Abstract
Elementary students demonstrate conflicts when measuring with rulers as well as measuring with discrete units. Their conflicts often lie in their focus on counting tick marks or focusing on the endpoint of objects on rulers or ignoring spaces with discrete units. In this study, we also investigated another area of conflict: students’ conceptions on where measurements should start on a ruler. Across two time points, 32 first graders and 37 third graders responded to a series of measurement tasks in the form of incorrect worked examples meant to expose students to these common conflicts of where the measurement starts and how to determine the overall length. Although students often said the objects should start at one, they were more likely to indicate that the object should be aligned with zero the closer the object was positioned to zero. However, many students said starting at the edge of the ruler (before zero) was okay, effectively equating the edge with either zero or one. Interestingly, on two related worked examples, their continuous and discrete measurements were not associated with each other. The incorrect worked examples helped illustrate important areas for further instruction, especially around their conceptions of rulers’ edges and the impact of spaces with discrete units.
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Notes
We needed to work with new first and third graders in the second year of our larger study, so we could not include second graders (who would be third graders the second year).
On Item 4-to-7, some students did not think the object was already at four, so they did not think it was “okay as is” and said it needed to be moved to four.
We used 3-letter animal names as pseudonyms to mask the participants’ identities from each classroom. Numbers refer to different children within each classroom.
References
Adams, D., McLaren, B. M., Durkin, K., Mayer, R. E., Rittle-Johnson, B., Isotani, S., & Van Velsen, M. (2012). Erroneous examples versus problem solving: Can we improve how middle school students learn decimals? In N. Miyakem, D. Peebles, & R. P. Coppers (Eds.), Proceedings of the Annual Meeting of the Cognitive Science Society (Vol. 34, pp. 1260–1265). https://escholarship.org/uc/item/1jn1x6g2
Antić, M. D., & Ðokić, O. J. (2019). The development of the components of the length measurement concept in the procedure of measurement using a ruler. Korean Mathematical Education Society Journal Series D: Mathematical Education Research, 22(4), 261–282. https://doi.org/10.7468/JKSMED.2019.22.4.261
Atkinson, R., Derry, S., Renkl, A., & Wortham, D. (2000). Learning from examples: Instructional principles from the worked examples research. Review of Educational Research, 70(2), 181–214. https://doi.org/10.3102/00346543070002181
Aqazade, M., Bofferding, L., & Farmer, S. (2016). Benefits of analyzing contrasting integer problems: The case of four second graders. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Proceedings of the 38th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 132-139). The University of Arizona.
Bofferding, L., Aqazade, M., & Farmer, S. (2017). Second graders’ integer addition understanding: Leveraging contrasting cases. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 243–250). Hoosier Association of Mathematics Teacher Educators.
Bofferding, L., & Hoffman, A. (2019). Children’s integer understanding and the effects of linear board games: A look at two measures. The Journal of Mathematical Behavior, 56, 1–20. https://doi.org/10.1016/j.jmathb.2019.100721
Barrett, J. E., Cullen, C., Sarama, J., Clements, D. H., Klanderman, D., Miller, A. L., & Rumsey, C. (2011). Children’s unit concepts in measurement: A teaching experiment spanning grades 2 through 5. ZDM, 43, 637–650. https://doi.org/10.1007/s11858-011-0368-8
Booth, J., Lange, K., Koedinger, K., & Newton, K. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34. https://doi.org/10.1016/j.learninstruc.2012.11.002
Bragg, P., & Outhred, L. (2001). So that’s what a centimeter looks like: Students’ understandings of linear units. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 209–216). Freudenthal Institute, Faculty of Mathematics and Computer Science, Utrecht University.
Bragg, P., & Outhred, L. (2000). What is taught versus what is learnt: The case of linear measurement. Paper presented at twenty-third annual conference of the Mathematics Education Research Group of Australasia, 2, 112–118.
Chan, C., Burtis, J., & Bereiter, C. (1997). Knowledge building as a mediator of conflict in conceptual change. Cognition and Instruction, 15, 1–40. https://doi.org/10.1207/s1532690xci1501_1
Chinn, C. A., & Brewer, W. F. (1993). The role of anomalous data in knowledge acquisition: A theoretical framework and implications for science instruction. Review of Educational Research, 63(1), 1–49. https://doi.org/10.3102/00346543063001001
Clements, D. H. (1999). Teaching length measurement: Research challenges. School Science and Mathematics, 99(1), 5–11. https://doi.org/10.1111/j.1949-8594.1999.tb17440.x
Clements, D. H., & Stephan, M. (2004). Measurement in pre-K to grade 2 mathematics. In D. H. Clements, J. Sarama, & A. M. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 299–317). Erlbaum.
Clements, D., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach (Studies in Mathematical Thinking and Learning Series). Routledge.
Congdon, E. L., Kwon, M. K., & Levine, S. C. (2018). Learning to measure through action and gesture: Children’s prior knowledge matters. Cognition, 180, 182–190. https://doi.org/10.1016/j.cognition.2018.07.002
Cullen, C., & Barrett, J. E. (2010). Strategy use indicative of an understanding of units of length. Paper presented at the 34th Annual Conference of the International Group for the Psychology of Mathematics in Education, 2, 281–288.
Dietiker, L. C., Gonulates, F., & Smith, J. P., III. (2011). Understanding linear measure. Teaching Children Mathematics, 18(4), 252–259. https://doi.org/10.5951/teacchilmath.18.4.0252
diSessa, A. (2014). A history of conceptual change research: Threads and fault lines. In R. Sawyer (Ed.), The Cambridge handbook of the learning sciences (second edition) (pp. 88–108). Cambridge University Press. https://doi.org/10.1017/CBO9781139519526.007
Drake, M. (2014). Learning to measure length: The problem with the school ruler. Australian Primary Mathematics Classroom, 19(3), 27–32.
Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206–214. https://doi.org/10.1016/j.learninstruc.2011.11.001
Gómezescobar, A., Guerrero, S., & Fernández-Cézar, R. (2020). How long is it? Difficulties with conventional ruler use in children aged 5 to 8. Early Childhood Education Journal, 48, 693–701. https://doi.org/10.1007/s10643-020-01030-y
Hatano, G., & Ito, Y. (1965). Development of length measuring behavior. Japanese Journal of Psychology, 36, 184–196. https://doi.org/10.4992/jjpsy.36.184
Hiebert, J. (1981). Cognitive development and learning linear measurement. Journal for Research in Mathematics Education, 12(3), 197–211.
Johnson, D. W., & Johnson, R. T. (2009). Energizing learning: The instructional power of conflict. Educational Researcher, 38(1), 37–51. https://doi.org/10.3102/0013189X08330540
Kamii, C. (1995, October). Why is the use of a ruler so hard? In D. T. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the 17th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 222–227). The Ohio State University.
Lehrer, R., Jenkins, M., & Osana, H. (1998). Longitudinal study of children’s conceptions of geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for develo** understanding of geometry and space (pp. 137–167). Lawrence Erlbaum Associates Inc.
Lehrer, R. (2003). Develo** understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 179–192). The National Council of Teachers of Mathematics.
Deitz, K., Huttenlocher, J., Kwon, M., Levine, S. C., & Ratliff, K. (2009). Children’s understanding of ruler measurement and units of measure: A training study. In N. A. Taatgen & H. van Rijn (Eds.), Proceedings of the 31st Annual Conference of the Cognitive Science Society (pp. 2391–2395). Cognitive Science Society. https://escholarship.org/uc/item/5cz2r7vj
Limón, M. (2001). On the cognitive conflict as an instructional strategy for conceptual change: A critical appraisal. Learning and Instruction, 11(4–5), 357–380. https://doi.org/10.1016/S0959-4752(00)00037-2
MacDonald, A., & Lowrie, T. (2011). Develo** measurement concepts within context: Children’s representations of length. Mathematics Education Research Journal, 23(1), 27–42. https://doi.org/10.1007/s13394-011-0002-7
Manson, E., & Ayres, P. (2021). Investigating how errors should be flagged and worked examples structured when providing feedback to novice learners of mathematics. Educational Psychology, 41(2), 153–171. https://doi.org/10.1080/01443410.2019.1650895
McDonough, A., & Sullivan, P. (2011). Learning to measure length in the first three years of school. Australasian Journal of Early Childhood, 36(3), 27–35. https://doi.org/10.1177/183693911103600305
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Retrieved December 2021 from http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf
Nührenbörger, M. (2001). Insights into children’s ruler concepts—Grade-2-students’ conceptions and knowledge of length measurement and paths of development. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 447–454). Freudenthal Institute.
Piaget, J. (1960). The child’s conception of geometry. Basic Books.
Piaget, J. (1985). The equilibration of cognitive structures: The central problem of intellectual development. University of Chicago Press.
Rittle-Johnson, B., & Star, J. R. (2011). The power of comparison in learning and instruction: Learning outcomes supported by different types of comparisons. Psychology of Learning and Motivation: Cognition in Education, 55, 199–225. https://doi.org/10.1016/B978-0-12-387691-1.00007-7
Sarama, J., Clements, D. H., Barrett, J. E., Cullen, C. J., Hudyma, A., & Vanegas, Y. (2022). Length measurement in the early years: Teaching and learning with learning trajectories. Mathematical Thinking and Learning, 24(4), 267–290. https://doi.org/10.1080/10986065.2020.1858245
Schwartz, D. L., Sears, D., & Chang, J. (2007). Reconsidering prior knowledge. In M. C. Lovett, & P. Shah (Eds.). Thinking with data (pp. 319–344). Lawrence Erlbaum Associates Publishers.
Sisman, G. T., & Aksu, M. (2016). A study on sixth grade students’ misconceptions and errors in spatial measurement: Length, area, and volume. International Journal of Science and Mathematics Education, 14, 1293–1319. https://doi.org/10.1007/s10763-015-9642-5
Solomon, T. L., Vasilyeva, M., Huttenlocher, J., & Levine, S. C. (2015). Minding the gap: Children’s difficulty conceptualizing spatial intervals as linear measurement units. Developmental Psychology, 51(11), 1564–1573. https://doi.org/10.1037/a0039707
Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 102(4), 408–426. https://doi.org/10.1016/j.jecp.2008.11.004
Stephan, M., & Clements, D. H. (2003). Linear and area measurement in prekindergarten to grade 2. In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement (pp. 3–16). National Council of Teachers of Mathematics.
Szilágyi, J., Clements, D. H., & Sarama, J. (2013). Young children’s understandings of length measurement: Evaluating a learning trajectory. Journal for Research in Mathematics Education, 44(3), 581–620. https://doi.org/10.5951/jresematheduc.44.3.0581
Vosniadou, S. (1996). Towards a revised cognitive psychology for new advances in learning and instruction. Learning and Instruction, 6(2), 95–109. https://doi.org/10.1016/0959-4752(96)00008-4
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This work was supported by the National Science Foundation under Grant #1759254.
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A portion of these results were presented as a poster at a conference for the North American Chapter of the Psychology of Mathematics Education.
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Bofferding, L., Haiduc, AM., Aqazade, M. et al. Using Incorrect Worked Examples to Investigate the Consistency of First and Third Graders’ Measurement Conceptions. Int J of Sci and Math Educ 21, 1913–1934 (2023). https://doi.org/10.1007/s10763-022-10334-x
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DOI: https://doi.org/10.1007/s10763-022-10334-x