Teachers' perspectives on students' metacognitive strategies during mathematical modelling processed - A case study

Book chapter

Wendt, Lisa, Vorhölter, Katrin and Kaiser, Gabriele. (2020). Teachers' perspectives on students' metacognitive strategies during mathematical modelling processed - A case study. In Mathematical modelling education and sense-making pp. 335-346 Springer Nature. https://doi.org/10.1007/978-3-030-37673-4_29
AuthorsWendt, Lisa, Vorhölter, Katrin and Kaiser, Gabriele
Abstract

The usage of metacognitive strategies can help students to avoid or to overcome cognitive barriers during modelling processes. For imparting relevant metacognitive strategies and supporting students during the modelling process, it is necessary that teachers act on a meta-meta level and monitor their students’ usage of metacognitive strategies. Therefore, the teachers’ perception of students’ metacognitive activities during modelling processes is of great importance. However, until now, there does not exist such a study. The study described in this chapter examines this topic in a qualitatively oriented intervention study. The results focus on one teacher, who showed a broadening of her perception of the students’ usage of metacognition in mathematical modelling processes.

Keywordsconditional knowledge; evaluating; metacognition; metacognitive knowledge; metacognitive strategies; modelling problems; monitoring; planning; regulating ; teachers’ perception
Page range335-346
Year2020
Book titleMathematical modelling education and sense-making
PublisherSpringer Nature
Place of publicationCham
Edition1st
SeriesInternational perspectives on the teaching and learning of mathematical modelling
ISBN9783030376727
Digital Object Identifier (DOI)https://doi.org/10.1007/978-3-030-37673-4_29
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All rights reserved
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Controlled
Output statusPublished
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Online15 May 2020
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Deposited27 Sep 2021
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On the occasion of the decennial publication of ZDM mathematics education by springer
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Perspectives on (future) teachers' professional competencies
Kaiser, Gabriele and Li, Yeping. (2017). Perspectives on (future) teachers' professional competencies. In B. Kaur, W. Kin Ho and T. L. Toh and B. H. Choy (Ed.). pp. 1 - 81
The teaching and learning of mathematical modeling
Kaiser, Gabriele. (2017). The teaching and learning of mathematical modeling. In In J. Cai (Ed.). Compendium for Research in Mathematics Education pp. 267 - 291 The National Council of Teachers of Mathematics, Inc..
A mixed-methods-evaluation approach for the study of macro-micro interactions: on the development of teaching- and learning-oriented beliefs of mathematics student teachers in the introductory phase of study
Buchholtz, Nils and Kaiser, Gabriele. (2017). A mixed-methods-evaluation approach for the study of macro-micro interactions: on the development of teaching- and learning-oriented beliefs of mathematics student teachers in the introductory phase of study. Koelner Zeitschrift fuer Soziologie und Sozialpsychologie. 69(Suppl 2), pp. 435 - 458. https://doi.org/10.1007/s11577-017-0465-y
Professional competences of teachers for fostering creativity and supporting high-achieving students
Hoth, Jessica, Kaiser, Gabriele, Busse, Andreas, Dohrmann, Martina, Konig, Johannes and Blomeke, Sigrid. (2017). Professional competences of teachers for fostering creativity and supporting high-achieving students. ZDM: The International Journal on Mathematics Education. 49(1), pp. 107 - 120. https://doi.org/10.1007/s11858-016-0817-5
Eine analyse der sogenannten schlussrechnung - die relevanz der ansatze von arnold kirsch fur aktuelle lernprozesse in der lehrerausbildung
Buchholtz, Nils, Schwarz, Bjorn and Kaiser, Gabriele. (2016). Eine analyse der sogenannten schlussrechnung - die relevanz der ansatze von arnold kirsch fur aktuelle lernprozesse in der lehrerausbildung. Journal fuer Mathematik-Didaktik. 37(1), pp. 31 - 53. https://doi.org/10.1007/s13138-016-0090-8
Uncovering predictors of disagreement: ensuring the quality of expert ratings
Hoth, Jessica, Schwarz, Björn, Kaiser, Gabriele, Busse, Andreas, König, Johannes and Blömeke, Sigrid. (2016). Uncovering predictors of disagreement: ensuring the quality of expert ratings. ZDM Mathematics Education. 48(1-2), pp. 83 - 95. https://doi.org/10.1007/s11858-016-0758-z
Fostering modeling competencies for complex situations
Stender, Peter and Kaiser, Gabriele. (2016). Fostering modeling competencies for complex situations. In In C. R. Hirsch (Ed.). Mathematical modeling and modeling mathematics pp. 107 - 115 National Council of Teachers of Mathematics.
