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November  2021, 1(4): 279-298. doi: 10.3934/steme.2021018

Examination of modelling in K-12 STEM teacher education: Connecting theory with practice

1. 

Faculty of Education, University of Windsor, 401 Sunset Ave., Windsor, ON N9B 3P4, Canada

2. 

Department of Curriculum and Pedagogy, University of British Columbia, 2125 Main Mall, Vancouver, BC V6T 1Z4, Canada; marina.milner-bolotin@ubc.ca (M.M.-B.)

* Correspondence: dragana@uwindsor.ca; Tel: +1-519-253-3000 ext. 3962

Academic Editor: William Guo

Received  September 2021 Revised  November 2021 Published  November 2021

The goal of this paper is to examine the place of modelling in STEM education and teacher education. First, we introduce modelling as a cyclical process of generating, testing, and applying knowledge while highlighting the epistemological commonalities and differences between the STEM disciplines. Second, we build on the four well-known frameworks, to propose an Educational Framework for Modelling in STEM, which describes both teacher and student roles in the modelling cycle. Third, we use this framework to analyze how modelling is presented in the new mathematics and science school curricula in two Canadian provinces (Ontario and British Columbia), and how it could be implemented in teacher education. Fourth, we emphasize the epistemological aspects of the Educational Framework for Modelling in STEM, as disciplinary epistemological foundations may seem too abstract to both teacher educators and teachers of STEM school subjects. Yet, epistemologies are the driving forces within each discipline and must be considered while teaching STEM as a unified field. To nurture critical thinkers and innovators, it is critical to pay attention to what knowledge is and how it is created and tested. The Educational Framework for Modelling in STEM may be helpful in introducing students and future teachers to the process of modelling, regardless of if they teach it in a single- or a multi-discipline course, such as STEM. This paper will be of interest to teacher educators, teachers, researchers, and policy makers working within and between the STEM fields and interested in promoting STEM education and its epistemological foundations.

Citation: Dragana Martinovic, Marina Milner-Bolotin. Examination of modelling in K-12 STEM teacher education: Connecting theory with practice. STEM Education, 2021, 1 (4) : 279-298. doi: 10.3934/steme.2021018
References:
[1] Government of Canada, The Government of Canada and STEM, Government of Canada, Ottawa, 2018. 
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Hallström, J. and K.J. Schönborn, Models and modelling for authentic STEM education: Reinforcing the argument. International Journal of STEM Education, 2019. 6: 22. https://doi.org/10.1186/s40594-019-0178-z. doi: 10.1186/s40594-019-0178-z.

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Martinovic, D. and M. Milner-Bolotin, Discussion: Teacher Professional Development in the Era of Change, in STEM Teachers and Teaching in the Era of Change: Professional expectations and advancement in 21st Century Schools, Y. Ben-David Kolikant, D. Martinovic, and M. Milner-Bolotin, Editors. 2020, pp. 185-197. Cham, Switzerland: Springer.

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Ben-David Kolikant, Y., D. Martinovic, and M. Milner-Bolotin, STEM Teachers and Teaching in the Digital Era: Professional expectations and advancement in 21st Century Schools, in STEM Teachers and Teaching in the Digital Era. 2020, pp. 325. Cham, Switzerland: Springer.

[26]

Yuan, Z. -Q., M. Milner-Bolotin, and D. Anderson, Lessons Learned from Educating STEM Teachers in Canadian Universities: The Case of the University of British Columbia. Journal of Mathematics Education, 2021. 30(6): 96-102.

[27]

British Columbia Ministry of Education, British Columbia New Curriculum, Government of British Columbia, 2020, Victoria, British Columbia, Canada.

[28]

Milner-Bolotin, M. and R. Zazkis, A study of future physics teachers' knowledge for teaching: A case of sound level and a decibel scale. LUMAT: International Journal on Math, Science and Technology Education, 2021. Submitted March 2021: 29.

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Milner-Bolotin, M., Increasing girls' participation in physics: Education research implications for practice. Physics in Canada, 2015. 71(2): 94-97.

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[35]

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[37]

Zazkis, R. and D. Zazkis, The significance of mathematical knowledge in teaching elementary methods courses: Perspectives of mathematics teacher educators. Educational Studies in Mathematics, 2011. 76(3): 247-263.

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Ma, L., Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and in the United States. Studies in mathematical thinking and learning series, ed. A.H. Schoenfeld. 1999, Mahwah, NJ: Lawrence Erlbaum Associates.

