12/2022 vol. 18
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RESEARCH PAPER
Pre-algebraic aspects in arithmetic strategies – The generalization and conceptual understanding of the ‘Auxiliary Task’
Taha Ertuğrul Kuzu 1
 
 
 
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University of Münster, Münster, GERMANY
 
 
Publication date: 2022-11-20
 
 
EURASIA J. Math., Sci Tech. Ed 2022;18(12):em2192
DOI: https://doi.org/10.29333/ejmste/12656
 
 
Article (PDF)
References (47)
 
KEYWORDS
Auxiliary Task
cognitive gap
design-based research
pre-algebraic thinking
 
ABSTRACT
In the last decades, a broad international reform approach was visible in support of mental calculation strategies: Instead of being solely ‘transition strategies’ for learning the standard algorithms, the understanding of numerical relations is essential for mental calculation strategies, making them highly important for a viable understanding of arithmetics. Yet, mental calculation strategies are not only important for understanding arithmetics, but highly relational strategies such as the ‘Auxiliary Task’ might have an important role in the emergence of a pre-algebraic understanding of numerical relations. In this qualitative study from Germany, 4th and 5th grade learners’ (n=18) processes of interpreting the ‘Auxiliary Task’ are examined by conducting linguistic and epistemological analyzes of their conceptual understanding of the ‘Auxiliary Task’ utilizing a design-based research framework. Insights are given into specific, language-related forms of pre-algebraic generalizations of the ‘Auxiliary Task’ as well as into developmental processes within the designed learning-environment.
REFERENCES (47)
1.
Akgun, L., & Ozdemir, M. E. (2006). Students’ understanding of the variable as general number and unknown: A case study. The Teaching of Mathematics, 9(1), 45-52.
Google Scholar
 
 
2.
Akinwunmi, K. (2012). Zur Entwicklung von Variablenkonzepten beim Verallgemeinern mathematischer Muster [Developing concepts of variables when generalizing mathematical patterns]. Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
3.
Bauersfeld, H. (1980). Hidden dimensions in the so-called reality of a mathematics classroom. Educational Studies in Mathematics, 11, 23-41. https://doi.org/10.1007/BF0036....
CrossRef
 
Google Scholar
 
 
4.
Blöte, A. W., Van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93, 627-638. https://doi.org/10.1037/0022-0....
CrossRef
 
Google Scholar
 
 
5.
Britt, M. S., & Irwin, K. C. (2008). Algebraic thinking with and without algebraic representation: A three-year longitudinal study. ZDM, 40, 39-53. https://doi.org/10.1007/s11858....
CrossRef
 
Google Scholar
 
 
6.
Britt, M. S., & Irwin, K. C. (2011). Algebraic thinking with and without algebraic representation: A pathway for learning. In J. Cai, & E. Knuth (Eds.), Early algebraization (pp. 137-159). Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
7.
Fischbein, E. (1989). Tacit models and mathematical reasoning. For the Learning of Mathematics, 9(2), 9-14.
Google Scholar
 
 
8.
Glade, M., & Prediger, S. (2017). Students’ individual schematization pathways-empirical reconstructions for the case of part-of-part determination for fractions. Educational Studies in Mathematics, 94(2), 185-203. ttps://doi.org/10.1007/s10649-016-9716-5.
Google Scholar
 
 
9.
Hammond, J., & Gibbons, P. (2005). Putting scaffolding to work: The contribution of scaffolding in articulating ESL education. Prospect, 20(1), 6-30.
Google Scholar
 
 
10.
Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27, 59-78. https://doi.org/10.1007/BF0128....
CrossRef
 
Google Scholar
 
 
11.
Kieran, C. (2011). Overall commentary on early algebraization: Perspectives for research and teaching. In J. Cai, & E. Knuth (Eds.), Early algebraization (pp. 579-593). Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
12.
Kieran, C. (2018). Teaching and learning algebraic thinking with 5- to 12-year-olds. Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
13.
Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early algebra: Research into its nature, its learning, its teaching. Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
14.
Korntreff, S., & Prediger, S. (2022). Students’ ideas about variables as generalizers and unknowns: Design research calling for more explicit comparisons of purposes. In J. Hodgen, & G. Bolondi (Eds.), Proceedings of 12th Congress of the European Society for Research in Mathematics Education (pp. 1-8). ERME.
Google Scholar
 
