Identifying and promoting young students’ early algebraic thinking

Sanna Wettergren

doi:10.31129/LUMAT.10.2.1617

Authors

Sanna Wettergren Faculty of Education and Welfare Studies, Åbo Akademi University, Vaasa, Finland https://orcid.org/0000-0002-1962-4126

DOI:

https://doi.org/10.31129/LUMAT.10.2.1617

Keywords:

early algebraic thinking;, learning activity;, mathematical thinking;, primary school;, Toulmin's argumentation model

Abstract

Algebraic thinking is an important part of mathematical thinking, and researchers agree that it is beneficial to develop algebraic thinking from an early age. However, there are few examples of what can be taken as indicators of young students’ algebraic thinking. The results contribute to filling that gap by analyzing and exemplifying young students’ early algebraic thinking when reasoning about structural aspects of algebraic expressions during a collective and tool-mediated teaching situation. The article is based on data from a research project exploring how teaching aiming to promote young students’ algebraic thinking can be designed. Along with teachers in grades 2, 3, and 4, the researchers planned and conducted research lessons in mathematics with a focus on argumentation and reasoning about algebraic expressions. The design of teaching situations and problems was inspired by Davydov’s learning activity, and Toulmin’s argumentation model was used when analyzing the students’ algebraic thinking. Three indicators of early algebraic thinking were identified, all non-numerical. What can be taken as indicators of early algebraic thinking appear in very short, communicative micro-moments during the lessons. The results further show that the use of learning models as mediating tools and collective reflections on a collective workspace support young students’ early algebraic thinking when reasoning about algebraic expressions.

References

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

Bråting, K., Hemmi, K., & Madej, L. (2018). Teoretiska och praktiska perspektiv på generaliserad aritmetik [Theoretical and practical perspectives on generalized arithmetic]. In J. Häggström, Y. Liljekvist, J. Bergman Ärlebäck, M. Fahlgren, & O. Olande (Eds.), Perspectives on professional development of mathematics teachers. Proceedings of MADIF 11 (pp. 27–36). The National Center for Mathematics Education & Swedish Association for Mathematics Didactic Research.

Bråting, K., Madej, L., & Hemmi, K. (2019). Development of algebraic thinking: opportunities offered by the Swedish curriculum and elementary mathematics textbooks. Nordic Studies in Mathematics Education, 24(1), 27–49. http://ncm.gu.se/wp-content/uploads/2020/06/24_1_027050_brating-1.pdf

Børne- og Undervisningsministeriet [Ministry of Children and Education]. (5 maj 2020). Læseplan for faget matematik [Syllabus for the subject mathematics]. https://emu.dk/grundskole/matematik/laeseplan-og-vejledning

Cai, J., & Knuth, E. (Eds.). (2011). Early algebraization: A global dialogue from multiple perspectives. Springer. 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

Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 669–705). Information Age Publishing.

Chick, H. (2009). Teaching the distributive law: Is fruit salad still on the menu? In R. Hunter, B. Bicknell & T. Burgess (Ed.), Crossing divides: proceedings of the 32nd annual MERGA conference. Mathematics Education Research Group of Australasia https://merga.net.au/Public/Public/Publications/Annual_Conference_Proceedings/2009_MERGA_CP.aspx

Davydov, V. V. (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula. Soviet Studies in Mathematics Education, 2, 2–222. NCTM. (Original work published 1972).

Davydov, V. V. (2008). Problems of developmental instruction: a theoretical and experimental psychological study. Nova Science Publishers, Inc. (Original work published 1986).

Davydov, V. V., & Rubtsov, V. V. (2018). Developing reflective thinking in the process of learning activity. Journal of Russian & East European Psychology, 55(4–6), 287–571. https://doi.org/10.1080/10610405.2018.1536008

diSessa, A. A., & Cobb, P. (2004). Ontological innovation and the role of theory in design experiments. Journal of the Learning Sciences, 13(1), 77–103. https://doi.org/10.1207/s15327809jls1301_4

Edwards, A. (2005). Relational agency: Learning to be a resourceful practitioner. International Journal of Educational Research, 43(3),168–182. https://doi.org/10.1016/j.ijer.2006.06.010