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Berlin, D.F. and A.L. White, A longitudinal look at attitudes and perceptions related to the integration of Mathematics, Science, and Technology education. School Science and Mathematics, 2012. 112(1): 20-30. https://doi.org/10.1111/j.1949-8594.2011.00111.x. doi: 10.1111/j.1949-8594.2011.00111.x.

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Lee, M. -H. and C. -C. Tsai, Exploring teachers' perceived self efficacy and Technological Pedagogical Content Knowledge with respect to educational use of the World Wide Web. Instructional Science, 2010. 38(1): 1-21.

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Martinovic, D. and M. Milner-Bolotin, Problematizing STEM: What it is, what it is not, and why it matters, in 15 Years of MACAS (Mathematics and its Connections to the Arts and Sciences), C. Michelsen, et al., Editors. 2022: Springer.

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[45]

Barquero, B., M. Bosch, and A. Romo, Mathematical modelling in teacher education: dealing with institutional constraints. ZDM, 2018. 50(1): 31-43. https://doi.org/10.1007/s11858-017-0907-z. doi: 10.1007/s11858-017-0907-z.

[46]

Frejd, P., Teachers' conceptions of mathematical modelling at Swedish Upper Secondary school. Journal of Mathematical Modelling and Application, 2012. 1(5): 17-40.

[47]

Ortiz, J. and A.D. Santos, Mathematical Modelling in Secondary Education: A Case Study, in Trends in Teaching and Learning of Mathematical Modelling, K. G., et al., Editors. 2011. Dordrecht. : Springer.

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show all references

References:
[1] Government of Canada, The Government of Canada and STEM, Government of Canada, Ottawa, 2018. 
[2] Government of the United Kingdom, STEM Strategy, London U.K., 2018. 
[3]

Timms, M., et al., Challenges in STEM learning in Australian schools: Literature and policy review, Australian Council for Educational Research (ACER), 2018, Camberwell, VIC.

[4]

Hallström, J. and K.J. Schönborn, Models and modelling for authentic STEM education: Reinforcing the argument. International Journal of STEM Education, 2019. 6: 22. https://doi.org/10.1186/s40594-019-0178-z. doi: 10.1186/s40594-019-0178-z.

[5]

Haines, C., P. Galbraith, and W. Blume. Mathematical Modelling: Education, Engineering and Economics - ICTMA 12. in Twelfth International Conference on the Teaching of Mathematical Modelling and Applications. 2007. City University London: Horwood Publishing.

[6]

Kaiser, G., M. Blomhøj, and B. Sriraman, Towards a didactical theory for mathematical modelling. ZDM, 2006. 38(2): 82-85. https://doi.org/10.1007/BF02655882. doi: 10.1007/BF02655882.

[7]

Blum, W. and D. Leiss. How to students and teachers deal with modelling problems? in Twelfth International Conference on the Teaching of Mathematical Modelling and Applications. 2005, pp. 222-231. London, UK: Hrowood Publishing.

[8] E. EtkinaD.T. Brookes and G. Planinsic, Investigative Science Learning Environment, Morgan & Claypool Publishers, 2019. 
[9]

Milner-Bolotin, M., Promoting Deliberate Pedagogical Thinking with Technology in physics teacher education: A teacher-educator's journey, in The Physics Educator: Tacit Praxes and Untold Stories T.G. Ryan and K.A. McLeod, Editors. 2016, pp. 112-141. Champaign, IL: Common Ground and The Learner.

[10]

Milner-Bolotin, M. Reimagining technology-enhanced STEM teacher education for 21st century: From more technology to increased quality of teaching and learning (Part 1). in Future Schools 2030. 2016. Beijing, China: Beijing Advanced Innovation Centre for Future Education: Beijing Normal University.

[11]

Martinovic, D., Z. Karadag, and D. McDougall, Proceedings of the Fifth North American GeoGebra Conference: Explorative learning with technology: GeoGebra-NA 2014 in GeoGebra_NA 2014. 2014, pp. 102. Toronto, ON, Canada: University of Toronto.

[12]

Blum, W. Quality Teaching of Mathematical Modelling: What Do We Know, What Can We Do? 2015, pp. 73-96. Cham: Springer International Publishing.

[13]

Windschitl, M., J. Thompson, and M. Braaten, Beyond the scientific method: Model-based inquiry as a new paradigm of preference for school science investigations. Science Education, 2008. 92(5): 941-967. https://doi.org/10.1002/sce.20259. doi: 10.1002/sce.20259.

[14]

Hofer, B.K., Personal epistemology research: Implications for learning and teaching. Educational Psychology Review, 2001. 13(4): 353-383. https://doi.org/10.1023/A:1011965830686. doi: 10.1023/A:1011965830686.

