Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-6bf8c574d5-7jkgd Total loading time: 0 Render date: 2025-02-16T21:37:55.602Z Has data issue: false hasContentIssue false

13 - Deeper Learning Mathematics

The case of algebraic questions

from Part II - Deeper Learning Episodes: First Steps towards Transforming Classrooms

Published online by Cambridge University Press:  04 February 2023

By
Susanne Prediger  and
Anna-Katharina Roos
Edited by
Do Coyle ,
Oliver Meyer  and
Susanne Staschen-Dielmann
Do Coyle
Affiliation:
University of Edinburgh
Oliver Meyer
Affiliation:
Johannes Gutenberg Universität Mainz, Germany
Get access

Summary

The mathematics chapter by Susanne Prediger and Anna-Katharina Roos offers profound insights into the nature of mathematical literacy. It documents a precise account of the way the knowledge and activity domains of doing, organising, explaining and arguing translate into algebraic activities and procedures. Using the example of transformation and transformation rules, they argue that algebraic rules that are not underpinned with meaning will become arbitrary and lead to typical student errors. They make the case for algebraic reasoning as a way of developing conceptual understanding and promoting deeper learning in the maths classroom. Based on their empirical classroom research, the authors propose three principles to inform the design of deeper learning episodes in mathematics: connecting multiple representations and languages, engaging learners in rich discourse practice and employing macro-level scaffolding that integrates mathematics and language learning.

Keywords

deeper learning in mathematicsmathematical literacymeaning-making in mathematicsmultiple representationsrich discourse practices, equation transformationalgebrabar model
Type
Chapter
Information
A Deeper Learning Companion for CLIL
Putting Pluriliteracies into Practice
 , pp. 262 - 287
Publisher: Cambridge University Press
Print publication year: 2023