 
15.
Kunsteller, J. (2018). Developing rules due to the use of family resemblances in classroom communication. In P. Ernest (Ed.), The philosophy of mathematics education today (pp. 365-377). Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
16.
Kuzu, T. (2019). Mehrsprachige Vorstellungsentwicklungsprozesse–Lernprozessstudie zum Anteilskonzept bei deutsch-türkischen Lernenden [Multilingual concept development processes-learning process study on the concept of participation in German-Turkish learners]. Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
17.
Kuzu, T. (2022a). Understanding the ‘Auxiliary Task’ conceptually – Discrete versus continuous cardinal objects. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proceedings of the 45th PME-Conference (Vol. 3, pp. 99-106). Alicante University Press (PME).
Google Scholar
 
 
18.
Kuzu, T. (2022b). Kreative Denkwege oder umständliches Denken? Einblicke in alternative Vorgehensweisen zur, Hilfsaufgabe’. In Gesellschaft für Didaktik der Mathematik (Eds.), Beiträge zum Mathematikunterricht 2022 (pp. 1-4). WTM. Paper accepted.
Google Scholar
 
 
19.
Kuzu, T., & Nührenbörger, M. (2021). The conceptual understanding of mental calculation strategies: Meaning-making in the case of the ‘Auxiliary task’. In J. Novotná & H. Moraová (Eds.), Proceedings of the International Symposium on Elementary Maths Teaching (pp. 270-280). Charles University, Faculty of Education.
Google Scholar
 
 
20.
Lew, K., Weber, K., & Mejía Ramos, J. P. (2020). Do generic proofs improve proof comprehension? Journal of Educational Research in Mathematics, 30, 229-248. https://doi.org/10.29275/jerm.....
CrossRef
 
Google Scholar
 
 
21.
Mayer, C. (2019). Zum algebraischen Gleichheitsverständnis von Grundschulkindern. Konstruktive und rekonstruktive Erforschung von Lernchancen [On the algebraic understanding of equality by elementary school children. Constructive and reconstructive exploration of learning opportunities]. Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
22.
Meyer, M. (2009). Abduction, induction–confusion. Notes on the logic of interpretative social research. Zeitschrift für Erziehungswissenschaft [Journal of Educational Science], 12, 302-320. https://doi.org/10.1007/s11618....
CrossRef
 
Google Scholar
 
 
23.
Nührenbörger, M., & Steinbring, H. (2009). Forms of mathematical interaction in different social settings: Examples from students’, teachers’ and teacher-students’ communication about mathematics. Journal of Mathematics Teacher Education, 12, 111-132. https://doi.org/10.1007/s10857....
CrossRef
 
Google Scholar
 
 
24.
Pöhler, B., & Prediger, S. (2015). Intertwining lexical and conceptual learning trajectories–A design research study on dual macro-scaffolding towards percentages. EURASIA Journal of Mathematics, 11(6), 1697-1722. https://doi.org/10.12973/euras....
CrossRef
 
Google Scholar
 
 
25.
Prediger, S. (2019). Theorizing in design research: Methodological reflections on developing and connecting theory elements for language-responsive mathematics classrooms. Avances de Investigación en Educación Matemática [Research Advances in Mathematics Education], 15, 5-27. https://doi.org/10.35763/aiem.....
CrossRef
 
Google Scholar
 
 
26.
Prediger, S., Clarkson, P., & Bose, A. (2016). Purposefully relating multilingual registers: Building theory and teaching strategies for bilingual learners based on an integration of three traditions. In R. Barwell, P. Clarkson, A. Halai, M. Kazima, J. N. Moschkovich, N. Planas, M. Phakeng, P. Valero, & M. Villavicencio-Ubillús (Eds.), Mathematics education and language diversity (pp. 193-215). Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
27.
Prediger, S., Gravemeijer, K., & Confrey, J. (2015). Design research with a focus on learning processes–an overview on achievements and challenges. ZDM Mathematics Education, 47(6), 877-891. https://doi.org/10.1007/s11858....
CrossRef
 