[15]

Kolb, D.A., Experiential learning: Experience as the source of learning and development. Vol. 1. 1984, Englewood Cliffs, NJ: Prentice-Hall.

[16]

Gardiner, P., Learning to think together: Creativity, interdisciplinary collaboration and epistemic control. Thinking Skills and Creativity, 2020. 38: 100749.

[17]

Carlson, M.A., et al., A case for mathematical modeling in the elementary school classroom, in Mathematical modeling and modeling mathematics, C.R. Hirsch and A.R. McDuffie, Editors. 2016, pp. 121-129. Reston, VA: National Council of Teachers of Mathematics.

[18]

Ben-David Kolikant, Y., D. Martinovic, and M. Milner-Bolotin, Introduction: STEM teachers and teaching in the era of change, in STEM Teachers and Teaching in the Era of Change: Professional expectations and advancement in 21st Century Schools, Y. Ben-David Kolikant, D. Martinovic, and M. Milner-Bolotin, Editors. 2020, pp. 1-18. Cham, Switzerland: Springer.

[19]

National Research Council, Next Generation Science Standards: For States, by States, ed. Q. Helen, S. Heidi, and K. Thomas. 2013, Washington DC: The National Academies Press, USA National Research Council.

[20]

Brown, J.R. Logic, Epistemology, Philosophy of Science The Canadian Encyclopedia. The Canadian Encyclopedia: Historica Canada 2012 August 24, 2014[cited 2021 September 3]; Available from: https://www.thecanadianencyclopedia.ca/en/article/logic-epistemology-philosophy-of-science.

[21]

Erduran, S., Nature of "STEM"? Science & Education, 2020. 29(4): 781-784. https://doi.org/10.1007/s11191-020-00150-6. doi: 10.1007/s11191-020-00150-6.

[22]

Reynante, B.M., M.E. Selbach-Allen, and D.R. Pimentel, Exploring the Promises and Perils of Integrated STEM Through Disciplinary Practices and Epistemologies. Science & Education, 2020. 29(4): 785-803. https://doi.org/10.1007/s11191-020-00121-x. doi: 10.1007/s11191-020-00121-x.

[23]

Kalman, C.S., The need to emphasize epistemology in teaching and research. Science & Education, 2009. 18: 325-348. https://doi.org/10.1007/s11191-007-9135-1. doi: 10.1007/s11191-007-9135-1.

[24]

Martinovic, D. and M. Milner-Bolotin, Discussion: Teacher Professional Development in the Era of Change, in STEM Teachers and Teaching in the Era of Change: Professional expectations and advancement in 21st Century Schools, Y. Ben-David Kolikant, D. Martinovic, and M. Milner-Bolotin, Editors. 2020, pp. 185-197. Cham, Switzerland: Springer.

[25]

Ben-David Kolikant, Y., D. Martinovic, and M. Milner-Bolotin, STEM Teachers and Teaching in the Digital Era: Professional expectations and advancement in 21st Century Schools, in STEM Teachers and Teaching in the Digital Era. 2020, pp. 325. Cham, Switzerland: Springer.

[26]

Yuan, Z. -Q., M. Milner-Bolotin, and D. Anderson, Lessons Learned from Educating STEM Teachers in Canadian Universities: The Case of the University of British Columbia. Journal of Mathematics Education, 2021. 30(6): 96-102.

[27]

British Columbia Ministry of Education, British Columbia New Curriculum, Government of British Columbia, 2020, Victoria, British Columbia, Canada.

[28]

Milner-Bolotin, M. and R. Zazkis, A study of future physics teachers' knowledge for teaching: A case of sound level and a decibel scale. LUMAT: International Journal on Math, Science and Technology Education, 2021. Submitted March 2021: 29.

[29] Ontario Ministry of Education, The Ontario Mathematics Curriculum: Elementary, Government of Ontario, Toronto, ON, 2020. 
[30]

Techbridge. Techbridge Girls. 2017; from: http://www.techbridgegirls.org/index.php?id=28.

[31] C. Annett, Girls and Women in Science, Technology, Engineering and Mathematics, Government of Canada, Ottawa, Canada, 2017. 
[32]

Milner-Bolotin, M., Increasing girls' participation in physics: Education research implications for practice. Physics in Canada, 2015. 71(2): 94-97.

[33]

Herranen, J.K., E.C. Fooladi, and M. Milner-Bolotin, Editorial: Special Issue "Promoting STEAM in Education". LUMAT: International Journal of Math, Science and Technology Education, 2021. 9(9): 1-8. https://doi.org/10.31129/LUMAT.9.2.1559. doi: 10.31129/LUMAT.9.2.1559.

