University of Groningen
“I just do not understand the logic of this”
Bronkhorst, Hugo
DOI:
10.33612/diss.171653189
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Publication date:
2021
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Bronkhorst, H. (2021). “I just do not understand the logic of this”: intervention study aimed at secondary
school students’ development of logical reasoning skills. University of Groningen.
https://doi.org/10.33612/diss.171653189
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Chapter 6
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curriculum reform proposals, we strongly recommend embedding critical thinking
and the corresponding logical reasoning skills much stronger throughout all levels
and years of secondary education. The OECD states that “the Dutch curriculum is
less articulate on this [critical thinking] construct than the CCM countries, …. The
highest difference is observed in ‘Mathematics’” (4% versus 11% of content items;
pp. 45-46). At the same time, the curriculum reform can be an excellent opportunity
to seek collaboration with other school subjects where proper reasoning and
analysing arguments is an essential part too. In the introductory chapter we already
mentioned that the Dutch language and the elective subject philosophy would be
ideal to emphasise cross-curricular components, but the list of possibilities is almost
endless, we mention: history, social studies, geography, other languages, and all
sciences subjects.
All these implications will only be effective if teachers are well prepared. We showed
in our results (see Chapters 4 and 5) that teachers need sufficient support in
organising classroom discussions and their provision of formative feedback, which
is also relevant for other topics and courses. We advised that the Thinking Through
a Lesson Protocol (TTLP; Smith et al., 2008) could support teachers in their
preparations. However, before teachers are able to provide feedback on possible
students’ biased reasoning, they should be able to recognise and prevent biased
reasoning themselves and be able to explain how to avoid flaws in observed
students’ reasoning. Results from Janssen (2020) show that teachers need training on
that and, luckily, can be trained as well. After all, for successful results in
strengthening critical thinking, teachers have a crucial role (Abrami et al., 2008;
Halpern, 1998).
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References
165
References
Abrami, P. C., Bernard, R. M., Borokhovski, E., Wade, A., Surkes, M. A., Tamim, R., & Zhang, D. (2008). Instructional interventions affecting critical thinking skills and dispositions: A stage 1 meta-analysis. Review of Educational Research, 78(4), 1102–1134. https://doi.org/10.3102/ 0034654308326084
Adey, P., & Shayer, M. (1993). An exploration of long-term far-transfer effects following an extended intervention program in the high school science curriculum. Cognition and Instruction, 11(1), 1–29. Adey, P., Shayer, M., & Yates, C. (1995). Thinking science: The curriculum materials of the Cognitive
Acceleration through Science Education (CASE) project (2nd ed.). Thomas Nelson and Sons Ltd. Altrichter, H. (2005). Curriculum implementation–limiting and facilitating factors. In Making it Relevant:
Context Based Learning of Science (pp. 35–62). Münster Waxmann. Aristotle. (2015). Topics. (W. A. Pickard-Cambridge, Trans.). Aeterna Press.
Attridge, N., Aberdein, A., & Inglis, M. (2016). Does studying logic improve logical reasoning? In C. Csíkos, A. Rausch, & J. Szitányi (Eds.), Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 27–34). PME.
Australian Curriculum, Assessment and Reporting Authority. (2016, June 30). Mathematics Foundation to Year 10 Curriculum by rows—The Australian Curriculum v8.2. http://www.australiancurriculum. edu.au/mathematics/curriculum/f-10?layout=1
Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Routledge. Binkley, M., Erstad, O., Herman, J., Raizen, S., Ripley, M., Miller-Ricci, M., & Rumble, M. (2012). Defining
twenty-first century skills. In P. Griffin, B. McGaw, & E. Care (Eds.), Assessment and Teaching of 21st Century Skills (pp. 17–66). Springer.
Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability (Formerly: Journal of Personnel Evaluation in Education), 21(1), 5. https://doi.org/10.1007/s11092-008-9068-5
Blair, J.A. (1999). [Review of the book Argumentation schemes for presumptive reasoning, by D. N. Walton] Argumentation, 13, 338-343.
Blair, J. A., & Johnson, R. H. (2000). Informal logic: An overview. Informal Logic, 20(2), 93–107. Bloem, F. (2018, February 20). Wiskunde C: volwaardig vak. Didactief, 48(3), 32–33.
Bronkhorst, H. (2006). Logica in de bovenbouw van het vwo [Master’s thesis]. University of Groningen. Bronkhorst, H. (2008). Als de eerste rood is, dan zijn ze allemaal rood. Euclides, 83(5), 274.
Bronkhorst, H., Roorda, G., Suhre, C., & Goedhart, M. (2018). Secondary students’ logical reasoning abilities. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, p. 27). PME. Bronkhorst, H., Roorda, G., Suhre, C., & Goedhart, M. (2020a). Logical reasoning in formal and everyday reasoning tasks. International Journal of Science and Mathematics Education, 18(8), 1673–1694. https://doi.org/10.1007/s10763-019-10039-8
Bronkhorst, H., Roorda, G., Suhre, C., & Goedhart, M. (2020b). Students’ use of formalisations for improved logical reasoning [Manuscript submitted for publication]. University of Groningen.
Brookhart, S. M. (2010). How to assess higher-order thinking skills in your classroom. ASCD. Bruner, J. S. (1966). Toward a theory of instruction. Belknap Press of Harvard University Press.
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References
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References
Abrami, P. C., Bernard, R. M., Borokhovski, E., Wade, A., Surkes, M. A., Tamim, R., & Zhang, D. (2008). Instructional interventions affecting critical thinking skills and dispositions: A stage 1 meta-analysis. Review of Educational Research, 78(4), 1102–1134. https://doi.org/10.3102/ 0034654308326084
Adey, P., & Shayer, M. (1993). An exploration of long-term far-transfer effects following an extended intervention program in the high school science curriculum. Cognition and Instruction, 11(1), 1–29. Adey, P., Shayer, M., & Yates, C. (1995). Thinking science: The curriculum materials of the Cognitive
Acceleration through Science Education (CASE) project (2nd ed.). Thomas Nelson and Sons Ltd. Altrichter, H. (2005). Curriculum implementation–limiting and facilitating factors. In Making it Relevant:
Context Based Learning of Science (pp. 35–62). Münster Waxmann. Aristotle. (2015). Topics. (W. A. Pickard-Cambridge, Trans.). Aeterna Press.
