En modell som stöd för att utforska ekvationer

Authors

DOI:

https://doi.org/10.31129/LUMAT.10.1.1581

Keywords:

lärandemodell, ekvation, del-helhetsstruktur, lärandeverksamhet, tematisk analys, learning model, equation, part-whole structure, learning activity, thematic analysis

Abstract

Syftet med artikeln är att lyfta fram om, och i sådant fall på vilka sätt, en specifik strukturell modell kan utgöra stöd när elever utforskar matematiska strukturer i ekvationer. Artikeln bygger på en empirisk forskningsstudie där elever utforskade matematiska strukturer med stöd av modellen, vilken är avsedd att visualisera strukturer. Lärare och forskare arbetade i en kollaborativ och intervenerande studie i iterativa processer. Sammantaget 149 elever från grundskolans årskurser 3, 8 och 9 deltog i filmade forskningslektioner utifrån forskningsansatsen learning study. Lektionerna designades med inspiration från ramverket lärandeverksamhet och eleverna utmanades i ett teoretiskt arbete. Analysen utfördes utifrån tematisk ansats och två kvalitativt skilda kärnteman identifierades: Formulär respektive Lärandemodell. I analysen framträdde att undervisningen behöver vara tillräckligt utmanande för att eleverna ska finna modellen meningsfull. Undervisningen behöver möjliggöra för eleverna att urskilja relationer mellan alla tal i en ekvation, där relationerna kan beskrivas som en del-helhetsstruktur.

A model to support exploring equations

The aim of the article is to highlight whether, and if so in what ways, a selected model can constitute support when students explore mathematical structures in equations. The article is based on an empirical research study where students explored mathematical structures with support by the model, which is intended to visualize structures. Teachers and researchers worked in a collaborative and interventional study in iterative processes. A total of 149 students from compulsory school grades 3, 8 and 9 participated in video recorded research lessons based on the research approach learning study. The lessons were designed with inspiration from the framework of learning activity and the students were challenged in a theoretical work. The analysis was performed on the basis of a thematic approach and two qualitatively different core themes were identified: Template respectively Learning model. In the analysis, it emerged that the teaching has to be challenging enough for the students to find the model meaningful. The teaching needs to enable students to discern relationships between all numbers in an equation, where the relationships can be described as a part-whole structure.

References

Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 93(4), 373–397. https://doi.org/10.1086/461730 DOI: https://doi.org/10.1086/461730

Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. P. (2014). Using order to reason about negative numbers: the case of Violet. Educational Studies in Mathematics, 86(1), 39–59. https://doi.org/10.1007/s10649-013-9519-x DOI: https://doi.org/10.1007/s10649-013-9519-x

Blanton, M., & Kaput, J. J. (2001). Algebraifying the elementary mathematics experience part II: Transforming practice on a district-wide scale. I H. Chik, K. Stacey, J. Vincent, & J. Vincent (Red.), Proceedings of the 12th ICMI study conference. The future of the teaching and learning of algebra (s. 87–95). University of Melbourne.

Blanton, M., Stephens, A., Knuth, E., Murphy Gardiner, A., Isler, I., & Kim, J. (2015). The development of children’s algebraic thinking: the impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1) 39–87. https://www.jstor.org/stable/pdf/10.5951/jresematheduc.46.1.0039 DOI: https://doi.org/10.5951/jresematheduc.46.1.0039

Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa DOI: https://doi.org/10.1191/1478088706qp063oa

Brown, C., Carpenter, T., Kouba, V., Lindquist, M., Silver, E., & Swafford, J. (1988). Secondary school results from the fourth NAEP mathematics assessment: Algebra, geometry, mathematical methods, and attitudes. Mathematics Teacher, 81(5), 337–347, 397. https://doi.org/10.5951/MT.81.5.0337 DOI: https://doi.org/10.5951/MT.81.5.0337

Bryman, A. (2018). Samhällsvetenskapliga metoder (2:a uppl.) (B. Nilsson övers.). Liber. (Originalutgåvan publicerad 2002)

Cai, K., & Knuth, E. (Red.), (2011). Early algebraization. A global dialogue from multiple perspectives. Springer. https://doi.org/10.1007/978-3-642-17735-4 DOI: https://doi.org/10.1007/978-3-642-17735-4

Carlgren, I. (2012). The learning study as an approach for “clinical” subject matter didactic research. International Journal for Lesson and Learning Studies, 1(2), 126–139. https://doi.org/10.1108/20468251211224172 DOI: https://doi.org/10.1108/20468251211224172

Carlgren, I. Eriksson, I., & Runesson, U. (2017). Learning study. I I. Carlgren (Red.), Undervisningsutvecklande forskning – exemplet learning study (s. 17–39). Gleerups.