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amit, M., & Fried, M. N. (2005). Multiple Representations in 8th Grade Algebra Lessons: Are Learners Really Getting It? Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, 2, 57–64.Google Scholar
Bednarz, N., Kieran, C., & Lee, L. (Eds.). (1996). Approaches to Algebra: Perspectives for Research and Teaching. Kluwer Academic Publishers.Google Scholar
Blum, W., & Borromeo Ferri, R. (2009). Mathematical Modelling: Can It Be Taught and Learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58.Google Scholar
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design Experiments in Educational Research. Educational Researcher, 32(1), 9–13. https://doi.org/10.3102/0013189X032001009Google Scholar
Coyle, D., & Meyer, O. (2021). Beyond CLIL: Pluriliteracies Teaching for Deeper Learning. Cambridge University Press.Google Scholar
Erath, K., Ingram, J., Moschkovich, J., & Prediger, S. (2021). Designing and Enacting Instruction that Enhances Language for Mathematics Learning: A Review of the State of Development and Research. ZDM – Mathematics Education, 53(2), 245–262. https://doi.org/10.1007/s11858-020-01213-2Google Scholar
Fong Ng, S., & Lee, K. (2009). The Model Method: Singapore Children’s Tool for Representing and Solving Algebraic Word Problems. Journal for Research in Mathematics Education, 40(3), 282–313. https://doi.org/10.5951/jresematheduc.40.3.0282Google Scholar
Friedlander, A., & Talbach, M. (2001). Promoting Multiple Representations in Algebra. In Cuoco, A. A. (Ed.), The Roles of Representation in School Mathematics: 2001 Yearbook of the National Council of the Teachers of Mathematics. NCTM, pp. 173–185.Google Scholar
Gibbons, P. (2002). Scaffolding Language, Scaffolding Learning: Teaching Second Language Learners in the Mainstream Classroom. Heinemann.Google Scholar
Götze, D., & Baiker, A. (2021). Language-Responsive Support for Multiplicative Thinking as Unitizing: Results of an Intervention Study in the Second Grade. ZDM – Mathematics Education, 53(2), 263–275. https://doi.org/10.1007/s11858-020-01206-1Google Scholar
Gravemeijer, K. (1998). Developmental Research as a Research Method. In Kilpatrick, J. & Sierpinska, A. (Eds.), What is Research in Mathematics Education and What Are Its Results? Kluwer.Google Scholar
Gravemeijer, K., & Cobb, P. (2006). Design Research from a Learning Design Perspective. In van den Akker, J., Gravemeijer, K., McKenney, S., & Nieveen, N. (Eds.), Educational Design Research. Routledge, pp. 17–51.Google Scholar
Grünkorn, J., Klieme, E., Praetorius, A.-K., & Schreyer, P., eds. (2020). Mathematikunterricht im internationalen Vergleich: Ergebnisse aus der TALIS-Videostudie Deutschland. DIPF | Leibniz-Institut für Bildungsforschung und Bildungsinformation.Google Scholar
Hiebert, J., & Carpenter, T. P. (1992). Learning and Teaching with Understanding. In Grouws, D. A. (Ed.), Handbook of Research on Mathematics Teaching and Learning. Macmillan, pp. 65–97.Google Scholar
Jablonka, E. (2003). Mathematical Literacy. In Bishop, A., Clements, M. A., Keitel, C., Kilpatrick, J., & Leung, F. K. S. (Eds.), Second International Handbook of Mathematics Education. Kluwer, pp. 77–104.Google Scholar
Kaput, J. J. (1998). Representations, Inscriptions, Descriptions and Learning: A Kaleidoscope of Windows. The Journal of Mathematical Behavior, 17(2), 265–281. https://doi.org/10.1016/S0364-0213(99)80062-7CrossRefGoogle Scholar
Kieran, C. (2004). The Core of Algebra: Reflections on Its Main Activities. In Stacey, K., Chick, H., & Kendal, M. (Eds.), The Future of the Teaching and Learning of Algebra: The 12th ICMI Study, vol. 8. Kluwer Academic Publishers, pp. 21–33. https://doi.org/10.1007/1-4020-8131-6_2Google Scholar
Kuhnke, K. (2013). Vorgehensweisen von Grundschulkindern beim Darstellungswechsel: Eine Untersuchung am Beispiel der Multiplikation im 2. Schuljahr. Springer.Google Scholar
Malle, G. (1993). Didaktische Probleme der elementaren Algebra. Vieweg.Google Scholar
McConachie, S. M., Petrosky, A. R., & Resnick, L. B. (Eds.). (2016). Content Matters: A Disciplinary Literacy Approach to Improving Student Learning. Jossey-Bass.Google Scholar
Moschkovich, J. (Ed.). (2010). Language and Mathematics Education. Information Age.Google Scholar
Moschkovich, J. (2013). Principles and Guidelines for Equitable Mathematics Teaching Practices and Materials for English Language Learners. Journal of Urban Mathematics Education, 6(1), 45–57.Google Scholar
Moschkovich, J. (2015). Academic Literacy in Mathematics for English Learners. The Journal of Mathematical Behavior, 40(A), 43–62. https://doi.org/10.1016/j.jmathb.2015.01.005Google Scholar
OECD. (2007). PISA 2006: Science Competencies for Tomorrow’s World: Volume 1: Analysis. OECD. https://doi.org/10.1787/9789264040014-enGoogle Scholar
Pimm, D. (1987). Speaking Mathematically: Communication in Mathematics Classrooms. Routledge.Google Scholar
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, Science and Technology Education, 11(6), 1697–1722. https://doi.org/10.12973/eurasia.2015.1497aGoogle Scholar
Polias, J. (2016). Apprenticing Students into Science: Doing, Talking & Writing Scientifically. Lexis Education.Google Scholar
Prediger, S. (2019a). Mathematische und sprachliche Lernschwierigkeiten: Empirische Befunde und Förderansätze am Beispiel des Multiplikationskonzepts. Lernen und Lernstörungen, 8(4), 247–260. https://doi.org/10.1024/2235-0977/a000268Google Scholar
Prediger, S. (2019b). Welche Forschung kann Sprachbildung im Fachunterricht empirisch fundieren? Ein Überblick zu mathematikspezifischen Studien und ihre forschungsstrategische Einordnung. In Ahrenholz, B., Jeuk, S., Lütke, B., Paetsch, J., & Roll, H. (Eds.), Fachunterricht, Sprachbildung und Sprachkompetenzen. De Gruyter, pp. 19–38. https://doi.org/10.1515/9783110570380-002Google Scholar
Prediger, S. (2020). Sprachbildender Mathematikunterricht in der Sekundarstufe – Ein forschungsbasiertes Praxisbuch. Cornelsen.Google Scholar
Prediger, S., & Hein, K. (2017). Learning to Meet Language Demands in Multi-Step Mathematical Argumentations: Design Research on a Subject-Specific Genre. European Journal of Applied Linguistics, 5(2), 309–335. https://doi.org/10.1515/eujal-2017-0010Google Scholar
Prediger, S., & Krägeloh, N. (2016). ‘X-Arbitrary Means Any Number, but You Do Not Know Which One’: The Epistemic Role of Languages while Constructing Meaning for the Variable as Generalizers. In Halai, A. & Clarkson, P. (Eds.), Teaching and Learning Mathematics in Multilingual Classrooms. SensePublishers, pp. 89–108. https://doi.org/10.1007/978-94-6300-229-5_7Google Scholar
Prediger, S., & Wessel, L. (2013). Fostering German Language Learners’ Constructions of Meanings for Fractions – Design and Effects of a Language- and Mathematics-Integrated Intervention. Mathematics Education Research Journal, 25(3), 435–456. https://doi.org/10.1007/s13394-013-0079-2Google Scholar
Prediger, S., & Zindel, C. (2017). School Academic Language Demands for Understanding Functional Relationships: A Design Research Project on the Role of Language in Reading and Learning. EURASIA Journal of Mathematics, Science and Technology Education, 13(7b), 4157–4188. https://doi.org/10.12973/eurasia.2017.00804aGoogle Scholar
Prediger, S., Erath, K., & Opitz, E. M. (2019). The Language Dimension of Mathematical Difficulties. In Fritz, A., Haase, V. G., & Räsänen, P. (Eds.), International Handbook of Mathematical Learning Difficulties: From the Laboratory to the Classroom. Springer, pp. 437–455. https://doi.org/10.1007/978-3-319-97148-3_27CrossRefGoogle Scholar
Usiskin, Z. (1988). Conceptions of School Algebra and Uses of Variables. In Coxford, A. F. & Shulte, A. P. (Eds.), The Ideas of Algebra, K-12. NCTM, pp. 8–19.Google Scholar
Warren, E. (2003). The Role of Arithmetic Structure in the Transition from Arithmetic to Algebra. Mathematics Education Research Journal, 15(2), 122–137. https://doi.org/10.1007/BF03217374Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×