Attridge, N., Aberdein, A., & Inglis, M. (2016). Does studying logic improve logical reasoning? In C. Csíkos, A. Rausch, & J. Szitányi (Eds.), Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 27–34). PME.
Australian Curriculum, Assessment and Reporting Authority. (2016, June 30). Mathematics Foundation to Year 10 Curriculum by rows—The Australian Curriculum v8.2. http://www.australiancurriculum. edu.au/mathematics/curriculum/f-10?layout=1
Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Routledge. Binkley, M., Erstad, O., Herman, J., Raizen, S., Ripley, M., Miller-Ricci, M., & Rumble, M. (2012). Defining
twenty-first century skills. In P. Griffin, B. McGaw, & E. Care (Eds.), Assessment and Teaching of 21st Century Skills (pp. 17–66). Springer.
Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability (Formerly: Journal of Personnel Evaluation in Education), 21(1), 5. https://doi.org/10.1007/s11092-008-9068-5
Blair, J.A. (1999). [Review of the book Argumentation schemes for presumptive reasoning, by D. N. Walton] Argumentation, 13, 338-343.
Blair, J. A., & Johnson, R. H. (2000). Informal logic: An overview. Informal Logic, 20(2), 93–107. Bloem, F. (2018, February 20). Wiskunde C: volwaardig vak. Didactief, 48(3), 32–33.
Bronkhorst, H. (2006). Logica in de bovenbouw van het vwo [Master’s thesis]. University of Groningen. Bronkhorst, H. (2008). Als de eerste rood is, dan zijn ze allemaal rood. Euclides, 83(5), 274.
Bronkhorst, H., Roorda, G., Suhre, C., & Goedhart, M. (2018). Secondary students’ logical reasoning abilities. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, p. 27). PME. Bronkhorst, H., Roorda, G., Suhre, C., & Goedhart, M. (2020a). Logical reasoning in formal and everyday reasoning tasks. International Journal of Science and Mathematics Education, 18(8), 1673–1694. https://doi.org/10.1007/s10763-019-10039-8
Bronkhorst, H., Roorda, G., Suhre, C., & Goedhart, M. (2020b). Students’ use of formalisations for improved logical reasoning [Manuscript submitted for publication]. University of Groningen.
Brookhart, S. M. (2010). How to assess higher-order thinking skills in your classroom. ASCD. Bruner, J. S. (1966). Toward a theory of instruction. Belknap Press of Harvard University Press.
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References
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Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: comparing two teaching sequences. Learning Disabilities Research & Practice, 18(2), 99–111. https://doi.org/10.1111/1540-5826.00066
Candela, A. G. (2016). Mathematics teachers’ perspectives on factors affecting the implementation of high cognitive demand tasks. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Proceedings of the 38th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. The University of Arizona.
Carrascal, B. (2011). Teaching logic in philosophy. In P. Blackburn, H. van Ditmarsch, M. Manzano, & F. Soler-Toscano (Eds.), Tools for Teaching Logic (Vol. 6680, pp. 38–45). Springer Berlin Heidelberg. Cerbin, B. (1988). The nature and development of informal reasoning skills in college students (ED298805). ERIC.
https://eric.ed.gov/?id=ED298805
Chasiotis, C. (1996). From common sense to formal logic: Use of logical games for the assessment, investigation and improvement of logical reasoning. In C. Keitel (Ed.), Mathematics (Education) and Common Sense. Proceedings of the 47th CIEAEM Meeting (pp. 440-447). Freie Universität Berlin. Cheng, P. W., Holyoak, K. J., Nisbett, Richard E., & Oliver, L. M. (1986). Pragmatic versus syntactic approaches to training deductive reasoning. Cognitive Psychology, 18(3), 293–328. https://doi.org/10.1016/0010-0285(86)90002-2
Chu, J., Rittle‐Johnson, B., & Fyfe, E. R. (2017). Diagrams benefit symbolic problem-solving. British Journal of Educational Psychology, 87(2), 273–287. https://doi.org/10.1111/bjep.12149
Clarke, D. (1994). Ten key principles from research for the professional development of mathematics teachers. In D. B. Aichele & A. F. Coxfors (Eds.), Professional development for teachers of mathematics (pp. 37–48). NCTM.
Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education (6th ed.). Routledge/Taylor & Francis Group.
Cohen, L., Manion, L., & Morrison, K. (2011). Research methods in education (7th ed). Routledge.
College voor Toetsen en Examens. (2013, June 13). Examen VWO 2013 wiskunde C (pilot), tijdvak 1.
https://www.examenblad.nl/examendocument/2013/cse-1/wiskunde-c-vwo/opgaven-2/2013/vwo/f=/vw-1026-f-13-1-o.pdf
College voor Examens. (2014). FILOSOFIE VWO | syllabus centraal examen 2016 (met eindtermen onderwerp scepticisme). https://www.examenblad.nl/examenstof/syllabus-2016-filosofie-vwo/2016/vwo/f=/ filosofie_vwo_def_versie_2016.pdf
College voor Examens. (2015). NEDERLANDS VWO (4F) | syllabus centraal examen 2016 nader vastgesteld. https://www.examenblad.nl/examenstof/syllabus-2016-nederlands-vwo-nader/2016/vwo/f=/ Nederlands_vwo_versie_2016_nader_vastgesteld.pdf
College voor Toetsen en Examens. (2016). WISKUNDE C VWO | syllabus centraal examen 2018 (Bij het nieuwe examenprogramma) nader vastgesteld 2. https://www.examenblad.nl/examenstof/syllabus -2018-wiskunde-c-vwo/2018/vwo/f=/
syllabus_wiskunde_C_2_versie_vwo_2018_nader_vastgesteld2_def.pdf
College voor Toetsen en Examens. (2017). Wiskunde C (pilot) examen vwo 2017. https://static. examenblad.nl/9336117/d/ex2017/VW-1026-f-17-1-o.pdf.
