HAL Id: hal-02398508
https://hal.archives-ouvertes.fr/hal-02398508
Submitted on 7 Dec 2019HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Planning for mathematical reasoning – Surprising
challenges in a design process
Helén Sterner
To cite this version:
Helén Sterner. Planning for mathematical reasoning – Surprising challenges in a design process. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht Univer-sity, Feb 2019, Utrecht, Netherlands. �hal-02398508�
Planning for mathematical reasoning – Surprising challenges in a
design process
Helén Sterner
Dalarna University and the University of Linnaeus, Sweden; hse@du.se
Keywords: Design Principles, Mathematical reasoning, Patterns
Introduction
Generalizing is a key element in mathematical reasoning as well in early algebra and algebraic thinking (Kaput, 2008). However, students find it difficult to understand the meaning of generalized arguments, which means that teachers need to support students to develop their generalized argument (Stylianides & Stylianides, 2017). A way to increase teachers’ possibilities to support students’ reasoning may be through intervention studies.
Stylianides and Stylianides (2013) write about the importance of research-based interventions of proof and proving in teaching. Since generalized arguments can be seen as a part of proof and proving (Stylianides & Silver, 2009), intervention studies also about generalized arguments should be important. Similar reflections have emerged in the Argumentation and proof group at CERME where a lack of design-based studies promoting investigation in the classroom and a need to shift research focus from the learners to the teacher is emphasized (Mariotti, Durand-Guerrier, & Stylianides, 2018). This poster will exemplify a part of a cyclically recurring intervention process by answering the question: What challenges do teachers meet when trying to understand a given Design Principle (DP) (McKenney & Reeves, 2012) and design and implement teaching based on it?
Mathematical reasoning and algebraic thinking
One part of mathematical reasoning and early algebra as well as of algebraic thinking is the structure and the understanding of relationships in quantities (Kaput, 2008); these relationships could be embedded in figural repeating patterns. Patterns and pattern identification could be seen as essential components in elementary school when working with the activity of reasoning-and-proving (Stylianides & Silver, 2009). Teaching patterns can provide students’ understanding and support their arguments why things work the way it do. Mulligan, Mitchelmore, English and Crevensten (2013) make similar assumptions, pointing out that teaching and learning mathematics through patterns and structure as well as generalized approaches may provide students with a deep understanding of mathematics.
Methodology
The frame of this study is based on educational design research, as well as an intervention inspired by classroom design research (Stephan, 2015). In the implementation of the intervention, DPs will be used to address the specific foci in this case, strategies for mathematical reasoning and proving in algebra. Three mathematics teachers (grade 1 and 6) and the author are part of this intervention. DPs are used to guide the content of the interventions and the DPs are used as a framework in the initial analyses. The DPs will guide and give the intervention a theoretical foundation in the ongoing process (three recurring
cyclical phases): first designing the intervention and the teaching, second the three teachers implementing the intervention and the third, the reflection after the implemented intervention. To stimulate mathematical reasoning, and proving in teaching and learning of pattern, six DPs have been created from previous research, and one of them is presented in this poster.
DP1 The students will have possibilities to identify a pattern, to structure the pattern and to generalize the pattern (Mulligan et al., 2013; Stylianides & Silver 2009).
Preliminary results from a part of the design research process
Preliminary results show the complexity involved in teachers’ understanding of teaching and learning patterns relating to DP1. The complexity becomes visible when teachers discuss and try to understand the concepts; identify, structure, and generalize patterns in relation to their own practice. Teachers try to exemplify what it means to identify, structure and generalize, and thereby become aware that the concepts overlap. Results also show that teachers’ understanding of generalization develops during the discussion; from equalizing generalization with general formulas to generalizations as something possible to show with different representations such as describing with words or showing with concrete material and pictures. In the classroom, teachers meet challenges when trying to challenge students’ understanding of a generalized pattern, for example by asking them to explain what is behind a general formula.
References
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 (5-17): Mahwah, NJ: Lawrence Erlbaum/Taylor & Francis Group; Reston, VA: NCTM.
Mariotti, M. A., Durand-Guerrier, V., & Stylianides, G. J. (2018). Argumentation and proof. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger & K. Ruthven (Eds.), Developing research in mathematics education. Twenty years of communication, cooperation and collaboration in Europe (75–89). Oxon, UK: Routledge.
McKenny, S., & Reeves, T. (2012). Conducting Educational Design Research. London: Routledge.
Mulligan, J. T., Mitchelmore, M. C., English, L. D., & Crevensten, N. (2013). Reconceptualizing Early Mathematics Learning: The Fundamental Role of Pattern and Structure. In L. D. English & J. T. Mulligan (Eds.) Reconceptualizing Early Mathematics Learning, Advances in Mathematics Education, New York: Springer.
Stephan, M. L. (2015). Conducting classroom design with teachers. ZDM - Mathematics Education. 47, 905–917.
Stylianides, G. J., & Silver, E. A. (2009). Reasoning-and-Proving in School Mathematics, the Case of Pattern Identification. In D. A. Stylianou, M. L. Blanton & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K–16 perspective. New York, NY: Routledge.
Stylianides, A. J., & Stylianides, G. J. (2013). Seeking research-grounded solutions to problems of practice: Classroom-based interventions in mathematics education. ZDM–The International Journal on Mathematics Education, 45(3), 333-341.
Stylianides, G. J., & Stylianides, A. J. (2017). Research-based interventions in the area of proof: the past, the present and the future. Educational Studies in Mathematics, 96, 119– 127.