Early Career Teachers’ ability to focus on typical students errors in relation to the complexity of a mathematical topic
Pankow, Lena, Kaiser, Gabriele, Busse, Andreas, König, Johannes, Blömeke, Sigrid, Hoth, Jessica and Döhrmann, Martina. (2016). Early Career Teachers’ ability to focus on typical students errors in relation to the complexity of a mathematical topic. ZDM: The International Journal on Mathematics Education. 48(1-2), pp. 55 - 67. https://doi.org/10.1007/s11858-016-0763-2
Theoretical and pedagogical considerations in promoting students' metacognitive modeling competencies
Vorholter, Katrin and Kaiser, Gabriele. (2016). Theoretical and pedagogical considerations in promoting students' metacognitive modeling competencies. In In C. R. Hirsch (Ed.). Mathematical modeling and modeling mathematics pp. 273 - 280 National Council of Teachers of Mathematics.
The relation between content-specific and general teacher knowledge and skills
Blömeke, Sigrid, Busse, Andreas, Kaiser, Gabriele, König, Johannes and Suhl, Ute. (2016). The relation between content-specific and general teacher knowledge and skills. Teaching and Teacher Education. 56, pp. 35 - 46. https://doi.org/10.1016/j.tate.2016.02.003
Early career teachers' ability to focus on typical students errors in relation to the complexity of a mathematical topic
Pankow, Lena, Kaiser, Gabriele, Busse, Andreas, Konig, Johannes, Blomeke, Sigrid, Hoth, Jessica and Dohrmann, Martina. (2016). Early career teachers' ability to focus on typical students errors in relation to the complexity of a mathematical topic. 48, pp. 55 - 67. https://doi.org/10.1007/s11858-016-0763-2
Diagnostic competence of primary school mathematics teachers during classroom situations
Hoth, Jessica, Döhrmann, Martina, Kaiser, Gabriele, Busse, Andreas, König, Johannes and Blömeke, Sigrid. (2016). Diagnostic competence of primary school mathematics teachers during classroom situations. ZDM: The International Journal on Mathematics Education. 48(1-2), pp. 41 - 53. https://doi.org/10.1007/s11858-016-0759-y
Early Career Mathematics Teachers’ General Pedagogical Knowledge and Skills: Do Teacher Education, Teaching Experience, and Working Conditions Make a Difference?
König, Johannes, Blömeke, Sigrid and Kaiser, Gabriele. (2015). Early Career Mathematics Teachers’ General Pedagogical Knowledge and Skills: Do Teacher Education, Teaching Experience, and Working Conditions Make a Difference? International Journal of Science and Mathematics Education. 13(2), pp. 331 - 350. https://doi.org/10.1007/s10763-015-9618-5
Effects of motivation on the belief systems of future mathematics teachers from a comparative perspective
Blömeke, Sigrid and Kaiser, Gabriele. (2015). Effects of motivation on the belief systems of future mathematics teachers from a comparative perspective. In In B. Pepin and B. Roesken-Winter (Ed.). From beliefs to dynamic affect systems in mathematics education: Exploring a mosaic of relationships and interactions pp. 227 - 243 Springer International Publishing.
Teacher Change During Induction: Development of Beginning Primary Teachers’ Knowledge, Beliefs and Performance
Blömeke, Sigrid, Hoth, Jessica, Döhrmann, Martina, Busse, Andreas, Kaiser, Gabriele and König, Johannes. (2015). Teacher Change During Induction: Development of Beginning Primary Teachers’ Knowledge, Beliefs and Performance. International Journal of Science and Mathematics Education. 13(2), pp. 287 - 308. https://doi.org/10.1007/s10763-015-9619-4
Modelling competencies: Past development and further perspectives
Kaiser, Gabriele and Brand, Susanne. (2015). Modelling competencies: Past development and further perspectives. In In G. A. Stillman, W. Blum and M. S. Biembengut (Ed.). Mathematical modelling in education research and practice: Cultural, social and cognitive influences pp. 129 - 149 Springer International Publishing.