Cook, T. D., & Campbell, D. T. (1979). Quasi-experimentation: Design & analysis issues for field settings. Rand McNally College Pub. Co.
cTWO. (2012). Denken & doen: Wiskunde op de havo en vwo per 2015. cTWO. https://www.fi.uu.nl/ctwo/ publicaties/ docs/CTWO-Eindrapport.pdf
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-wiskunde/uitwerking-rekenen-wiskunde/
Curriculum.nu – Wat moeten onze leerlingen kennen en kunnen? (2020, September 30). https://www. curriculum.nu/
Daniel, D. B., & Klaczynski, P. A. (2006). Developmental and individual differences in conditional reasoning: Effects of logic instructions and alternative antecedents. Child Development, 77(2), 339– 354. https://doi.org/10.1111/j.1467-8624.2006.00874.x
Davidson, N., & Kroll, D. L. (1991). An overview of research on cooperative learning related to mathematics. Journal for Research in Mathematics Education, 22(5), 362–365. https:// doi.org/10.2307/749185
De Lange, J. (1998). Wiskunde C&M. Schets van een mogelijke invulling van het vak wiskunde binnen het Profiel Cultuur en Maatschappij. (p. 59). Freudenthal Instituut.
Denscombe, M. (2014). The good research guide: For small-scale social research projects (5. ed). Open University Press.
Department of Education UK. (2014, July 16). National curriculum in England: Mathematics programmes of study - GOV.UK. https://www.gov.uk/government/publications/national-curriculum-in-england- mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study
De Pater, W. A., & Vergauwen, R. (2005). Logica: formeel en informeel. Universitaire pers.
Dienst Uitvoering Onderwijs. (2018). Examenmonitor VO 2018. Ministerie van Onderwijs, Cultuur en Wetenschap.
Dienst Uitvoering Onderwijs. (2020). Examenmonitor VO 2019. Ministerie van Onderwijs, Cultuur en Wetenschap.
Dijkhuis, J., Admiraal, C. J., Verbeek, J. A., De Jong, G., Houwing, H., Kuis, J. D., Ten Klooster, F., De Waal, S., Van Braak, J., Liesting-Maas, J. H. M., Wieringa, M., Van Maarseveen, M., Hiele, R. D., Romkes, R. D., Haneveld, M., Voets, S., & Cornelisse, I. (2017). Getal & Ruimte 11e editie vwo C deel 4. Noordhoff Uitgevers.
Doorman, M., & Roodhardt, A. (2011). Wiskunde: Meer een filosofie dan wetenschap? Nieuwe Wiskrant, 30(4), 43–48.
Draper, S. (2019, February 17). Effect size. http://www.psy.gla.ac.uk/~steve/best/effect.html
Durand-Guerrier, V. (2003). Which notion of implication is the right one? From logical considerations to a didactic perspective. Educational Studies in Mathematics, 53(1), 5.
EP-Nuffic. (2015). Education system The Netherlands. https://www.epnuffic.nl/en/publications/find-a-publication/education-system-the-netherlands.pdf
European Union. (2002). Key competencies: A developing concept in general compulsory education. Eurydice. Evans, J. St. B. T. (2002). Logic and human reasoning: An assessment of the deduction paradigm.
Psychological Bulletin, 128(6), 978–996. https://doi.org/10.1037/0033-2909.128.6.978
Febriana, D. F., Amin, S. M., & Wijayanti, P. (2019). Concreteness fading process of elementary school students based on mathematical ability. Journal of Physics: Conference Series, 1157(4). https://doi.org/10.1088/1742-6596/1157/4/042049
Folmer, E., Kuiper, W., Bruning, L., & Ottevanger, W. (2012). Evaluatie examenpilot wiskunde C vwo 2009-2012. SLO.
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References
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Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: comparing two teaching sequences. Learning Disabilities Research & Practice, 18(2), 99–111. https://doi.org/10.1111/1540-5826.00066
Candela, A. G. (2016). Mathematics teachers’ perspectives on factors affecting the implementation of high cognitive demand tasks. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Proceedings of the 38th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. The University of Arizona.
Carrascal, B. (2011). Teaching logic in philosophy. In P. Blackburn, H. van Ditmarsch, M. Manzano, & F. Soler-Toscano (Eds.), Tools for Teaching Logic (Vol. 6680, pp. 38–45). Springer Berlin Heidelberg. Cerbin, B. (1988). The nature and development of informal reasoning skills in college students (ED298805). ERIC.
https://eric.ed.gov/?id=ED298805
Chasiotis, C. (1996). From common sense to formal logic: Use of logical games for the assessment, investigation and improvement of logical reasoning. In C. Keitel (Ed.), Mathematics (Education) and Common Sense. Proceedings of the 47th CIEAEM Meeting (pp. 440-447). Freie Universität Berlin. Cheng, P. W., Holyoak, K. J., Nisbett, Richard E., & Oliver, L. M. (1986). Pragmatic versus syntactic approaches to training deductive reasoning. Cognitive Psychology, 18(3), 293–328. https://doi.org/10.1016/0010-0285(86)90002-2
Chu, J., Rittle‐Johnson, B., & Fyfe, E. R. (2017). Diagrams benefit symbolic problem-solving. British Journal of Educational Psychology, 87(2), 273–287. https://doi.org/10.1111/bjep.12149
Clarke, D. (1994). Ten key principles from research for the professional development of mathematics teachers. In D. B. Aichele & A. F. Coxfors (Eds.), Professional development for teachers of mathematics (pp. 37–48). NCTM.
Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education (6th ed.). Routledge/Taylor & Francis Group.
Cohen, L., Manion, L., & Morrison, K. (2011). Research methods in education (7th ed). Routledge.
