Investigation of Middle School Students' Conceptual Understanding of Circle: Flipped Classroom Approaches with the 5E Model
Yıl 2022,
Cilt: 9 Sayı: 1, 110 - 133, 01.05.2022
Şule Özcan
,
Mehmet Demir
,
Nazlı Aksu
,
Selin Urhan
,
Yılmaz Zengin
Öz
This study examines middle school students' conceptual understanding of the circle in terms of transformation of semiotic representations within the context of the flipped classroom approaches with the 5E inquiry model. The participants of the research are six seventh graders in a public school. The teaching experiment method was used in the study. The mathematical tasks prepared by the researchers, the GeoGebra files created by the students, video and audio recordings, and task-based interviews were used as data collection tools. Discourse analysis was used to examine the students' conceptual understanding in detail. The results revealed that the video in the out-of-class process of the engagement phase enhanced the students to remember their prior knowledge and the videos in the out-of-class process of the explanation phase helped the students to repeat the subject. In in-class activity of the evaluation phase, the students made a transformation among at least two representation systems to solve the problem situation. The results revealed that in the GeoGebra-supported tasks implemented within the scope of the flipped classroom approaches with the 5E inquiry model, the students made a transformation of semiotic representations using different representation registers, which has contributed to their conceptual understanding.
Kaynakça
- Abeysekera, L., & Dawson, P. (2015). Motivation and cognitive load in the Flipped classroom: Definition, rationale and a call for research. Higher Education Research & Development, 34(1), 1-14. https://doi.org/10.1080/07294360.2014.934336
- Akyuz, D. (2016). Mathematical practices in a technological setting: A design research experiment for teaching circle properties. International Journal of Science and Mathematics Education. 14(3), 549–573. https://doi.org/10.1007/s10763-014-9588-z
- Bergmann, J., & Sams, A. (2012). Flipp your classroom. Reach every student in every class every day. ISTE.
- Bhagat, K. K., Chang, C. N., & Chang, C. Y. (2016). The impact of the flipped classroom on mathematics concept learning in high school. Educational Technology & Society, 19(3), 124-132. www.jstor.org/stable/jeductechsoci.19.3.134
- Blair, E., Maharaj, C., & Primus, S. (2016). Performance and perception in the flipped classroom. Education and Information Technologies. 21(6), 1465-1482. https://doi.org/10.1007/s10639-015-9393-5
- Brooks, J. G., & Brooks, M. G. (1993). The case for constructivist classrooms. ASCD Alexandria.
- Bybee, R. W., Taylor, J. A., Gardner, A., van Scotter, P., Powell, J. C.,Westbrook, A., & Landes, N. (2006). The BSCS 5E instructional model: Origins and effectiveness. BSCS.
- Cantimer, G. G., Şengül, S. (2017). Ortaokul 7. ve 8. sınıf öğrencilerinin çember konusundaki kavram yanılgıları ve hataları. Gazi Eğitim Bilimleri Dergisi, 3(1), 17-27. https://dergipark.org.tr/tr/pub/gebd/issue/35207/390665
- Dienes, Z.P. (1960). Building up mathematics. Hutchinson Educational.
- Diković, L. (2009). Implementing dynamic mathematics resources with geogebra at the college level. International Journal of Emerging Technologies in Learning, 4(3), 51-54. https://doi.org/10.3991/ijet.v4i3.784
- Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. ERIC.
- Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103-131. https://doi.org/10.1007/s10649-006-0400-z
- Falcade, R., Laborde, C., & Mariotti, M. A. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317-333. https://doi.org/10.1007/s10649-006-9072-y
- Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2003). Perspectives on technology mediated learning in secondary school mathematics classrooms. The Journal of Mathematical Behavior, 22(1), 73–89. https://doi.org/10.1016/S0732-3123(03)00005-1
- Hiebert, J., & Carpenter, T. (1992). Learning and Teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-100). NCTM.
- Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). NCTM.
- Hitt, F. & González-Martín, A. S. (2015). Covariation between variables in a modellling process: The ACODESA (collaborative learning, scientific debate and slf-reflection) method. Educational Studies in Mathematics, 88(2), 201-219. http://dx.doi.org/10.1007/s10649-014-9578-7
- Hölzl, R. (1995). Between drawing and figüre. In R. Sutherland and J. Mason (Eds.), Exploiting mental ımagery with computers in mathematics education. Springer.
- Hwang, G.J., Lai, C. L., & Wang, S. Y. (2015). Seamless flipped learning: A mobile technology-enhanced flipped classroom with effective learning strategies. Journal of Computers in Education, 2(4), 449-473. https://doi.org/10.1007/s40692-015-0043-0
- Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpratations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44(1-2), 55-85. https://doi.org/10.1023/A:1012789201736
- Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press.