Carpenter, T. P., & Moser, J. M. (1982). The development of addition and subtraction problem-solving skills. I T. P. Carpenter, J. M. Moser, & T. A. Romberg (Red.), Addition and subtraction: A cognitive perspective (s. 9–24). Lawrence Erlbaum. https://doi.org/10.4324/9781003046585 DOI: https://doi.org/10.1201/9781003046585-2

Davydov, V. V. (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula. (J. Teller övers.). Soviet Studies in Mathematics Education, 2, 2–222. NCTM. (Originalutgåvan publicerad 1972)

Davydov, V. V. (2008). Problems of developmental instruction. A theoretical and experimental psychological study. Nova Science. (Originalutgåvan publicerad 1986) https://ebookcentral-proquest-com.ezp.sub.su.se/lib/sub/detail.action?docID=3021729

Eriksson, H., & Eriksson, I. (2020). Learning actions indicating algebraic thinking in multilingual classrooms. Educationals Studies in Mathematics, 106, 363–378. https://doi.org/10.1007/s10649-020-10007-y DOI: https://doi.org/10.1007/s10649-020-10007-y

Eriksson, I. (2017). Lärandeverksamhet som redskap i en Learning study. I I. Carlgren (Red.), Undervisningsutvecklande forskning - exemplet Learning study (s. 61–81). Gleerups.

Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25. https://www.jstor.org/stable/40247950

Gorbov, S. F., & Chudinova, E. V. (2000). The effect of modeling on the students’ learning (Regarding problem formulation). Psykologisk Vetenskap och Utbildning [Psychological Science and Education], 2, 96–110.

Hemmi, K., Bråting, K., & Lepik, M. (2021). Curricular approaches to algebra in Estonia, Finland and Sweden – a comparative study. Mathematical Thinking and Learning, 23(1), 49–71. https://doi.org/10.1080/10986065.2020.1740857 DOI: https://doi.org/10.1080/10986065.2020.1740857

Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326. https://doi.org/10.1007/BF00311062 DOI: https://doi.org/10.1007/BF00311062

Kieran, C. (1990). Cognitive processes involved in learning school algebra. I P. Nesher & J. Kilpatrick (Red.), ICMI study series. Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education (s. 96–112). Cambridge University Press. https://doi.org/10.1017/CBO9781139013499.007 DOI: https://doi.org/10.1017/CBO9781139013499.007

Kieran, C. (2018). Teaching and learning algebraic thinking with 5- to 12-year-olds. The global evolution of an emerging field of research and practice. Springer. https://www.springer.com/gp/book/9783319683508 DOI: https://doi.org/10.1007/978-3-319-68351-5

Kilhamn, C. (2011). Making sense of negative numbers. (Doktorsavhandling). Göteborgs universitet: Institutionen för didaktik och pedagogisk profession. https://gupea.ub.gu.se/handle/2077/24151?locale=sv

Kilpatrick, J., Swafford, J., & Findell, B. (Red.), (2001). Adding it up: Helping children learn mathematics. National Research Council. National Academy Press. https://doi.orf/10.17226/9822

Kiselman, C., & Mouwitz, L. (2008). Matematiktermer för skolan. Nationellt centrum för matematikutbildning.

Kullberg, A. (2010). What is taught and what is learned. Professional insights gained and shared by teachers of mathematics. (Doktorsavhandling). Göteborg: Acta Universitatis Gothoburgensis. https://gupea.ub.gu.se/handle/2077/22180?locale=sv

Llinares, S., & Roig, A. I. (2005). Secondary school students’ construction and use of mathematical models in solving word problems. International Journal of Science and Mathematics Education, 6, 505–532. https://doi.org/10.1007/s10763-006-9055-6 DOI: https://doi.org/10.1007/s10763-006-9055-6

Marton, F. (2015). Necessary conditions of learning. Routledge. DOI: https://doi.org/10.4324/9781315816876

Mason, J., Graham, A., & Johnston-Wilder, S. (2012). Developing thinking in algebra. SAGE Publications Ltd. (Originalet publicerat 2005)

Matthews, P. G., & Fuchs, L. S. (2020). Keys to the gate? Equal sign knowledge at second grade predicts fourth-grade algebra competence. Child Development, 91(1), e14–e28. 10.1111/cdev.13144 DOI: https://doi.org/10.1111/cdev.13144

McAuliffe, S., Tambara, C., & Simsek, E. (2020). Young students’ understanding of mathematical equivalence across different schools in South Africa. South African Journal of Childhood Education, 10(1) a 807. https://doi.org/10.4102/sajce.v10i1.807 DOI: https://doi.org/10.4102/sajce.v10i1.807