College voor Toetsen en Examens. (2013, June 13). Examen VWO 2013 wiskunde C (pilot), tijdvak 1.
https://www.examenblad.nl/examendocument/2013/cse-1/wiskunde-c-vwo/opgaven-2/2013/vwo/f=/vw-1026-f-13-1-o.pdf
College voor Examens. (2014). FILOSOFIE VWO | syllabus centraal examen 2016 (met eindtermen onderwerp scepticisme). https://www.examenblad.nl/examenstof/syllabus-2016-filosofie-vwo/2016/vwo/f=/ filosofie_vwo_def_versie_2016.pdf
College voor Examens. (2015). NEDERLANDS VWO (4F) | syllabus centraal examen 2016 nader vastgesteld. https://www.examenblad.nl/examenstof/syllabus-2016-nederlands-vwo-nader/2016/vwo/f=/ Nederlands_vwo_versie_2016_nader_vastgesteld.pdf
College voor Toetsen en Examens. (2016). WISKUNDE C VWO | syllabus centraal examen 2018 (Bij het nieuwe examenprogramma) nader vastgesteld 2. https://www.examenblad.nl/examenstof/syllabus -2018-wiskunde-c-vwo/2018/vwo/f=/
syllabus_wiskunde_C_2_versie_vwo_2018_nader_vastgesteld2_def.pdf
College voor Toetsen en Examens. (2017). Wiskunde C (pilot) examen vwo 2017. https://static. examenblad.nl/9336117/d/ex2017/VW-1026-f-17-1-o.pdf.
Cook, T. D., & Campbell, D. T. (1979). Quasi-experimentation: Design & analysis issues for field settings. Rand McNally College Pub. Co.
cTWO. (2012). Denken & doen: Wiskunde op de havo en vwo per 2015. cTWO. https://www.fi.uu.nl/ctwo/ publicaties/ docs/CTWO-Eindrapport.pdf
References
167 Curriculum.nu. (2019). Leergebied rekenen & wiskunde. https://www.curriculum.nu/voorstellen/rekenen
-wiskunde/uitwerking-rekenen-wiskunde/
Curriculum.nu – Wat moeten onze leerlingen kennen en kunnen? (2020, September 30). https://www. curriculum.nu/
Daniel, D. B., & Klaczynski, P. A. (2006). Developmental and individual differences in conditional reasoning: Effects of logic instructions and alternative antecedents. Child Development, 77(2), 339– 354. https://doi.org/10.1111/j.1467-8624.2006.00874.x
Davidson, N., & Kroll, D. L. (1991). An overview of research on cooperative learning related to mathematics. Journal for Research in Mathematics Education, 22(5), 362–365. https:// doi.org/10.2307/749185
De Lange, J. (1998). Wiskunde C&M. Schets van een mogelijke invulling van het vak wiskunde binnen het Profiel Cultuur en Maatschappij. (p. 59). Freudenthal Instituut.
Denscombe, M. (2014). The good research guide: For small-scale social research projects (5. ed). Open University Press.
Department of Education UK. (2014, July 16). National curriculum in England: Mathematics programmes of study - GOV.UK. https://www.gov.uk/government/publications/national-curriculum-in-england- mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study
De Pater, W. A., & Vergauwen, R. (2005). Logica: formeel en informeel. Universitaire pers.
Dienst Uitvoering Onderwijs. (2018). Examenmonitor VO 2018. Ministerie van Onderwijs, Cultuur en Wetenschap.
Dienst Uitvoering Onderwijs. (2020). Examenmonitor VO 2019. Ministerie van Onderwijs, Cultuur en Wetenschap.
Dijkhuis, J., Admiraal, C. J., Verbeek, J. A., De Jong, G., Houwing, H., Kuis, J. D., Ten Klooster, F., De Waal, S., Van Braak, J., Liesting-Maas, J. H. M., Wieringa, M., Van Maarseveen, M., Hiele, R. D., Romkes, R. D., Haneveld, M., Voets, S., & Cornelisse, I. (2017). Getal & Ruimte 11e editie vwo C deel 4. Noordhoff Uitgevers.
Doorman, M., & Roodhardt, A. (2011). Wiskunde: Meer een filosofie dan wetenschap? Nieuwe Wiskrant, 30(4), 43–48.
Draper, S. (2019, February 17). Effect size. http://www.psy.gla.ac.uk/~steve/best/effect.html
Durand-Guerrier, V. (2003). Which notion of implication is the right one? From logical considerations to a didactic perspective. Educational Studies in Mathematics, 53(1), 5.
EP-Nuffic. (2015). Education system The Netherlands. https://www.epnuffic.nl/en/publications/find-a-publication/education-system-the-netherlands.pdf
European Union. (2002). Key competencies: A developing concept in general compulsory education. Eurydice. Evans, J. St. B. T. (2002). Logic and human reasoning: An assessment of the deduction paradigm.
Psychological Bulletin, 128(6), 978–996. https://doi.org/10.1037/0033-2909.128.6.978
Febriana, D. F., Amin, S. M., & Wijayanti, P. (2019). Concreteness fading process of elementary school students based on mathematical ability. Journal of Physics: Conference Series, 1157(4). https://doi.org/10.1088/1742-6596/1157/4/042049
Folmer, E., Kuiper, W., Bruning, L., & Ottevanger, W. (2012). Evaluatie examenpilot wiskunde C vwo 2009-2012. SLO.
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References
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Franks, B. A., Therriault, D. J., Buhr, M. I., Chiang, E. S., Gonzalez, C. M., Kwon, H. K., Schelbe, J. L., & Wang, X. (2013). Looking back: reasoning and metacognition with narrative texts. Metacognition and Learning, 8(2), 145–171.
Fyfe, E. R., McNeil, N. M., & Borjas, S. (2015). Benefits of “concreteness fading” for children’s mathematics understanding. Learning and Instruction, 35, 104–120. https://doi.org/10.1016/ j.learninstruc.2014.10.004
Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: a systematic review. Educational Psychology Review, 26(1), 9–25. https://doi.org/10.1007/s10648-014-9249-3
Gademan, J., & Tolboom, J. (2018, March). Hoe verloopt de invoering van Wiskunde C? Euclides, 93(5), 18–21.
Galotti, K. M. (1989). Approaches to studying formal and everyday reasoning. Psychological Bulletin, 105(3), 331–351.
Galotti, K. M. (2017). Cognitive development: infancy through adolescence (2nd ed.). Sage.
Geerlings, J. (2008, April). Werkstuk vwo-5 wa1/wa1,2 “Logisch redeneren.” http://www.fi.uu.nl/ctwo/ lesmateriaaldir/ExperimenteelLesmateriaal/VWO%20Wiskunde%20C/Logisch%20Redeneren/O ud%20materiaal/toetsideeen/20080404_Gregorius_WerkstukLogica_vwo5_a1_a12.doc
Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). Erlbaum.