Scaffolding in complex modelling situations
Stender, Peter and Kaiser, Gabriele. (2015). Scaffolding in complex modelling situations. ZDM: The International Journal on Mathematics Education. 47(7), pp. 1255 - 1267. https://doi.org/10.1007/s11858-015-0741-0
Promotion of mathematical modelling competencies in the context of modelling projects
Kaiser, Gabriele and Grünewald, Susanne. (2015). Promotion of mathematical modelling competencies in the context of modelling projects. In In N. H. Lee and K. E. D. Ng (Ed.). Mathematical modelling from theory to practice pp. 21 - 39 World Scientific Publishing.
About the Complexities of Video-Based Assessments: Theoretical and Methodological Approaches to Overcoming Shortcomings of Research on Teachers’ Competence
Kaiser, Gabriele, Busse, Andreas, Hoth, Jessica, König, Johannes and Blömeke, Sigrid. (2015). About the Complexities of Video-Based Assessments: Theoretical and Methodological Approaches to Overcoming Shortcomings of Research on Teachers’ Competence. International Journal of Science and Mathematics Education. 13(2), pp. 369 - 387. https://doi.org/10.1007/s10763-015-9616-7
Early career mathematics teachers' general pedagogical knowledge and skills: do teacher education, teaching experience, and working conditions make a difference?
Konig, Johannes, Blomeke, Sigrid and Kaiser, Gabriele. (2015). Early career mathematics teachers' general pedagogical knowledge and skills: do teacher education, teaching experience, and working conditions make a difference? International Journal of Science and Mathematics Education. 13(2), pp. 331 - 350. https://doi.org/10.1007/s10763-015-9618-5
Erfassung berufsbezogener kompetenzen von studierenden: Ein kommentar
Kaiser, Gabriele. (2015). Erfassung berufsbezogener kompetenzen von studierenden: Ein kommentar. Zeitschrift fuer Paedagogik. Beiheft. 61, pp. 136 - 143.
Wissen und Fähigkeiten in Fachdidaktik und Pädagogik
Busse, Andreas and Kaiser, Gabriele. (2015). Wissen und Fähigkeiten in Fachdidaktik und Pädagogik. Zeitschrift für Pädagogik. 61(3), pp. 328 - 344.
Preface for the special issue on "video-based research on teacher expertise"
Blömeke, Sigrid, Kaiser, Gabriele and Clarke, David J.. (2015). Preface for the special issue on "video-based research on teacher expertise". International Journal of Science and Mathematics Education. 13(2), pp. 257 - 266. https://doi.org/10.1007/s10763-015-9629-2
The conceptualisation of mathematics competencies in the International Teacher Education Study TEDS-M
Döhrmann, Martina, Kaiser, Gabriele and Blömeke, Sigird. (2014). The conceptualisation of mathematics competencies in the International Teacher Education Study TEDS-M. In In S. Blömeke, F.-J. Hsieh and G. Kaiser & W. H. Schmidt (Ed.). International perspectives on teacher knowledge, beliefs and opportunities to learn: TEDS-M results pp. 431 - 456 Springer. https://doi.org/10.1007/978-94-007-6437-8_20
Resolving the chicken-or-egg causality dilemma: The longitudinal interplay of teacher knowledge and teacher beliefs
Blömeke, Sigrid, Buchholz, Nils, Suhl, Ute and Kaiser, Gabriele. (2014). Resolving the chicken-or-egg causality dilemma: The longitudinal interplay of teacher knowledge and teacher beliefs. Teaching and Teacher Education. 37, pp. 130 - 139. https://doi.org/10.1016/j.tate.2013.10.007
Von der Lehrerausbildung in den Beruf – Fachbezogenes Wissen als Voraussetzung für Wahrnehmung, Interpretation und Handeln im Unterricht
Blömeke, Sigrid, König, Johannes, Busse, Andreas, Suhl, Ute, Benthien, Jessica, Döhrmann, Martina and Kaiser, Gabriele. (2014). Von der Lehrerausbildung in den Beruf – Fachbezogenes Wissen als Voraussetzung für Wahrnehmung, Interpretation und Handeln im Unterricht. Zeitschrift für Erziehungswissenschaft. 17(3), pp. 509 - 542. https://doi.org/10.1007/s11618-014-0564-8
The cultural dimension of beliefs: An investigation of future primary teachers' epistemological beliefs concerning the nature of mathematics in 15 countries
