The language involved in de-contextualized integer comparisons poses challenges, as students may interpret “most” based on absolute values rather than on order. Using the context of temperature, we explored how students’ integer value comparisons differed based on question phrasing (which temperature is hottest, most hot, least hot, coldest, most cold, least cold) and on numbers presented (positive, negative, mixed). Participants included 88 second graders and 70 fourth graders from a rural school district in the Midwestern USA, and each student solved 36 integer comparisons. For comparisons with positive number choices, students had more difficulty with “coldest” than “hottest”; however, the results were reversed for comparisons with only negative number choices. When working with mixed comparisons, students often chose the least of the cold as opposed to the least cold, suggesting that they saw hot and cold as categorical opposites rather than opposites on a continuum, with zero as a boundary.
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Adetula, L. O. (1990). Language factor: Does it affect children’s performance on word problems? Educational Studies in Mathematics, 21(4), 351–365. https://doi.org/10.1007/BF00304263
Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373–397.
Barner, D., & Snedeker, J. (2008). Compositionality and statistics in adjective acquisition: 4-year-olds interpret tall and short based on the size distributions of novel noun referents. Child Development, 79(3), 594–608. https://doi.org/10.1111/j.1467-8624.2008.01145.x
Bell, A. (1984). Short and long term learning—Experiments in diagnostic teaching design. In B. Southwell (Ed.), Proceedings of the Eighth International Conference for the Psychology of Mathematics Education (pp. 55–62). Sydney, Australia: International Group for the Psychology of Mathematics Education.
Bofferding, L. (2014). Negative integer understanding: Characterizing first graders’ mental models. Journal for Research in Mathematics Education, 45(2), 194–245. https://doi.org/10.5951/jresematheduc.45.2.0194.
Bofferding, L. & Hoffman, A. (2015). Comparing negative integers: Issues of language. In K. Beswick, T. Muir, & J. Wells (Eds.), Proceedings of the 39th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 150). Hobart, Australia: PME.
Case, R. (1996). Introduction: Reconceptualizing the nature of children’s conceptual structures and their development in middle childhood. Monographs of the Society for Research in Child Development, 61(1–2), 1–26.
Cheshire, J. (1998). Double negatives are illogical. In L. Bauer & P. Trudgill (Eds.), Language myths (pp. 113–122). New York, NY: Penguin Putnam, Inc..
Clark, E. V. (1971). On the acquisition of the meaning of before and after. Journal of Verbal Learning and Verbal Behavior, 10(3), 266–275. https://doi.org/10.1016/S0022-5371(71)80054-3
Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach (2nd ed.). New York, NY: Routledge.
Donaldson, M., & Balfour, G. (1968). Less is more: A study of language comprehension in children. British Journal of Psychology, 59(4), 461–471. https://doi.org/10.1111/j.2044-8295.1968.tb01163.x
Dougherty, B. J. (2010). Developing essential understanding of number and numeration for teaching mathematics in prekindergarten–grade 2. Reston, VA: National Council of Teachers of Mathematics.
Fuson, K. C., Carroll, W. M., & Landis, J. (1996). Level in conceptualizing and solving addition and subtraction compare problems. Cognition and Instruction, 14(3), 345–371. https://doi.org/10.1207/s1532690xci1403_3
Gobbo, C., & Agnoli, F. (1985). Comprehension of two types of negative comparisons in children. Journal of Psycholinguistic Research, 14(3), 301–316. https://doi.org/10.1007/BF01068088
Griffiths, J. A., Shantz, C. A., & Sigel, I. E. (1967). A methodological problem in conservation studies: The use of relational terms. Child Development, 38(3), 841–848. https://doi.org/10.2307/1127261
Klatzky, R. L., Clark, E. V., & Macken, M. (1973). Asymmetries in the acquisition of polar adjectives: Linguistic or conceptual? Journal of Experimental Child Psychology, 16(1), 32–46. https://doi.org/10.1016/0022-0965(73)90060-X
Lean, G. A., Clements, M. A., & Del Campo, G. (1990). Linguistic and pedagogical factors affecting children’s understanding of word problems: A comparative study. Educational Studies in Mathematics, 21(2), 165–191. https://doi.org/10.1007/BF00304900
Murray, J. C. (1985). Children’s informal conceptions of integer arithmetic. In L. Streefland (Ed.), Proceedings of the Ninth Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 147–153). Noordwijkerhout, The Netherlands: International Group for the Psychology of Mathematics Education.
Murray, P. L., & Mayer, R. E. (1988). Preschool children’s judgments of number magnitude. Journal of Educational Psychology, 80(2), 206–209. https://doi.org/10.1037/0022-06220.127.116.11
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Retrieved from http://www.nctm.org/flipbooks/standards/pssm/index.html.
National Governors Association Center for Best Practices & Council of Chief State School Officers (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/Math/Content/K/introduction
National Research Council (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Committee on Early Childhood Mathematics. In C.T. Cross, T.A. Woods & H. Schweingruber (Eds.), Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press.