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177. https://doi.org/10.1207/s15327833mtl0102_4 Gravemeijer, K. (2020). A socio-constructivist elaboration of realistic mathematics education. In M. V. den
Heuvel-Panhuizen (Ed.), National reflections on the Netherlands didactics of mathematics: Teaching and learning in the context of realistic mathematics education (pp. 217–233). Springer International Publishing. https://doi.org/10.1007/978-3-030-33824-4
Grossen, B. (1991). The fundamental skills of higher order thinking. Journal of Learning Disabilities, 24(6), 343–353.
Grouws, D. A., & Cebulla, K. J. (2000). Improving student achievement in mathematics. Educational practices series 4 (ED445925). ERIC. https://eric.ed.gov/?id=ED445925
Haber, J. (2020). Critical thinking. MIT Press.
Halpern, D. F. (1998). Teaching critical thinking for transfer across domains. Dispositions, skills, structure training, and metacognitive monitoring. American Psychologist, 53(4), 449–455.
Halpern, D. F. (2014). Thought and knowledge: An introduction to critical thinking (Fifth edition). Psychology Press.
Hansen, H., & Cohen, D. (2011). Are there methods of informal logic? In F. Zenker (Ed.), Argumentation: Cognition and Community. Proceedings of the 9th International Conference of the Ontario Society for the Study of Argumentation (CD ROM, pp. 1-13). OSSA, Windsor.
Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Corwin.
Hayes, A. F., & Krippendorff, K. (2007). Answering the call for a standard reliability measure for coding data. Communication Methods and Measures, 1(1), 77–89. https://doi.org/10.1080/ 19312450709336664
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169 Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684–689. https://doi.org/10.1037/0022-0663.91.4.684
Hintikka, J. (2001). Is logic the key to all good reasoning? Argumentation, 15(1), 35–57.
Hitchcock, D. (2018). Critical Thinking. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2020 ed.). https://plato.stanford.edu/archives/fall2020/entries/critical-thinking/
Hof: Wilders moet worden vervolgd. (2009, January 21). NRC Handelsblad. https://www.nrc.nl/nieuws/ 2009/01/21/hof-wilders-moet-worden-vervolgd-11671820-a414693
Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6(2), 65–70. JSTOR.
Hoyles, C., & Küchemann, D. (2002). Students’ understandings of logical implication. Educational Studies in Mathematics, 51(3), 193-223.
IAE. (2010). Pedagogy and ICT use in schools around the world: Findings from the IEA sites 2006 study. (N. Law, W. J. Pelgrum, & T. Plomp, Eds.). Springer.
Inglis, M., & Simpson, A. (2006). The role of mathematical context in evaluating conditional statements. Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 3, 337–343.
Jäder, J., Sidenvall, J., & Sumpter, L. (2017). Students’ mathematical reasoning and beliefs in non -routine task solving. International Journal of Science and Mathematics Education, 15(4), 759–776. https://doi.org/10.1007/s10763-016-9712-3
Janssen, E. (2020). Teaching critical thinking in higher education. Avoiding, detecting, and explaining bias in reasoning [Doctoral dissertation]. Utrecht University. https://doi.org/10.33540/351
Johnson, R. H., & Blair, J. A. (2006). Logical self-defence. Central European Uni Press. Kahneman, D. (2016). Ons feilbare denken. Business Contact.
Keijzer, R., & Terwel, J. (2003). Kansen voor formaliseren als wiskundige activiteit in het basisonderwijs. Nieuwe wiskrant, 23(1), 25–28.
Kim, H. (2020). Concreteness fading strategy: a promising and sustainable instructional model in mathematics classrooms. Sustainability, 12(6), 2211. https://doi.org/10.3390/su12062211
Klein Kranenbarg, L. M. (2020). Het beeld van wiskunde C (bijgesteld). [Master’s thesis]. University of Twente. http://purl.utwente.nl/essays/80606
KNAW. (2009). Rekenonderwijs op de basisschool: Analyse en sleutels tot verbetering: advies (p. 113). Koninklijke Nederlandse Akademie van Wetenschappen.
Koelewijn, R. (2016). Het begint bij een glas per dag. NRC Media. https://www.nrc.nl/nieuws/2016/10/04/het-begint-bij-een-glas-per-dag-4637333-a1524797.
Kuhn, D. (1991). The skills of argument. Cambridge University Press.
Kuiper, W., Van der Hoeven, M., Folmer, E., Van Graft, M., & Van den Akker, J. (2010). Leerplankundige analyse van PISA-trends. SLO.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. J. Worrall & E. Zahar (Eds.). Cambridge University Press.
Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4. https://doi.org/10.3389/ fpsyg.2013.00863
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Franks, B. A., Therriault, D. J., Buhr, M. I., Chiang, E. S., Gonzalez, C. M., Kwon, H. K., Schelbe, J. L., & Wang, X. (2013). Looking back: reasoning and metacognition with narrative texts. Metacognition and Learning, 8(2), 145–171.
Fyfe, E. R., McNeil, N. M., & Borjas, S. (2015). Benefits of “concreteness fading” for children’s mathematics understanding. Learning and Instruction, 35, 104–120. https://doi.org/10.1016/ j.learninstruc.2014.10.004
Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: a systematic review. Educational Psychology Review, 26(1), 9–25. https://doi.org/10.1007/s10648-014-9249-3
Gademan, J., & Tolboom, J. (2018, March). Hoe verloopt de invoering van Wiskunde C? Euclides, 93(5), 18–21.
Galotti, K. M. (1989). Approaches to studying formal and everyday reasoning. Psychological Bulletin, 105(3), 331–351.
Galotti, K. M. (2017). Cognitive development: infancy through adolescence (2nd ed.). Sage.
Geerlings, J. (2008, April). Werkstuk vwo-5 wa1/wa1,2 “Logisch redeneren.” http://www.fi.uu.nl/ctwo/ lesmateriaaldir/ExperimenteelLesmateriaal/VWO%20Wiskunde%20C/Logisch%20Redeneren/O ud%20materiaal/toetsideeen/20080404_Gregorius_WerkstukLogica_vwo5_a1_a12.doc
Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). Erlbaum.