Google Scholar
 
 
28.
Radford, L. (2010). Layers of generality and types of generalization in pattern activities. PNA, 4(2), 37-62.
Google Scholar
 
 
29.
Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26, 257-277. https://doi.org/10.1007/s13394....
CrossRef
 
Google Scholar
 
 
30.
Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 3-27). Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
31.
Rathgeb-Schnierer, E., & Rechtsteiner, C. (2018). Rechnen lernen und Flexibilität entwickeln: Grundlagen–Förderung–Beispiele [Learning arithmetic and developing flexibility: Basics-support-examples]. Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
32.
Schütte, M., Friesen, R. A., & Jung, J. (2019). Interactional analysis: A method for analyzing mathematical learning processes in interactions. In G. Kaiser, & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 101-129). Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
33.
Schwarzkopf, 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-....
CrossRef
 
Google Scholar
 
 
34.
Selter, C., Prediger, S., Nührenbörger, M., & Hußmann, S. (2012). Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies in Mathematics, 79(3), 389-408. https://doi.org/10.1007/s10649....
CrossRef
 
Google Scholar
 
 
35.
Serrazina, L., & Rodrigues, M. (2021). Number sense and flexibility of calculation: A common focus on number relations. In A. G. Spinillo, S. L. Lautert, & R. E. S. R. Borba (Eds.), Mathematical reasoning of children and adults (pp. 19-40). Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
36.
Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction. An epistemological perspective. Springer.
Google Scholar
 
 
37.
Steinbring, H. (2006). What makes a sign a mathematical sign? An epistemological perspective on mathematical interaction. Educational Studies in Mathematics, 61, 133-162. https://doi.org/10.1007/s10649....
CrossRef
 
Google Scholar
 
 
38.
Steinbring, H., & Nührenbörger, M. (2010). Mathematisches Wissen als Gegenstand von Lehr-/Lerninteraktionen. Eigenständige Schülerinteraktionen in Differenz zu Lehrerinterventionen [Mathematical knowledge as a subject of teaching/learning interactions. Independent school interactions in difference to teacher interventions]. In U. Dausendschön-Gay, C. Domke, & S. Ohlhus (Eds.), Wissen in (Inter-)Aktion. Verfahren der Wissensgenerierung in unterschiedlichen Praxisfeldern [Knowledge in (inter)action. Methods of knowledge generation in different fields of practice] (pp. 161-188). De Gruyter.
Google Scholar
 
 
39.
Steinweg, A. S. (2013). Algebra in der Grundschule. Muster und Strukturen–Gleichungen–funktionale Beziehungen [Algebra in elementary school. Patterns and structures–equations–functional relationships]. Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
40.
Steinweg, A. S., Akinwunmi, K., & Lenz, D. (2018). Making implicit algebraic thinking explicit: Exploiting national characteristics of German approaches. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 283-307). Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
41.
Swain, M. (2006). Languaging, agency and collaboration in advanced second language proficiency. In H. Byrnes (Ed.), Advanced language learning (pp. 95-108). Continuum.
Google Scholar
 
 
42.
Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50, 29-47. https://doi.org/10.1023/A:1020....
CrossRef
 
Google Scholar
 
 
43.
Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83-94. https://doi.org/10.1159/000202....
CrossRef
 
Google Scholar
 
 
44.
Warren, E., & Cooper, T. (2008). Generalizing the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67, 171-185. https://doi.org/10.1007/s10649....
CrossRef
 
Google Scholar
 
 
45.
Warren, E., Cooper, T., & Lamb, J. (2006). Investigating functional thinking in the elementary classroom: Foundations of early algebra reasoning. The Journal of Mathematical Behavior, 25(3), 208-223. https://doi.org/10.1016/j.jmat....
CrossRef
 
Google Scholar
 
 
46.
Wessel, L. (2020). Vocabulary in learning processes towards conceptual understanding of equivalent fractions—specifying students’ language demands on the basis of lexical trace analyses. Mathematics Education Research Journal, 32, 653-681. https://doi.org/10.1007/s13394....
CrossRef
 
Google Scholar
 
 
47.
Wittmann, E. C. (2021). Connecting mathematics and mathematics education. Collected papers on mathematics education as a design science. Springer. https://doi.org/10.1007/978-3-....
CrossRef
 
Google Scholar
 
 
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