Peled, I., Mukhoadhyay, S., & Resnick, L. (1989). Formal and informal sources of mental models for negative numbers. In G. Vergnaud, J. Rogalski, & M. Artique (Eds.), The international group for the psychology of mathematics education (Vol. 3, pp. 106–110). Paris: International Group for the Psychology of Mathematics Education.
Pratt, D., & Simpson, A. (2004). McDonald’s vs Father Christmas. Australian Primary Mathematics Classroom, 9(3), 4–9.
Ryalls, B. O. (2000). Dimensional adjectives: Factors affecting children’s ability to compare objects using novel words. Journal of Experimental Child Psychology, 76(1), 26–49. https://doi.org/10.1006/jecp.1999.2537
Schwartz, B. B., Kohn, A. S., & Resnick, L. B. (1993-1994). Positives about negatives: A case study of an intermediate model for signed numbers. The Journal of the Learning Sciences, 3(1), 37–92.
Shire, B., & Durkin, K. (1989). Junior school children’s responses to conflict between the spatial and numerical meanings of ‘up’ and ‘down’. Educational Psychology, 9(2), 141–147. https://doi.org/10.1080/0144341890090206
Siegler, R. S., & Robinson, M. (1982). The development of numerical understandings. In H. W. Reese & L. P. Lisitt (Eds.), Advances in child development and behavior (Vol. 16, pp. 242–312). New York, NY: Academic Press.
Smith, L. B., Rattermann, M. J., & Sera, M. (1988). “Higher” and “lower”: Comparative and categorical interpretations by children. Cognitive Development, 3(4), 341–357.
Sophian, C. (1987). Early developments in children’s use of counting to solve quantitative problems. Cognition and Instruction, 4(2), 61–90.
Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428–464. https://doi.org/10.5951/jresematheduc.43.4.0428
Swanson, P. E. (2010). The intersection of language and mathematics. Mathematics Teaching in the Middle School, 15(9), 516–523.
Vamvakoussi, X., & Vosnaidou, S. (2012). Bridging the gap between the dense and the discrete: The number line and the “rubber line” bridging analogy. Mathematical Thinking and Learning, 14(4), 265–284. https://doi.org/10.1080/10986065.2012.717378
Varma, S., & Schwartz, D. L. (2011). The mental representation of integers: An abstract-to-concrete shift in the understanding of mathematical concepts. Cognition, 121, 363–385. https://doi.org/10.1016/j.cognition.2011.08.005
Verschaffel, L. (1994). Using retelling data to study elementary school children’s representations and solutions of compare problems. Journal for Research in Mathematics Education, 25(2), 141–165. https://doi.org/10.2307/749506
Vosnaidou, S., & Brewer, W. F. (1992). Mental models of the earth: A study of conceptual change in childhood. Cognitive Psychology, 24(4), 535–585. https://doi.org/10.1016/0010-0285(92)90018-W
Vosniadou, S., Vamvakoussi, X., & Skopeliti, I. (2008). The framework theory approach to the problem of conceptual change. In S. Vosniadou (Ed.), International handbook of research on conceptual change (pp. 3–34). New York, NY: Routledge.
Wessman-Enzinger, N. M., & Mooney, E. S. (2014). Making sense of integers through storytelling. Mathematics Teaching in the Middle School, 20(4), 202–205. https://doi.org/10.5951/mathteacmiddscho.20.4.0202
Whitacre, I., Azuz, B., Lamb, L. L. C., Bishop, J., Schappelle, B. P., & Philipp, R. A. (2017). Integer comparisons across the grades: Students’ justifications and ways of reasoning. The Journal of Mathematical Behavior, 45, 47–62. https://doi.org/10.1016/j.jmathb.2016.11.001
Whitacre, I., Bishop, J. P., Philipp, R. A., Lamb, L. L., & Schappelle, B. P. (2015). Dollars & sense: Students’ integer perspectives. Mathematics Teaching in the Middle School, 20(2), 84–89. https://doi.org/10.5951/mathteacmiddscho.20.2.0084
Widjaja, W., Stacey, K., & Steinle, V. (2011). Locating negative decimals on the number line: Insights into the thinking of pre-service primary teachers. The Journal of Mathematical Behavior, 30, 80–91. https://doi.org/10.1016/j.jmathb.2010.11.004
This research was supported by NSF CAREER award DRL-1350281. The authors would like to thank the schools, teachers, and students involved in the research for their participation and support. The authors give thanks especially to Mahtob Aqazade for her review of the paper.
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Bofferding, L., Farmer, S. Most and Least: Differences in Integer Comparisons Based on Temperature Comparison Language. Int J of Sci and Math Educ 17, 545–563 (2019). https://doi.org/10.1007/s10763-018-9880-4
- Number comparisons