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177. https://doi.org/10.1207/s15327833mtl0102_4 Gravemeijer, K. (2020). A socio-constructivist elaboration of realistic mathematics education. In M. V. den
Heuvel-Panhuizen (Ed.), National reflections on the Netherlands didactics of mathematics: Teaching and learning in the context of realistic mathematics education (pp. 217–233). Springer International Publishing. https://doi.org/10.1007/978-3-030-33824-4
Grossen, B. (1991). The fundamental skills of higher order thinking. Journal of Learning Disabilities, 24(6), 343–353.
Grouws, D. A., & Cebulla, K. J. (2000). Improving student achievement in mathematics. Educational practices series 4 (ED445925). ERIC. https://eric.ed.gov/?id=ED445925
Haber, J. (2020). Critical thinking. MIT Press.
Halpern, D. F. (1998). Teaching critical thinking for transfer across domains. Dispositions, skills, structure training, and metacognitive monitoring. American Psychologist, 53(4), 449–455.
Halpern, D. F. (2014). Thought and knowledge: An introduction to critical thinking (Fifth edition). Psychology Press.
Hansen, H., & Cohen, D. (2011). Are there methods of informal logic? In F. Zenker (Ed.), Argumentation: Cognition and Community. Proceedings of the 9th International Conference of the Ontario Society for the Study of Argumentation (CD ROM, pp. 1-13). OSSA, Windsor.
Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Corwin.
Hayes, A. F., & Krippendorff, K. (2007). Answering the call for a standard reliability measure for coding data. Communication Methods and Measures, 1(1), 77–89. https://doi.org/10.1080/ 19312450709336664
References
169 Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684–689. https://doi.org/10.1037/0022-0663.91.4.684
Hintikka, J. (2001). Is logic the key to all good reasoning? Argumentation, 15(1), 35–57.
Hitchcock, D. (2018). Critical Thinking. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2020 ed.). https://plato.stanford.edu/archives/fall2020/entries/critical-thinking/
Hof: Wilders moet worden vervolgd. (2009, January 21). NRC Handelsblad. https://www.nrc.nl/nieuws/ 2009/01/21/hof-wilders-moet-worden-vervolgd-11671820-a414693
Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6(2), 65–70. JSTOR.
Hoyles, C., & Küchemann, D. (2002). Students’ understandings of logical implication. Educational Studies in Mathematics, 51(3), 193-223.
IAE. (2010). Pedagogy and ICT use in schools around the world: Findings from the IEA sites 2006 study. (N. Law, W. J. Pelgrum, & T. Plomp, Eds.). Springer.
Inglis, M., & Simpson, A. (2006). The role of mathematical context in evaluating conditional statements. Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 3, 337–343.
Jäder, J., Sidenvall, J., & Sumpter, L. (2017). Students’ mathematical reasoning and beliefs in non -routine task solving. International Journal of Science and Mathematics Education, 15(4), 759–776. https://doi.org/10.1007/s10763-016-9712-3
Janssen, E. (2020). Teaching critical thinking in higher education. Avoiding, detecting, and explaining bias in reasoning [Doctoral dissertation]. Utrecht University. https://doi.org/10.33540/351
Johnson, R. H., & Blair, J. A. (2006). Logical self-defence. Central European Uni Press. Kahneman, D. (2016). Ons feilbare denken. Business Contact.
Keijzer, R., & Terwel, J. (2003). Kansen voor formaliseren als wiskundige activiteit in het basisonderwijs. Nieuwe wiskrant, 23(1), 25–28.
Kim, H. (2020). Concreteness fading strategy: a promising and sustainable instructional model in mathematics classrooms. Sustainability, 12(6), 2211. https://doi.org/10.3390/su12062211
Klein Kranenbarg, L. M. (2020). Het beeld van wiskunde C (bijgesteld). [Master’s thesis]. University of Twente. http://purl.utwente.nl/essays/80606
KNAW. (2009). Rekenonderwijs op de basisschool: Analyse en sleutels tot verbetering: advies (p. 113). Koninklijke Nederlandse Akademie van Wetenschappen.
Koelewijn, R. (2016). Het begint bij een glas per dag. NRC Media. https://www.nrc.nl/nieuws/2016/10/04/het-begint-bij-een-glas-per-dag-4637333-a1524797.
Kuhn, D. (1991). The skills of argument. Cambridge University Press.
Kuiper, W., Van der Hoeven, M., Folmer, E., Van Graft, M., & Van den Akker, J. (2010). Leerplankundige analyse van PISA-trends. SLO.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. J. Worrall & E. Zahar (Eds.). Cambridge University Press.
Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4. https://doi.org/10.3389/ fpsyg.2013.00863
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Ledoux, G., Meijer, J., Van der Veen, I., & Breetvelt, I. (2013). Meetinstrumenten voor sociale competenties, metacognitie en advanced skills. Kohnstamm Instituut.
Lehman, D. R., Lempert, R. O., & Nisbett, R. E. (1988). The effects of graduate training on reasoning: Formal discipline and thinking about everyday-life events. American Psychologist, 43(6), 431–442. https://doi.org/10.1037/0003-066X.43.6.431
Lehman, D. R., & Nisbett, R. E. (1990). A longitudinal study of the effects of undergraduate training on reasoning. Developmental Psychology, 26(6), 952–960. https://doi.org/10.1037/0012-1649.26.6.952 Liu, H., Ludu, M., & Holton, D. (2015). Can K-12 math teachers train students to make valid logical
reasoning? In X. Ge, D. Ifenthaler, & J. M. Spector (Eds.), Emerging technologies for STEAM education: Full STEAM ahead (pp. 331–353). Springer International Publishing. https://doi.org/ 10.1007/978-3-319-02573-5_18
Look, B. C. (2013). Gottfried Wilhelm Leibniz. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Spring 2014 ed.). http://plato.stanford.edu/archives/spr2014/entries/leibniz/
McChesney, J. (2017). Searching the New Zealand curriculum landscape for clarity and coherence: Some tensions in mathematics and statistics. Curriculum Matters, 13, 115–131.
McKendree, J., Small, C., Stenning, K., & Conlon, T. (2002). The role of representation in teaching and learning critical thinking. Educational Review, 54(1), 57–67. https://doi.org/10.1080/ 00131910120110884
McNeil, N. M., & Fyfe, E. R. (2012). “Concreteness fading” promotes transfer of mathematical knowledge. Learning and Instruction, 22(6), 440–448. https://doi.org/10.1016/j.learninstruc.2012.05.001 Milbou, L., Deprez, J., & Laenens, E. (2013). A study on the reintroduction of logic in secondary schools .