- Kim, D. J., Choi, S., & Lim, W. (2017). Stard’s commognitive frework as a method of discourse anlysis in mathematics. World Academy of Science, Engineering and Technology International Journal of Cognitive and Language Sciences, 11(11), 481-485. https://doi.org/10.5281/zenodo.1132727
- Lesh, R., Mierkiewicz D., & Kantowski, M. (1979). Applied mathematical problem solving. OH: Columbus.
- Love, B., Hodge, A., Corritore, C., & Ernst, D. ( 2015). Inquiry-based learning and the flipped classroom model. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 25(8), 745-762. https://doi.org/10.1080/10511970.2015.1046005
- Love, B., Hodge, A., Grandgenett, N., & Swift, A. W. (2014). Student learning and perceptions in a flipped linear algebra course. International Journal of Mathematical Education in Science and Technology, 45(3), 317-324. https://doi.org/10.1080/0020739X.2013.822582
- McMillan, J. H., & Schumacher, S. (2010). Research in education: Evidence-based inquiry (7th ed.). Pearson.
- Muir, T. (2020). Self-determination theory and the flipped classroom: a case study of a senior secondary mathematics class. Mathematics Education Research Journal, advance online publication. https://doi.org/10.1007/s13394-020-00320-3
- National Council of Teachers of Mathematics [NCTM] (2000). Principles and standards for school mathematics. NCTM.
- Prediger, S. (2013). Focussing structural relations in the bar board—a design research study for fostering all students’ conceptual understanding of fractions. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education. Ankara. (pp. 343–352).
- Shahbari, J. A. & Tabach, M. (2020). Features of modeling processes that elicit mathematical models represented at different semiotic registers. Educational Studies in Mathematic, 105(2), 115-135. https://doi.org/10.1007/s10649-020-09971-2
- Sahin, A., Cavlazoğlu, B., & Zeytuncu, Y. E. (2015). Flipping a college calculus course: A casy study. Educational Technology & Society, 18(3), 142-152. http://www.jstor.org/stable/jeductechsoci.18.3.142
- Schallert, S., Lavicza, Z., & Vandervieren, E. (2020). Merging flipped classroom approaches with the 5E inquiry model: A design heuristic. International Journal of Mathematical Education in Science and Technology. Advance online publication. https://doi.org/10.1080/0020739X.2020.1831092
- Schallert, S., Lavicza, Z., & Vandervieren, E. (2021). Towards inquiry-based flipped classroom scenarios: A design heuristic and principles for lesson planning. International Journal of Science and Mathematics Education. Advance online publication. https://doi.org/10.1007/s10763-021-10167-0
- Smith, M. S., & Stein, M. K. (1998). Reflections on practice: Selecting and creating mathematical tasks: From research to practice. Mathematics teaching in the middle school, 3(5), 344-350.https://doi.org/10.5951/MTMS.3.5.0344
- Song, Y. & Kapur, M. (2017). How to flip the classroom-“Productive failure or traditional flipped classroom” pedagogical design?. İnternational Forum of Educational Technology & Society, 20(1), 292-305. http://www.jstor.org/stable/jeductechsoci.20.1.292
- Steffe, L.P., & Thompson, P. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh and A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267-306). Lawrence Erlbaum Associates.
- Talbert, R. (2017). Flipped learning: A guide for higher education faculty. Stylus.