Eriksson, I. (2018). Lärares medverkan i praktiknära forskning: Förutsättningar och hinder [Teachers' participation in practice-relevant research: Prerequisites and obstacles]. Utbildning & Lärande [Education & Learning], 12(1), 27–40. https://www.du.se/sv/forskning/utbildning-och-larande/tidskriften-utbildning--larande/tidigare-nummer/

Eriksson, I., & Jansson, A. (2017). Designing algebraic tasks for 7-year-old students – a pilot project inspired by Davydov’s learning activity. International Journal for Mathematics Teaching and Learning, 18(2), 257–272. https://www.cimt.org.uk/ijmtl/index.php/IJMTL/issue/view/6

Eriksson, I., Wettergren, S., Fred, J., Nordin, A.-K., Nyman, M., & Tambour, T. (2019). Materialisering av algebraiska uttryck i helklassdiskussioner med lärandemodeller som medierande redskap i årskurs 1 och 5 [Materialization of algebraic expressions in whole-class discussions with learning models as mediating tools in Grades 1 and 5]. Nordic Studies in Mathematics Education, 24(3–4), 86–106. http://ncm.gu.se/wp-content/uploads/2021/10/24_34_081106_eriksson.pdf

Goos, M., & Kaya, S. (2020). Understanding and promoting students’ mathematical thinking: a review of research published in ESM. Educational Studies in Mathematics, 103(1), 7–25. https://doi.org/10.1007/s10649-019-09921-7

Gorbov, S. F., & Chudinova, E. V. (2000). Deystviye modelirovaniya v uchebnoy deyatel'nosti shkol'nikov (k postanovke problemy) [The effect of modeling on students’ learning (regarding problem formulation)]. Psychological Science and Education, 2, 96–110.

Greer, B. (2008). Algebra for all? The Mathematics Enthusiast, 5(2/3), 423–428. https://scholarworks.umt.edu/tme/vol5/iss2/23

Hansson, Å. (2011). Ansvar för matematiklärande. Effekter av undervisningsansvar i det flerspråkiga klassrummet [Responsibility for mathematics learning. Effects of instructional responsibility in the multilingual classroom]. [Doctoral dissertation, University of Gothenburg].

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

Hemmi, K., Lepik, M., & Viholainen, A. (2013). Analysing proof-related competences in Estonian, Finnish and Swedish mathematics curricula—towards a framework of developmental proof. Journal of Curriculum Studies, 45(3), 354–378. https://doi.org/10.1080/00220272.2012.754055

James, G., & James R. C. (1976). Mathematics dictionary. van Nostrand Reinhold.

Johansson, M. (2006). Teaching mathematics with textbooks: a classroom and curricular perspective. [Doctoral dissertation, Luleå University of Technology].

Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Routledge. https://doi.org/10.4324/9781410602619

Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher & M. Blanton (Eds.), Algebra in the early grades (pp. 5–17). Routledge. https://doi-org.ezp.sub.su.se/10.4324/9781315097435

Kaput, J. J., Carraher, D., & Blanton, M. (2008). Algebra in the early grades. Routledge. https://doi.org/10.4324/9781315097435

Kieran, C. (2004). Algebraic thinking in the early grades. What is it? The Mathematics Educator, 8(1), 139–151. https://gpc-maths.org/data/documents/kieran2004.pdf

Kieran, C. (2006). Research on the learning and teaching of algebra: A broadening of sources of meaning. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: past, present and future (pp. 11–49). Sense Publishers.

Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester, Jr., (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Information Age.

Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early algebra research into its nature, its learning, its teaching. Springer International Publishing. https://doi.org/10.1007/978-3-319-32258-2

Kilhamn, C., & Röj-Lindberg, A.-S. (2019). Algebra teachers’ questions and quandaries – Swedish and Finnish algebra teachers discussing practice. Nordic Studies in Mathematics Education, 24(3–4), 153–171.

Kilhamn, C., Röj-Lindberg, A.-S., & Björkqvist, O. (2019). School algebra. In C. Kilhamn & R. Säljö (Eds.), Encountering algebra: A comparative study of classrooms in Finland, Norway, Sweden, and the USA (pp. 3–11). Springer.