In Proceedings of the International Conference on The Future of Education (3rd edition).
Moshman, D. (1996). The development of metalogical understanding. In L. Smith (Ed.), Critical Readings on Piaget. Routledge.
National Academies of Sciences, Engineering, and Medicine. (2018). How people learn II: learners, contexts, and cultures. The National Academies Press. https://doi.org/10.17226/24783
National Research Council. (1999). How people learn: Brain, mind, experience, and school (J. Bransford, A. L. Brown, & R. R. Cocking, Eds.). National Academies Press.
National Research Council. (2001). Adding it up: Helping children learn mathematics (B. Findell, J. Swafford, & J. Kilpatrick, Eds.). National Academy Press.
NCTM. (2009). Focus in high school mathematics. Reasoning and sense making. National Council of Teachers of Mathematics.
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NGA Center and CCSSO. (2016). Common core state standards for mathematics. http://www. corestandards.org/wp-content/uploads/Math_Standards1.pdf
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Nunes, T. (2012). Logical reasoning and learning. In N. M. Seel (Ed.), Encyclopedia of the sciences of learning (pp. 2066–2069). Springer. https://doi.org/10.1007/978-1-4419-1428-6_790
O’Brien, T. C., Shapiro, B. J., & Reali, N. C. (1971). Logical thinking—Language and context. Educational Studies in Mathematics, 4(2), 201–219.
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Ledoux, G., Meijer, J., Van der Veen, I., & Breetvelt, I. (2013). Meetinstrumenten voor sociale competenties, metacognitie en advanced skills. Kohnstamm Instituut.
Lehman, D. R., Lempert, R. O., & Nisbett, R. E. (1988). The effects of graduate training on reasoning: Formal discipline and thinking about everyday-life events. American Psychologist, 43(6), 431–442. https://doi.org/10.1037/0003-066X.43.6.431
Lehman, D. R., & Nisbett, R. E. (1990). A longitudinal study of the effects of undergraduate training on reasoning. Developmental Psychology, 26(6), 952–960. https://doi.org/10.1037/0012-1649.26.6.952 Liu, H., Ludu, M., & Holton, D. (2015). Can K-12 math teachers train students to make valid logical
reasoning? In X. Ge, D. Ifenthaler, & J. M. Spector (Eds.), Emerging technologies for STEAM education: Full STEAM ahead (pp. 331–353). Springer International Publishing. https://doi.org/ 10.1007/978-3-319-02573-5_18
Look, B. C. (2013). Gottfried Wilhelm Leibniz. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Spring 2014 ed.). http://plato.stanford.edu/archives/spr2014/entries/leibniz/
McChesney, J. (2017). Searching the New Zealand curriculum landscape for clarity and coherence: Some tensions in mathematics and statistics. Curriculum Matters, 13, 115–131.
McKendree, J., Small, C., Stenning, K., & Conlon, T. (2002). The role of representation in teaching and learning critical thinking. Educational Review, 54(1), 57–67. https://doi.org/10.1080/ 00131910120110884
McNeil, N. M., & Fyfe, E. R. (2012). “Concreteness fading” promotes transfer of mathematical knowledge. Learning and Instruction, 22(6), 440–448. https://doi.org/10.1016/j.learninstruc.2012.05.001 Milbou, L., Deprez, J., & Laenens, E. (2013). A study on the reintroduction of logic in secondary schools .
In Proceedings of the International Conference on The Future of Education (3rd edition).
Moshman, D. (1996). The development of metalogical understanding. In L. Smith (Ed.), Critical Readings on Piaget. Routledge.
National Academies of Sciences, Engineering, and Medicine. (2018). How people learn II: learners, contexts, and cultures. The National Academies Press. https://doi.org/10.17226/24783
National Research Council. (1999). How people learn: Brain, mind, experience, and school (J. Bransford, A. L. Brown, & R. R. Cocking, Eds.). National Academies Press.
National Research Council. (2001). Adding it up: Helping children learn mathematics (B. Findell, J. Swafford, & J. Kilpatrick, Eds.). National Academy Press.
NCTM. (2009). Focus in high school mathematics. Reasoning and sense making. National Council of Teachers of Mathematics.
Nesmith, S. J. (2008). Mathematics and literature: Educators’ perspectives on utilizing a reformative approach to bridge two cultures. Forum on Public Policy Online, 2008(2) (EJ1099543). ERIC. https://eric.ed.gov/?id=EJ1099543
NGA Center and CCSSO. (2016). Common core state standards for mathematics. http://www. corestandards.org/wp-content/uploads/Math_Standards1.pdf
Nieveen, N., & Folmer, E. (2013). Formative evaluation in educational design research. In Educational Design Research / Part A: An introduction. SLO.
Nunes, T. (2012). Logical reasoning and learning. In N. M. Seel (Ed.), Encyclopedia of the sciences of learning (pp. 2066–2069). Springer. https://doi.org/10.1007/978-1-4419-1428-6_790
O’Brien, T. C., Shapiro, B. J., & Reali, N. C. (1971). Logical thinking—Language and context. Educational Studies in Mathematics, 4(2), 201–219.
References
171 OECD. (2019a). Education 2030 Curriculum content mapping: An analysis of the Netherlands curriculum proposal. OECD Publishing. https://www.oecd.org/education/2030-project/contact/ E2030_CCM_analysis_NLD_curriculum_proposal. pdf
OECD. (2019b). PISA 2018 results: What students know and can do. OECD Publishing.
Ottmar, E., & Landy, D. (2017). Concreteness fading of algebraic instruction: effects on learning. Journal of the Learning Sciences, 26(1), 51–78. https://doi.org/10.1080/10508406.2016.1250212
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Traditional teaching methods still dominant in maths classrooms. (2012, June 9). https://www.manchester.ac.uk/discover/news/traditional-teaching-methods-still-dominant-in-maths-classrooms/
UNESCO. (2014). UNESCO roadmap for implementing the global action programme on education for sustainable development. United Nations Educational, Scientific and Cultural Organization. https://unesdoc.unesco.org/ark:/48223/pf0000230514
Van Aalten, P., & De Waard, K. (2001). Denklessen. APS.