- Tapan-Broutin, M. S. (2014). Matematiksel nesnelerin yapısı ve temsiller: Klasik semiyotik üçgenin geometri öğretimine yansımalarının analizi. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 27(1), 255-281. https://doi.org/10.19171/uuefd.49474
- Topuz, F. & Birgin, O. (2020). Yedinci sınıf “çember ve daire” konusunda geliştirilen geogebra destekli öğretim materyaline ve öğrenme ortamına ilişkin öğrenci görüşleri. Journal of Computer and Education Research, 8(15), 1-27. https://doi.org/10.18009/jcer.638142
- Voigt, M., Fredriksen, H., & Rasmussen, C. (2020). Leveraging the design heuristics of realistic mathematics education and culturally responsive pedagogy to create a richer flipped classroom calculus curriculum. ZDM, 52(5), 1051-1062. https://doi.org/10.1007/s11858-019-01124-x
- Wasserman, N. H., Quint, C., Norris, S. A., & Carr, T. (2017). Exploring flipped classroom instruction in calculus III. International Journal of Science and Mathematics Education, 15(3), 545–568. https://doi.org/10.1007/s10763-015-9704-8
- Wei, X., Cheng, IL., Chen, NS., Yang, X., Liu, Y., Dong, Y., Zhal, X., & Kinshuk (2020). Effect of the flipped classroom on the mathematics performance of middle school students. Education Tech Research Development, 68(3), 1461-1484. https://doi.org/10.1007/s11423-020-09752-x
- Yorgancı, S. (2020). A flipped learning approach to improving students’ learning performance in mathematics courses. Necatibey Faculty of Education Electronic Journal of Science and Mathematics Education, 14(1), 348-371. https://doi.org/10.17522/balikesirnef.657197
- Zengin, Y. (2017). Investigating the use of the Khan Academy and mathematics Software with a flipped classroom approach in mathematics teaching. Educational Technology & Society, 20(2), 89-100. https://www.jstor.org/stable/90002166
- Zengin, Y. (2019). Development of mathematical connection skills in a dynamic learning environment. Education and Information Technologies, 24(3), 2175-2194. https://doi.org/10.1007/s10639-019-09870-x
- Zheng, L., Bhagat, K. K., Zhen, Y., & Zhang, X. (2020). The effectiveness of the flipped classroom on students’ learning achievement and learning motivation: A meta-analysis. Educational Technology & Society, 23(1), 1-15. https://www.jstor.org/stable/26915403
- Zhuang, Y., & Conner, A. (2018). Analysis of teachers’ questioning in supporting mathematical argumentation by integrating Habermas’ rationality and Toulmin’s model. In T. Hodges, G. Roy, & A. Tyminski (Eds.), Proceedings of the 40th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1323–1330). University of South Carolina & Clemson University.
Ortaokul Öğrencilerinin Çember Konusundaki Kavramsal Anlamalarının İncelenmesi: 5E Öğrenme Modeli ile Ters Yüz Edilmiş Sınıf Yaklaşımı
Yıl 2022,
Cilt: 9 Sayı: 1, 110 - 133, 01.05.2022
Şule Özcan
,
Mehmet Demir
,
Nazlı Aksu
,
Selin Urhan
,
Yılmaz Zengin
Öz
Bu çalışmada, sorgulamaya dayalı 5E öğrenme modeli ile ters yüz edilmiş sınıf yaklaşımı bağlamında ortaokul öğrencilerinin çember konusundaki kavramsal anlamaları temsil dönüşümü açısından incelenmektedir. Araştırmanın katılımcılarını bir devlet okulunda yedinci sınıfta öğrenimine devam eden altı öğrenci oluşturmaktadır. Çalışmada nitel araştırma yöntemlerinden öğretim deneyi yöntemi benimsenmiştir. Veri toplama aracı olarak araştırmacıların hazırladığı matematiksel etkinlikler, öğrencilerin oluşturmuş olduğu GeoGebra dosyaları, uygulama sırasında alınan görüntü ve ses kayıtları ve uygulama sonrası yapılan etkinlik temelli görüşmeler kullanılmıştır. Öğrencilerin kavramsal anlamalarının ayrıntılı incelenmesi için veriler söylem analizi yoluyla analiz edilmiştir. Araştırmanın sonucunda, giriş aşamasının sınıf dışı süreci için hazırlanan videonun öğrencilerin çemberin ve çember parçasının uzunluğuna ilişkin ön bilgilerini hatırlamasını; açıklama aşamasının sınıf dışı süreci için hazırlanan videoların ise öğrencilerin konuyu tekrar etmesini sağladığı belirlenmiştir. Değerlendirme aşamasının sınıf içi sürecinde öğrencilerin problem durumuna çözüm üretirken en az iki temsil sistemi arasında dönüşüm yapabildiği görülmüştür. Çalışmada sorgulamaya dayalı 5E öğrenme modeli ile ters yüz edilmiş sınıf yaklaşımı kapsamında uygulanan GeoGebra destekli etkinliklerde öğrencilerin farklı temsil sistemlerini kullanarak temsiller arası dönüşüm gerçekleştirmelerinin kavramsal anlama süreçlerine katkı sağladığı belirlenmiştir.
Kaynakça
- Abeysekera, L., & Dawson, P. (2015). Motivation and cognitive load in the Flipped classroom: Definition, rationale and a call for research. Higher Education Research & Development, 34(1), 1-14. https://doi.org/10.1080/07294360.2014.934336
- Akyuz, D. (2016). Mathematical practices in a technological setting: A design research experiment for teaching circle properties. International Journal of Science and Mathematics Education. 14(3), 549–573. https://doi.org/10.1007/s10763-014-9588-z
- Bergmann, J., & Sams, A. (2012). Flipp your classroom. Reach every student in every class every day. ISTE.
