific data management and stewardship . Scientific Data, 3(1), 1–9.https://www.nature.com/articles/sdata201618 . 10.1038/sdata.2016.18