Kiselman, C., & Mouwitz, L. (2008). Matematiktermer för skolan [Math terms for school]. The National Center for Mathematics Education.

Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229–270). Lawrence Erlbaum.

Küchemann, D. (2019). Cuisenaire rods and symbolic algebra. Mathematics Teaching, 265, 34–37.

Larsson, M., & Ryve, A. (2012). Balancing on the edge of competency-oriented versus procedural-oriented practices: orchestrating whole-class discussions of complex mathematical problems. Mathematics Education Research Journal, 24(4), 447–465. https://doi.org/10.1007/s13394-012-0049-0

Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem-solving. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems: Advances and new perspectives (Vol. 1–Book, Section, pp. 361–386). Springer.

Linell, P. (1994). Transkription av tal och samtal: teori och praktik [Transcription of speech and conversation: theory and practice]. Linköping University.

Lins, R., & Kaput, J. J. (2004). The early development of algebraic reasoning: The current state of the field. In K. Stacey, H. Chick & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 45–70). Springer. https://doi.org/10.1007/1-4020-8131-6_4

MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1–19. https://doi.org/10.1023/A:1002970913563

Marton, F. (2005). Om praxisnära grundforskning [About practice-based fundamental research]. In I. Carlgren, I. Josefson & C. Liberg (Eds.), Forskning av denna världen 2 – om teorins roll i praxisnära forskning [Research of this world 2 - on the role of theory in practice-based research] (pp. 105–122). Vetenskapsrådet.

Marton, F. (2015). Necessary conditions of learning. Routledge.

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. https://doi.org/10.1111/cdev.13144

Nordin, A.-K., & Boistrup, L. B. (2018). A framework for identifying mathematical arguments as supported claims created in day-to-day classroom interactions. The Journal of Mathematical Behavior, 51, 15–27. https://doi.org/10.1016/j.jmathb.2018.06.005

Radford, L. (2008a). Culture and cognition: Towards an anthropology of mathematical thinking. In L. English (Ed.), Handbook of International Research in Mathematics Education, 2nd Edition (pp. 439–464). Routledge.

Radford, L. (2008b). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G. Schubring & F. Seeger (Eds.), Semiotics in mathematics education: epistemology, history, classroom, and culture (pp. 215–234). Sense Publishers. https://doi.org/10.1163/9789087905972_013

Radford, L. (2010). Signs, gestures, meanings: Algebraic thinking from a cultural semiotic perspective. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the sixth conference of European research in mathematics education (CERME 6) (pp. XXXIII–LIII). Université Claude Bernard, Lyon, France.

Repkin, V. V. (2003). Developmental teaching and learning activity. Journal of Russian & East European Psychology, 41(5), 10–33. https://doi.org/10.2753/RPO1061-0405410510

Roth, W.-M., & Radford, L. (2011). A cultural-historical perspective on mathematics teaching and learning. Sense Publishers. https://doi.org/10.1007/978-94-6091-564-2

Runesson, U. (2017). Variationsteori som redskap för att analysera lärande och designa undervisning [Variation theory as a tool for analyzing learning and designing teaching]. In I. Carlgren (Ed.), Undervisningsutvecklande forskning. Exemplet Learning study [Practice- development research. The example of Learning study] (pp. 45–60). Gleerups.

Röj-Lindberg, A.-S. (2017). Skolmatematisk praktik i förändring – en fallstudie [School mathematics practice in change – a case study]. [Doctoral dissertation, Åbo Akademi University].

Röj-Lindberg, A.-S., Partanen, A.-M., & Hemmi, K. (2017). Introduction to equation solving for a new generation of algebra learners. In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education, CERME10 (pp. 495–503). Dublin City University, Institute of Education and European Society for Research in Mathematics Education.

Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, 19(1), 19–43. https://doi.org/10.1007/BF03173235

Schmittau, J. (2005). The development of algebraic thinking. ZDM – the International Journal on Mathematics Education, 37(1), 16–22. https://doi-org.ezp.sub.su.se/10.1007/bf02655893

Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V. V. Davydov. The Mathematics Educator, 8(1), 60–87. http://lchc.ucsd.edu/MCA/Mail/xmcamail.2011_06.dir/pdfn8bJa6mo4c.pdf

Schoenfeld, A. H. (1995). Is thinking about ‘algebra’ a misdirection? In C. Lacampagne, W. Blair, & J. Kaput (Eds.), The algebra initiative colloquium. Volume 2: Working group papers (pp. 83–86). U.S. Department of Education, Office of Educational Research and Improvement, National Institute on Student Achievement, Curriculum and Assessment. http://files.eric.ed.gov/fulltext/ED385437.pdf

Skolverket [Swedish National Agency for Education]. (2019). Läroplan för grundskolan, förskoleklassen och fritidshemmet 2011: reviderad 2019 [Curriculum for the compulsory school, preschool class and school-age educare (revised 2019)]. Skolverket.

Stacey, K., & Chick, H. (2004). Solving the problem with algebra. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 1–20). Springer. https://doi.org/10.1007/1-4020-8131-6_1

Stacey, K., & MacGregor, M. (1999). Ideas about symbolism that students bring to algebra. In B. Moses (Ed.), Algebraic Thinking Grades K-12 (pp. 308–312). National Council of Teachers of Mathematics.

Toulmin, S. (2003). The uses of argument. (Updated ed.). Cambridge University Press.

Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford & A. P. Shulte (Eds.), Ideas of algebra: K–12. 1988 Yearbook of the National Council of Teachers of Mathematics (pp. 8–19). National Council of Teachers of Mathematics.

Utbildningsstyrelsen [Finnish National Agency for Education]. (5 maj 2020). Grunderna för läroplanen för den grundläggande utbildningen 2014 [The core curriculum for basic education 2014]. https://www.oph.fi/sv/utbildning-och-examina/grunderna-laroplanen-den-grundlaggande-utbildningen

Utdanningsdirektoratet [Norwegian Directorate for Education and Training]. (5 maj 2020). Læreplan i matematikk 1.–10. [Syllabus in mathematics for Grades 1–10]. https://www.udir.no/LK20/mat01-05

Venenciano, L., & Dougherty, B. (2014). Addressing priorities for elementary school mathematics. For the Learning of Mathematics, 34(1), 18–24. https://www.jstor.org/stable/43894872

Venenciano, L. C., Yagi, S. L., Zenigami, F. K., & Dougherty, B. J. (2020). Supporting the development of early algebraic thinking, an alternative approach to number. Investigations in Mathematics Learning, 12(1), 38–52. https://doi.org/10.1080/19477503.2019.1614386

Ventura, A. C., Brizuela, B. M., Blanton, M., Sawrey, K., Murphy Gardiner A., & Newman-Owens, A. (2021). A learning trajectory in kindergarten and first grade students’ thinking of variable and use of variable notation to represent indeterminate quantities. The Journal of Mathematical Behavior, 62, 1–17. https://doi.org/10.1016/j.jmathb.2021.100866

Vygotsky, L. (1986). Thought and language. MIT Press.

Wahlström, R., Dahlgren, L. O., Tomson, G., Diwan, V. K., & Beermann, B. (1997). Changing primary care doctors’ conceptions: A qualitative approach to evaluating an intervention. Advances in Health Sciences Education, 2(3), 221–236. https://doi.org/10.1023/A:1009763521278

Warren, E., & Cooper, T. J. (2009) Developing mathematics understanding and abstraction: The case of equivalence in the elementary years. Mathematics Education Research Journal, 21(2), 76–95. https://doi.org/10.1007/BF03217546

Wettergren, S., Eriksson, I., & Tambour, T. (2021). Yngre elevers uppfattningar av det matematiska i algebraiska uttryck [Younger students’ conceptions of the mathematics in algebraic expressions]. LUMAT: International Journal on Math, Science and Technology Education, 9(1), 1–28. https://doi.org/10.31129/LUMAT.9.1.1377

Zuckerman, G. (2003). The learning activity in the first years of schooling. In A. Kozulin, B. Gindis, V. S. Ageyev, & S. M. Miller (Eds.), Vygotsky's educational theory in cultural context (pp. 39–64). Cambridge University Press.

Zuckerman, G. (2004). Development of reflection through learning activity. European Journal of Psychology of Education, 19(1), 9–18. https://doi-org.ezp.sub.su.se/10.1007/bf03173234