Felbrich, Anja, Kaiser, Gabriele and Schmotz, Christiane. (2014). The cultural dimension of beliefs: An investigation of future primary teachers' epistemological beliefs concerning the nature of mathematics in 15 countries. In In S. Blömeke, F.-J. Hsieh and G. Kaiser & W. H. Schmidt (Ed.). pp. 209 - 229 Springer. https://doi.org/10.1007/978-94-007-6437-8_10
Mathematical modeling in school education: Mathematical, cognitive, curricular, instructional and teacher educational perspectives
Cai, Jinfa, Cirillo, Michelle, Pelesko, John, Bommero Ferri, Rita, Borba, Marcelo, Geiger, Vince, Stillman, Gloria, English, Lyn, Wake, Geoff and Kaiser, Gabriele. (2014). Mathematical modeling in school education: Mathematical, cognitive, curricular, instructional and teacher educational perspectives. In P Liljedahl, S Oesterle and C Nicol (Ed.). Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education. Canada: Springer. pp. 145 - 172
Learning from the Eastern and the Western Debate: The case of mathematics teacher education
Kaiser, Gabriele and Blömeke, Sigrid. (2014). Learning from the Eastern and the Western Debate: The case of mathematics teacher education. In In S. Blömeke, F.-J. Hsieh and G. Kaiser & W. H. Schmidt (Ed.). International perspectives on teacher knowledge, beliefs and opportunities to learn: TEDS-M results pp. 517 - 539 Routledge. https://doi.org/10.1007/978-94-007-6437-8_24
Is teachers' general pedagogical knowledge a premise for noticing and interpreting classroom situations? A video-based assessment approach
König, Johannes, Blömeke, Sigrid, Klein, Patricia, Suhl, Ute, Busse, Andreas and Kaiser, Gabriele. (2014). Is teachers' general pedagogical knowledge a premise for noticing and interpreting classroom situations? A video-based assessment approach. Teaching and Teacher Education. 38, pp. 76 - 88. https://doi.org/10.1016/j.tate.2013.11.004
Family background, entry selectivity and opportunities to learn: What matters in primary teacher education? An international comparison of fifteen countries
Blömeke, Sigrid, Suhl, Ute, Kaiser, Gabriele and Döhrmann, Martina. (2014). Family background, entry selectivity and opportunities to learn: What matters in primary teacher education? An international comparison of fifteen countries. In In S. Blömeke, F.-J. Hsieh and G. Kaiser & W. H. Schmidt (Ed.). International perspectives on teacher knowledge, beliefs and opportunities to learn: TEDS-M results pp. 327 - 353 Routledge. https://doi.org/10.1007/978-94-007-6437-8_15
Overcoming the gap between university and school mathematics: The impact of an innovative program in mathematics teacher education at the Justus-Liebig University in Giessen
Kaiser, Gabriele. (2014). Overcoming the gap between university and school mathematics: The impact of an innovative program in mathematics teacher education at the Justus-Liebig University in Giessen. In In S. Rezat, M. Hattermann and A. Peter-Koop (Ed.). Transformation — A fundamental Idea of mathematics education pp. 85 - 105 Springer. https://doi.org/10.1007/978-1-4614-3489-4_5
Modelling in mathematics classroom instruction an innovative approach for transforming mathematics education
Vorhölter, Kartin, Kaiser, Gabriele and Borromeo Ferri, Rita. (2014). Modelling in mathematics classroom instruction an innovative approach for transforming mathematics education. In In Y. Li, E. A. Silver and S. Li (Ed.). Transforming mathematics instruction: Multiple approaches and practices pp. 21 - 36 Springer. https://doi.org/10.1007/978-3-319-04993-9_3
Teacher Education Effectiveness: Quality and Equity of Future Primary Teachers’ Mathematics and Mathematics Pedagogical Content Knowledge
Blömeke, Sigrid, Suhl, Ute and Kaiser, Gabriele. (2014). Teacher Education Effectiveness: Quality and Equity of Future Primary Teachers’ Mathematics and Mathematics Pedagogical Content Knowledge. In International Perspectives on Teacher Knowledge, Beliefs and Opportunities to Learn: TEDS-M Results pp. 91 - 114 Springer. https://doi.org/10.1007/978-94-007-6437-8_5