Van Bergen, R. (2010). Logica binnen wiskunde C. Nieuwe Wiskrant, 30(2), 41–46.
Van den Akker, J., Bannan, B., Kelly, A. E., Nieveen, N., & Plomp, T. (2013). Educational design research. Part A: An introduction. SLO.
Van Eemeren, F. H., Garssen, B., Krabbe, E. C. W., Henkemans, A. F. S., Verheij, B., & Wagemans, J. H. M. (2014). Informal logic. In Handbook of argumentation theory (pp. 373–423). Springer Netherlands. Van Gelder, T. (2005). Teaching critical thinking: Some lessons from cognitive science. College Teaching,
53(1), 41–48.
Van Peppen, L. M. (2020). Fostering critical thinking: Generative processing strategies to learn to avoid bias in reasoning [Doctoral dissertation]. Erasmus University Rotterdam. http://hdl.handle.net/ 1765/130461
Van Someren, M. W., Barnard, Y. F., & Sandberg, J. A. C. (1994). The think aloud method: A practical guide to modelling cognitive processes. Academic Press.
Vincent-Lancrin, S., González-Sancho, C., Bouckaert, M., Luca, F. de, Fernández-Barrerra, M., Jacotin, G., Urgel, J., & Vidal, Q. (2019). Fostering students’ creativity and critical thinking: What it means in school. Centre for Educational Research and Innovation. https://www.oecd-ilibrary.org/conten t/ publication/62212c37-en
Voss, J. F., Perkins, D. N., & Segal, J. W. (Eds.). (1991). Informal reasoning and education. L. Erlbaum Associates.
Vygotskiĭ, L. S. (1978). Mind in society: The development of higher psychological processes (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Eds.). Harvard University Press.
Wagner-Egger, P. (2007). Conditional reasoning and the Wason selection task: Biconditional interpretation instead of reasoning bias. Thinking and Reasoning, 13(4), 484–505. https://doi.org/10.1080/13546780701415979
Walton, D. N. (1996). Argumentation schemes for presumptive reasoning. Erlbaum.
Walton, D. N., Reed, C., & Macagno, F. (2008). Argumentation schemes. Cambridge University Press. Wason, P. C. (1968). Reasoning about a rule. The Quarterly Journal of Experimental Psychology, 20(3), 273–
281.
Witzel, B., Riccomini, P., & Schneider, E. (2008). Implementing CRA with secondary students with learning disabilities in mathematics. Intervention in School and Clinic, 43(5), 270–276. https://doi.org/10.1177/1053451208314734
Yackel, E., Cobb, P., & Wood, T. (1991). Small-group interactions as a source of learning opportunities in second-grade mathematics. Journal for Research in Mathematics Education, 22(5), 390–408. https://doi.org/10.2307/749187
References
173 Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics. National Council of Teachers of Mathematics.
Zalta, E. N. (2016). Gottlob Frege. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Winter 2016 ed.). http://plato.stanford.edu/archives/win2016/entries/frege/
Zohar, A., & Dori, Y. J. (2003). Higher order thinking skills and low-achieving students: Are they mutually exclusive? Journal of the Learning Sciences, 12(2), 145–181.
Zohar, A. (2006). The nature and development of teachers’ metastrategic knowledge in the context of teaching higher order thinking. Journal of the Learning Sciences, 15(3), 331–377. https://doi.org/10.1207/s15327809jls1503_2
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Samenvatting
(in Dutch)
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Samenvatting
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Introductie
Sinds de herziening van de curricula voor het wiskundeonderwijs in 2015 in
Nederland is er bij wiskunde C voor het profiel Cultuur & Maatschappij (C&M) op
het vwo een apart domein logisch redeneren opgenomen. Het C&M-profiel bereidt
leerlingen voor op universitaire studies in de sectoren Taal en Cultuur, de sector
Recht en de sectoren Gedrag en Maatschappij (SLO, 2015). Het programma
wiskunde C is hierop aangepast met profielspecifieke onderdelen die wiskunde in
verband brengen met kunst en filosofie. Logisch redeneren is een van die
profielspecifieke onderdelen. De gedachte is dat dit onderdeel een bijdrage kan
leveren aan het redeneren op allerlei gebieden in de maatschappij. In de eindtermen
staat bijvoorbeeld: “De kandidaat kan de correctheid van redeneringen en daarbij
horende conclusies, zoals gebruikt in het maatschappelijk debat, verifiëren en
analyseren” (College voor Examens, 2016, p. 14). Zowel in het dagelijks leven als in
diverse beroepen, zoals arts, jurist, rechter of politicus, moeten immers redeneringen
opgezet, geanalyseerd en beoordeeld worden. Het is bijvoorbeeld van
maatschappelijk belang dat een rechter uitspraken goed kan wegen en
onderbouwen met bewijs en onjuiste veroordelingen voorkomt. Mede vanwege de
maatschappelijke relevantie is er in het onderwijs meer aandacht voor redeneer- en
argumentatievaardigheden gekomen.
Het is echter onduidelijk hoe onderwijs in logisch redeneren vorm moet
krijgen om aan te kunnen sluiten bij de al aanwezige redeneervaardigheden van
leerlingen; vooral met het oog op het doel dat de ontwikkeling van logisch
redeneervaardigheden moet bijdragen aan situaties buiten het klaslokaal. Omdat
wiskunde C alleen wordt gegeven in de bovenbouw van het vwo aan leerlingen met
een C&M-profiel, vormden zij de doelgroep van dit onderzoek waarvoor we een
interventie hebben ontwikkeld met als doel hun redeneervaardigheden te
verbeteren.
Theoretische achtergrond
De belangstelling voor logisch redeneren in het Nederlandse onderwijs staat niet op
zichzelf. Allerlei internationale organisaties (zie bijv. European Union, 2002; OECD,
2019a; Thijs et al., 2004) beschrijven vaardigheden die nodig zijn om leerlingen voor
te bereiden op de huidige informatiemaatschappij. De 21e-eeuwse vaardigheden
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Samenvatting
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