Mellone, M., & Ramploud, A. (2015). Additive structure: An educational experience of cultural transposition. I X. Sun, B. Kaur, & J. Novotná. (Red.), Proceedings of the ICMI Study 23, 567–574. University of Macau. http://dx.doi.org/10.1142/978-99965-1-066-3

Polotskaia, E. (2014). How elementary students learn to mathematically analyze word problems: The case of addition and subtraction. McGill University. 10.13140/RG.2.1.3525.3289

Polotskaia, E. (2017). How the relational paradigm can transform the teaching and learning of mathematics: Experiment in Quebec. International Journal for Mathematics Teaching and Learning, 18(2), 161–180.

Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. I C. Kieran (Red.), Teaching and learning algebraic thinking with 5- to 12-year-olds (s. 3–25). Springer. https://www.springer.com/gp/book/9783319683508 DOI: https://doi.org/10.1007/978-3-319-68351-5_1

Röj-Lindberg, A.-S., & Partanen, A.-M. (2019). Learning to solve equations in three Swedish-speaking classrooms in Finland. I C. Kilhamn, & R. Säljö (Red.), Encountering algebra. A comparative study of classrooms in Finland, Norway, Sweden and the USA (s. 111–138). Springer. https://doi.org/10.1007/978-3-030-17577-1_6 DOI: https://doi.org/10.1007/978-3-030-17577-1_6

Schmittau, J. (2011). The role of theoretical analysis in developing algebraic thinking: A Vygotskian perspective. I J. Cai, & E. Knuth (Red.), Early algebraization: A global dialogue from multiple perspectives (s. 71–85). Springer. https://doi.org/10.1007/978-3-642-17735-4_5 DOI: https://doi.org/10.1007/978-3-642-17735-4_5

Schubring, G. (2005). Conflicts between generalization, rigor and intuition. Number concepts underlying the development of analysis in 17-19th century France and Germany. Springer. https://link.springer.com/content/pdf/10.1007%2F0-387-28273-4.pdf

Sherman, J., & Bisanz, J. (2009). Equivalence in symbolic and nonsymbolic contexts: Benefit of solving problems with manipulatives. Journal of Educational Psychology, 101(1), 85–100. http://dx.doi.org/10.1037/a0013156 DOI: https://doi.org/10.1037/a0013156

Skolverket (2020). TIMSS 2019. Svenska grundskoleelevers kunskaper i matematik och naturvetenskap i ett internationellt perspektiv. Internationella studier 2020:8. https://www.skolverket.se/getFile?file=7592

Tuominen, J., Andersson, C., & Boistrup, L. B. (2021). Critical aspects of equations when ex-plored as a part-whole structure. Proceedings of MADIF 12. The twelfth research seminar of the Swedish Society for Research in Mathematics Education (s. 223–233). Linnéuniversitetet. http://matematikdidaktik.org/wp-content/uploads/2021/03/MADIF12_dokumentation.pdf

Tuominen, J., Andersson, C., & Boistrup, L. B., & Eriksson, I. (2018). Relate before calculate: Students’ experiences of relationships of quantities. Didactica Mathematicae, 40, 51–79. https://wydawnictwa.ptm.org.pl/index.php/didactica-mathematicae/issue/view/426

van Oers, B. (2001). Educational forms of initiation in mathematical culture. Educational Studies in Mathematics, 46, 59–85. https://doi.org/10.1023/A:1014031507535 DOI: https://doi.org/10.1023/A:1014031507535

Vergnaud, G. (1982). A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. I T. P. Carpenter, J. M. Moser, & T. A. Romberg (Red.), Addition and subtraction: A cognitive perspective (s. 39–59). Lawrence Erlbaum. 10.1201/9781003046585-4 DOI: https://doi.org/10.4324/9781003046585-4

Wettergren, S., Eriksson, I., & Tambour, T. (2021). Yngre elevers uppfattningar av det matematiska i algebraiska uttryck. LUMAT General Issue, 9, 1–28. https://doi.org/10.31129/LUMAT.9.1.1377 DOI: https://doi.org/10.31129/LUMAT.9.1.1377

Downloads

Published

2022-05-16

How to Cite

Andersson, C., & Tuominen, J. (2022). En modell som stöd för att utforska ekvationer. LUMAT: International Journal on Math, Science and Technology Education, 10(1), 182–209. https://doi.org/10.31129/LUMAT.10.1.1581

Similar Articles

<< < 8 9 10 11 12 13 14 15 16 17 > >> 

You may also start an advanced similarity search for this article.