Theoretical Framework, Study Design and Main Results of TEDS-M
Blömeke, Sigrid and Kaiser, Gabriele. (2014). Theoretical Framework, Study Design and Main Results of TEDS-M. In International Perspectives on Teacher Knowledge, Beliefs and Opportunities to Learn: TEDS-M Results pp. 19 - 47 Springer. https://doi.org/10.1007/978-94-007-6437-8_2
Homogeneity or Heterogeneity? Profiles of Opportunities to Learn in Primary Teacher Education and Their Relationship to Cultural Context and Outcomes
Blömeke, Sigrid and Kaiser, Gabriele. (2014). Homogeneity or Heterogeneity? Profiles of Opportunities to Learn in Primary Teacher Education and Their Relationship to Cultural Context and Outcomes. In In S. Blömeke, G. Kaiser and F. Hsieh & W. H. Schmidt (Ed.). International Perspectives on Teacher Knowledge, Beliefs and Opportunities to Learn: TEDS-M Results pp. 299 - 325 Springer. https://doi.org/10.1007/978-94-007-6437-8_14
Mathematical modelling: Connecting to teaching and research practices - the impact of globalisation
Stillman, Gloria, Kaiser, Gabriele, Blum, Werner and Brown, Jill P.. (2013). Mathematical modelling: Connecting to teaching and research practices - the impact of globalisation. In In GA Stillman, G Kaiser and W Blum & JP Brown (Ed.). Teaching mathematical modelling: connecting to research and practice pp. 1 - 24 Springer. https://doi.org/10.1007/978-94-007-6540-5_1
Future mathematics teachers' professional knowledge of elementary mathematics from an advanced standpoint
Buchholtz, Nils, Leung, Frederick KS, Ding, Lin, Kaiser, Gabriele, Park, Kyungmee and Schwarz, Bjorn. (2013). Future mathematics teachers' professional knowledge of elementary mathematics from an advanced standpoint. ZDM - International Journal on Mathematics Education. https://doi.org/10.1007/s11858-012-0462-6
Authentic complex modelling problems in mathematics education
Kaiser, Gabriele, Bracke, Martin, Göttlich, Simone and Kaland, Christine. (2013). Authentic complex modelling problems in mathematics education. In In A. Damlamian, J. F. Rodrigues and R. Sträßer (Ed.). Educational interfaces between mathematics and industry: Report on an ICMI-ICIAM-Study pp. 287 - 297 Springer. https://doi.org/10.1007/978-3-319-02270-3_29
Complex modelling problems in co-operative, self-directed learning environments
Kaiser, Gabriele and Stender, Peter. (2013). Complex modelling problems in co-operative, self-directed learning environments. In In G. A. Stillman, G. Kaiser and W. Blum & J. P. Brown (Ed.). pp. 277 - 293 Springer. https://doi.org/10.1007/978-94-007-6540-5_23
Educational interfaces between mathematics and industry at school level: Report on WG 4
Kaiser, Gabriele, van der Kooij, Henk and Wake, Geoff. (2013). Educational interfaces between mathematics and industry at school level: Report on WG 4. In In A. Damlamian, J. F. Rodrigues and R. Sträßer (Ed.). Educational interfaces between mathematics and industry: Report on an ICMI-ICIAM-Study pp. 263 - 270 Springer. https://doi.org/10.1007/978-3-319-02270-3_26
Improving mathematics teacher education in Germany: empirical results from a longitudinal evaluation of innovative programs
Bucholtz, Nils and Kaiser, Gabriele. (2013). Improving mathematics teacher education in Germany: empirical results from a longitudinal evaluation of innovative programs. International Journal of Science and Mathematics Education. 11(4), pp. 949 - 977. https://doi.org/10.1007/s10763-013-9427-7
Preface to Section IV
Kaiser, Gabriele. (2013). Preface to Section IV. In In Y. Li and R. Huang (Ed.). How Chinese Teach Mathematics and Improve Teaching pp. 384 - 390 Routledge. https://doi.org/10.4324/9780203110119
Spiegelt sich pädagogisches Wissen in den Kompetenzselbsteinschätzungen angehender Lehrkräfte?