- Bhagat, K. K., Chang, C. N., & Chang, C. Y. (2016). The impact of the flipped classroom on mathematics concept learning in high school. Educational Technology & Society, 19(3), 124-132. www.jstor.org/stable/jeductechsoci.19.3.134
- Blair, E., Maharaj, C., & Primus, S. (2016). Performance and perception in the flipped classroom. Education and Information Technologies. 21(6), 1465-1482. https://doi.org/10.1007/s10639-015-9393-5
- Brooks, J. G., & Brooks, M. G. (1993). The case for constructivist classrooms. ASCD Alexandria.
- Bybee, R. W., Taylor, J. A., Gardner, A., van Scotter, P., Powell, J. C.,Westbrook, A., & Landes, N. (2006). The BSCS 5E instructional model: Origins and effectiveness. BSCS.
- Cantimer, G. G., Şengül, S. (2017). Ortaokul 7. ve 8. sınıf öğrencilerinin çember konusundaki kavram yanılgıları ve hataları. Gazi Eğitim Bilimleri Dergisi, 3(1), 17-27. https://dergipark.org.tr/tr/pub/gebd/issue/35207/390665
- Dienes, Z.P. (1960). Building up mathematics. Hutchinson Educational.
- Diković, L. (2009). Implementing dynamic mathematics resources with geogebra at the college level. International Journal of Emerging Technologies in Learning, 4(3), 51-54. https://doi.org/10.3991/ijet.v4i3.784
- Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. ERIC.
- Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103-131. https://doi.org/10.1007/s10649-006-0400-z
- Falcade, R., Laborde, C., & Mariotti, M. A. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317-333. https://doi.org/10.1007/s10649-006-9072-y
- Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2003). Perspectives on technology mediated learning in secondary school mathematics classrooms. The Journal of Mathematical Behavior, 22(1), 73–89. https://doi.org/10.1016/S0732-3123(03)00005-1
- Hiebert, J., & Carpenter, T. (1992). Learning and Teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-100). NCTM.
- Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). NCTM.
- Hitt, F. & González-Martín, A. S. (2015). Covariation between variables in a modellling process: The ACODESA (collaborative learning, scientific debate and slf-reflection) method. Educational Studies in Mathematics, 88(2), 201-219. http://dx.doi.org/10.1007/s10649-014-9578-7
- Hölzl, R. (1995). Between drawing and figüre. In R. Sutherland and J. Mason (Eds.), Exploiting mental ımagery with computers in mathematics education. Springer.
- Hwang, G.J., Lai, C. L., & Wang, S. Y. (2015). Seamless flipped learning: A mobile technology-enhanced flipped classroom with effective learning strategies. Journal of Computers in Education, 2(4), 449-473. https://doi.org/10.1007/s40692-015-0043-0
- Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpratations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44(1-2), 55-85. https://doi.org/10.1023/A:1012789201736
- Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press.
- Kim, D. J., Choi, S., & Lim, W. (2017). Stard’s commognitive frework as a method of discourse anlysis in mathematics. World Academy of Science, Engineering and Technology International Journal of Cognitive and Language Sciences, 11(11), 481-485. https://doi.org/10.5281/zenodo.1132727
- Lesh, R., Mierkiewicz D., & Kantowski, M. (1979). Applied mathematical problem solving. OH: Columbus.
- Love, B., Hodge, A., Corritore, C., & Ernst, D. ( 2015). Inquiry-based learning and the flipped classroom model. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 25(8), 745-762. https://doi.org/10.1080/10511970.2015.1046005
- Love, B., Hodge, A., Grandgenett, N., & Swift, A. W. (2014). Student learning and perceptions in a flipped linear algebra course. International Journal of Mathematical Education in Science and Technology, 45(3), 317-324. https://doi.org/10.1080/0020739X.2013.822582
- McMillan, J. H., & Schumacher, S. (2010). Research in education: Evidence-based inquiry (7th ed.). Pearson.
- Muir, T. (2020). Self-determination theory and the flipped classroom: a case study of a senior secondary mathematics class. Mathematics Education Research Journal, advance online publication. https://doi.org/10.1007/s13394-020-00320-3
- National Council of Teachers of Mathematics [NCTM] (2000). Principles and standards for school mathematics. NCTM.
- Prediger, S. (2013). Focussing structural relations in the bar board—a design research study for fostering all students’ conceptual understanding of fractions. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education. Ankara. (pp. 343–352).
