The problematic equal sign
DOI:
https://doi.org/10.7146/nomad.v28i1-2.149263Abstract
In this study, we investigated how 2,544 Norwegian students in the fifth, sixth, eighth and ninth grades of different schools answered a missing addend task that required a relational understanding of the equal sign. Only 50 % of the students correctly solved the task (30 % in grade 5 and 80 % in grade 9). We then selected the students who managed to correctly solve two tasks that did not have an explicit equal sign but required an assessment of equality. The percentage of correct answers in the missing addend task increased to 71 %, but even among those students who successfully handled the concept of equality, a substantial portion did not solve the equal sign task correctly. Our results indicate that equality and the equal sign cannot be treated as equivalent concepts.
References
Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M. & Stephens, A. C. (2007). A longitudinal examination of middle school students' understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9 (3), 221-247. https://doi.org/10.1080/10986060701360902
Barlow, A. T. & Harmon, S. E. (2012). Problem contexts for thinking about equality: an additional resource. Childhood Education, 88 (2), 96-101. https://doi.org/10.1080/00094056.2012.662121
Bergem, O. K. (2016). Hovedresultater i matematikk. In O. K. Bergem, H. Kaarstein & T. Nilsen (Eds.), Vi kan lykkes i realfag - resultater og analyser fra TIMSS 2015 (pp. 22-43). Universitetsforlaget. https://doi.org/10.18261/97882150279999-2016-03
Falkner, K. P., Levi, L. & Carpenter, T. P. (1999). Children's understanding of equality: a foundation for algebra. Teaching Children Mathematics, 6 (4), 232-236. https://doi.org/10.5951/TCM.6.4.0232
Grønmo, L. S., Onstad, T., Nilsen, T., Hole, A., Aslaksen, H. & Borge, I. C. (2012). Framgang, men langt fram: norske elevers prestasjoner i matematikk og naturfag i TIMSS 2011. Akademika forlag.
Haug, P. (Ed.). (2017). Spesialundervisning: innhald og funksjon. Samlaget.
Izard, V., Pica, P., Spelke, E. & Dehaene, S. (2008). Exact equality and successor function: two key concepts on the path towards understanding exact numbers. Philosophical Psychology, 21 (4), 491-491. https://doi.org/10.1080/09515080802285354
Izard, V., Streri, A. & Spelke, E. S. (2014). Toward exact number: young children use one-to-one correspondence to measure set identity but not numerical equality. Cognitive Psychology, 72, 27-53. https://doi.org/10.1016/j.cogpsych.2014.01.004
Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12 (3), 317-326. https://doi.org/10.1007/BF00311062
Knuth, E. J., Alibali, M. W., Hattikudur, S., McNeil, N. M. & Stephens, A. C. (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13 (9), 514-519. https://doi.org/10.5951/MTMS.13.9.0514
Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A. & Stephens, A. C. (2005). Middle school students' understanding of core algebraic concepts: equivalence & variable. ZDM, 37 (1), 68-76. https://doi.org/10.1007/BF02655899
Knuth, E. J., Stephens, A. C., McNeil, N. M. & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37 (4), 297-312. https://doi.org/10.2307/30034852
Leavy, A., Hourigan, M. & McMahon, Á. (2013). Early understanding of equality. Teaching Children Mathematics, 20 (4), 246-252. https://doi.org/10.5951/teacchilmath.20.4.0246
Li, X., Ding, M., Capraro, M. M. & Capraro, R. M. (2008). Sources of differences in children's understanding of mathematical equality: comparative analysis of teacher guides and student texts in China and the United States. Cognition and Instruction, 26 (2), 195-217. https://doi.org/10.1080/07370000801980845
Ma, L. (1999). Knowing and teaching elementary mathematics: teachers' understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum Associates. https://doi.org/10.4324/9781410602589
Madej, L. (2022). Primary school students' knowledge of the equal sign - the Swedish case. International Journal of Science and Mathematics Education, 20 (2), 321-343. https://doi.org/10.1007/s10763-020-10144-z
Matthews, P., Rittle-Johnson, B., McEldoon, K. & Taylor, R. (2012). Measure for measure: what combining diverse measures reveals about children's understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43 (3), 220-254. https://doi.org/10.5951/jresematheduc.43.3.0316
McNeil, N. M. (2014). A change-resistance account of children's difficulties understanding mathematical equivalence. Child Development Perspectives, 8(1), 42-47. https://doi.org/10.1111/cdep.12062
McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S. & Krill, D. E. (2006). Middle-school students' understanding of the equal sign: the books they read can't help. Cognition and Instruction, 24 (3), 367-385. https://doi.org/10.1207/s1532690xci2403_3
Opsal, H. (2019). How students in 5th and 8th grade in Norway understand the equal sign. In U. Jankvist, M. van den Heuvel-Panhuizen & M. Veldhuis (Eds.), Proceedings of CERME 11 (pp. 638-645). European Society for Research in Mathematics Education.
Opsvik, F. & Skorpen, L. B. (2017). Utvikling av kartleggingsprøver i matematikk. In P. Haug (Ed.), Spesialundervisning, innhald og funksjon (pp. 256-271). Samlaget.
Prediger, S. (2010). How to develop mathematics for teaching and for understanding: the case of meanings of the equal sign. Journal of Mathematics Teacher Education, 13 (1), 73-93. https://doi.org/10.1007/s10857-009-9119-y
Schwartzkopf, R., Nührenbörger, M. & Mayer, C. (2018). Algebraic understanding of equalities in primary classes. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 195-212). Springer. https://doi.org/10.1007/978-3-319-68351-5_8
Stephens, A. C., Knuth, E. J., Blanton, M. L., Isler, I., Gardiner, A. M. & Marum, T. (2013). Equation structure and the meaning of the equal sign: the impact of task selection in eliciting elementary students' understandings. Journal of Mathematical Behavior, 32 (2), 173-182. https://doi.org/10.1016/j.jmathb.2013.02.001
Swedish national agency of education (2018). Curriculum for the compulsory school, preschool class and school-age education (revised 2018). Skolverket. https://www.skolverket.se/download/18.31c292d516e7445866a218f/1576654682907/pdf3984.pdf
Tøgersen, I. V. (2015). Tolkninger av likhetstegnet [Master theses]. Institutt for matematiske fag, Norges teknisk-naturvitenskaplige universitet.
Utdanningsdirektoratet. (2013). Curriculum for the common core subject of mathematics (MAT1-04). https://www.udir.no/kl06/MAT1-04?lplang=http://data.udir.no/kl06/eng
Utdanningsdirektoratet. (2020). Curriculum for mathematics year 1-10 (MAT01‑05). https://www.udir.no/lk20/MAT01-05?lang=eng
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