König, Johannes, Kaiser, Gabriele and Felbrich, Anja. (2012). Spiegelt sich pädagogisches Wissen in den Kompetenzselbsteinschätzungen angehender Lehrkräfte? Zeitschrift für Pädagogik. 58(4), pp. 476 - 491.
CERME7 Working Group 6: Applications and modelling
Kaiser, Gabriele, Carreira, Susana, Lingefjard, Thomas and Wake, Geoff. (2012). CERME7 Working Group 6: Applications and modelling. Research in Mathematics Education. 14(2), pp. 195 - 196. https://doi.org/10.1080/14794802.2012.694287
Reflections and future prospects
Kaiser, Gabriele and Li, Yeping. (2011). Reflections and future prospects. In In Y. Li and G. Kaiser (Ed.). Expertise in mathematics instruction: An international perspective pp. 343 - 353 Springer. https://doi.org/10.1007/978-1-4419-7707-6_16
Equity and the quality of the language used in mathematics education
Schütte, Marcus and Kaiser, Gabriele. (2011). Equity and the quality of the language used in mathematics education. In In B. Atweh, M. Graven and W. Secada & P. Valero (Ed.). Mapping equity and quality in mathematics education pp. 237 - 251 Springer. https://doi.org/10.1007/978-90-481-9803-0_17
Expertise in mathematics instruction: Advancing research and practice from an international perspective
Li, Yeping and Kaiser, Gabriele. (2011). Expertise in mathematics instruction: Advancing research and practice from an international perspective. In In Y. Li and G. Kaiser (Ed.). Expertise in mathematics instruction: An international perspective pp. 3 - 15 Springer. https://doi.org/10.1007/978-1-4419-7707-6_1
Bedingungsfaktoren des fachbezogenen Kompetenzerwerbs von Lehrkraften. Zum Einfluss von Ausbildungs-, Personlichkeits- und Kompositionsmerkmalen in der Mathematiklehrerausbildung fur die Sekundarstufe I
Blömeke, Sigrid, Kaiser, Gabriele and Döhrmann, Martina. (2011). Bedingungsfaktoren des fachbezogenen Kompetenzerwerbs von Lehrkraften. Zum Einfluss von Ausbildungs-, Personlichkeits- und Kompositionsmerkmalen in der Mathematiklehrerausbildung fur die Sekundarstufe I. Zeitschrift für Pädagogik. 57, pp. 77 - 103.
Trends in Teaching and Learning of Mathematical Modelling (ICTMA 14)
Stillman, Gloria, Ferri, Rita Borromeo, Blum, Werner and Kaiser, Gabriele. (2011). Trends in Teaching and Learning of Mathematical Modelling (ICTMA 14) Springer.
Competencies in argumentation and proof :A comparative study of future mathematics teachers from Australia, Germany and Hong Kong
Schwarz, Bjorn, Vale, Colleen, Brown, Jill Patricia, Stillman, Gloria, Kaiser, Gabriele, Buchholtz, Nils and Leung, Issic. (2008). Competencies in argumentation and proof :A comparative study of future mathematics teachers from Australia, Germany and Hong Kong. In Gabriele Kaiser (Ed.). Discussion Group 14: International Comparisons in Mathematics Education. Monterrey, Mexico: International Commission on Mathematical Instruction. pp. 1 - 6
Future teachers' professional knowledge on argumentation and proof: A case study from universities in three countries
Schwarz, Bjorn, Leung, Issic K. C., Bucholtz, Nils, Kaiser, Gabriele, Stillman, Gloria, Brown, Jill Patricia and Vale, Colleen. (2008). Future teachers' professional knowledge on argumentation and proof: A case study from universities in three countries. ZDM. 40(5), pp. 791 - 811. https://doi.org/10.1007/s11858-008-0150-8
The case of mathematical proof in lower secondary school :Knowledge and competencies of pre-service teachers
Brown, Jill Patricia, Stillman, Gloria, Schwarz, Bjorn and Kaiser, Gabriele. (2008). The case of mathematical proof in lower secondary school :Knowledge and competencies of pre-service teachers. In Merrilyn Goos, Ray Brown and Katie Makar (Ed.). Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia. Brisbane, Australia: Mathematics Education Research Group of Australasia. pp. 85 - 91