- Shahbari, J. A. & Tabach, M. (2020). Features of modeling processes that elicit mathematical models represented at different semiotic registers. Educational Studies in Mathematic, 105(2), 115-135. https://doi.org/10.1007/s10649-020-09971-2
- Sahin, A., Cavlazoğlu, B., & Zeytuncu, Y. E. (2015). Flipping a college calculus course: A casy study. Educational Technology & Society, 18(3), 142-152. http://www.jstor.org/stable/jeductechsoci.18.3.142
- Schallert, S., Lavicza, Z., & Vandervieren, E. (2020). Merging flipped classroom approaches with the 5E inquiry model: A design heuristic. International Journal of Mathematical Education in Science and Technology. Advance online publication. https://doi.org/10.1080/0020739X.2020.1831092
- Schallert, S., Lavicza, Z., & Vandervieren, E. (2021). Towards inquiry-based flipped classroom scenarios: A design heuristic and principles for lesson planning. International Journal of Science and Mathematics Education. Advance online publication. https://doi.org/10.1007/s10763-021-10167-0
- Smith, M. S., & Stein, M. K. (1998). Reflections on practice: Selecting and creating mathematical tasks: From research to practice. Mathematics teaching in the middle school, 3(5), 344-350.https://doi.org/10.5951/MTMS.3.5.0344
- Song, Y. & Kapur, M. (2017). How to flip the classroom-“Productive failure or traditional flipped classroom” pedagogical design?. İnternational Forum of Educational Technology & Society, 20(1), 292-305. http://www.jstor.org/stable/jeductechsoci.20.1.292
- Steffe, L.P., & Thompson, P. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh and A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267-306). Lawrence Erlbaum Associates.
- Talbert, R. (2017). Flipped learning: A guide for higher education faculty. Stylus.
- Tapan-Broutin, M. S. (2014). Matematiksel nesnelerin yapısı ve temsiller: Klasik semiyotik üçgenin geometri öğretimine yansımalarının analizi. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 27(1), 255-281. https://doi.org/10.19171/uuefd.49474
- Topuz, F. & Birgin, O. (2020). Yedinci sınıf “çember ve daire” konusunda geliştirilen geogebra destekli öğretim materyaline ve öğrenme ortamına ilişkin öğrenci görüşleri. Journal of Computer and Education Research, 8(15), 1-27. https://doi.org/10.18009/jcer.638142
- Voigt, M., Fredriksen, H., & Rasmussen, C. (2020). Leveraging the design heuristics of realistic mathematics education and culturally responsive pedagogy to create a richer flipped classroom calculus curriculum. ZDM, 52(5), 1051-1062. https://doi.org/10.1007/s11858-019-01124-x
- Wasserman, N. H., Quint, C., Norris, S. A., & Carr, T. (2017). Exploring flipped classroom instruction in calculus III. International Journal of Science and Mathematics Education, 15(3), 545–568. https://doi.org/10.1007/s10763-015-9704-8
- Wei, X., Cheng, IL., Chen, NS., Yang, X., Liu, Y., Dong, Y., Zhal, X., & Kinshuk (2020). Effect of the flipped classroom on the mathematics performance of middle school students. Education Tech Research Development, 68(3), 1461-1484. https://doi.org/10.1007/s11423-020-09752-x
- Yorgancı, S. (2020). A flipped learning approach to improving students’ learning performance in mathematics courses. Necatibey Faculty of Education Electronic Journal of Science and Mathematics Education, 14(1), 348-371. https://doi.org/10.17522/balikesirnef.657197
- Zengin, Y. (2017). Investigating the use of the Khan Academy and mathematics Software with a flipped classroom approach in mathematics teaching. Educational Technology & Society, 20(2), 89-100. https://www.jstor.org/stable/90002166
- Zengin, Y. (2019). Development of mathematical connection skills in a dynamic learning environment. Education and Information Technologies, 24(3), 2175-2194. https://doi.org/10.1007/s10639-019-09870-x
- Zheng, L., Bhagat, K. K., Zhen, Y., & Zhang, X. (2020). The effectiveness of the flipped classroom on students’ learning achievement and learning motivation: A meta-analysis. Educational Technology & Society, 23(1), 1-15. https://www.jstor.org/stable/26915403
- Zhuang, Y., & Conner, A. (2018). Analysis of teachers’ questioning in supporting mathematical argumentation by integrating Habermas’ rationality and Toulmin’s model. In T. Hodges, G. Roy, & A. Tyminski (Eds.), Proceedings of the 40th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1323–1330). University of South Carolina & Clemson University.