[34]

Perignat, E. and J. Katz-Buonincontro, STEAM in practice and research: An integrative literature review. Thinking Skills and Creativity, 2019. 31: 31-43. https://doi.org/10.1016/j.tsc.2018.10.002. doi: 10.1016/j.tsc.2018.10.002.

[35]

Ge, X., D. Ifenhaler, and J.M. Spector, Emerging Technologies for STEAM Education: Full STEAM ahead. Educational Communications and Technologies: Issues and Innovations. 2015, New York: Springer.

[36]

Hourigan, M. and J. Donaghue, The challenges facing initial teacher education: Irish prospective elementary teachers' mathematics subject matter knowledge. International Journal of Mathematical Education in Science and Technology, 2013. 44(1): 36-58. https://doi.org/10.1080/0020739X.2012.690897. doi: 10.1080/0020739X.2012.690897.

[37]

Zazkis, R. and D. Zazkis, The significance of mathematical knowledge in teaching elementary methods courses: Perspectives of mathematics teacher educators. Educational Studies in Mathematics, 2011. 76(3): 247-263.

[38]

Ma, L., Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and in the United States. Studies in mathematical thinking and learning series, ed. A.H. Schoenfeld. 1999, Mahwah, NJ: Lawrence Erlbaum Associates.

[39]

Berlin, D.F. and A.L. White, A longitudinal look at attitudes and perceptions related to the integration of Mathematics, Science, and Technology education. School Science and Mathematics, 2012. 112(1): 20-30. https://doi.org/10.1111/j.1949-8594.2011.00111.x. doi: 10.1111/j.1949-8594.2011.00111.x.

[40]

Lee, M. -H. and C. -C. Tsai, Exploring teachers' perceived self efficacy and Technological Pedagogical Content Knowledge with respect to educational use of the World Wide Web. Instructional Science, 2010. 38(1): 1-21.

[41]

Martinovic, D. and M. Milner-Bolotin, Problematizing STEM: What it is, what it is not, and why it matters, in 15 Years of MACAS (Mathematics and its Connections to the Arts and Sciences), C. Michelsen, et al., Editors. 2022: Springer.

[42]

Marder, M., A problem with STEM. CBE Life Sciences Education, 2013. 12(2): 148-150. https://doi.org/10.1187/cbe.12-12-0209. doi: 10.1187/cbe.12-12-0209.

[43] M.E. Martinez, Learning and Cognition: The Design of the Mind, Pearson, 2010. 
[44]

Ortiz-Revilla, J., A. Adúriz-Bravo, and I.M. Greca, A framework for epistemological discussion on integrated STEM education. Science & Education, 2020. 29: 857-880. https://doi.org/10.1007/s11191-020-00131-9. doi: 10.1007/s11191-020-00131-9.

[45]

Barquero, B., M. Bosch, and A. Romo, Mathematical modelling in teacher education: dealing with institutional constraints. ZDM, 2018. 50(1): 31-43. https://doi.org/10.1007/s11858-017-0907-z. doi: 10.1007/s11858-017-0907-z.

[46]

Frejd, P., Teachers' conceptions of mathematical modelling at Swedish Upper Secondary school. Journal of Mathematical Modelling and Application, 2012. 1(5): 17-40.

[47]

Ortiz, J. and A.D. Santos, Mathematical Modelling in Secondary Education: A Case Study, in Trends in Teaching and Learning of Mathematical Modelling, K. G., et al., Editors. 2011. Dordrecht. : Springer.

[48]

Ärlebäck, J.B. and C. Bergsten, On the Use of Realistic Fermi Problems in Introducing Mathematical Modelling in Upper Secondary Mathematics, in Modeling Students' Mathematical Modeling Competencies: ICTMA 13, R. Lesh, et al., Editors. 2013, pp. 597-609. Dordrecht: Springer Netherlands.

[49]

Hestenes, D., Modeling Theory for Math and Science Education, in Modeling Students' Mathematical Modeling Competencies: ICTMA 13, R. Lesh, et al., Editors. 2010, pp. 13-41. Boston, MA: Springer US.

[50]

English, L.D. and N.G. Mousoulides, Engineering-Based Modelling Experiences in the Elementary and Middle Classroom, in Models and Modeling: Cognitive Tools for Scientific Enquiry, M.S. Khine and I.M. Saleh, Editors. 2011, pp. 173-194. Dordrecht: Springer Netherlands.

[51]

Gil, E. and A.L. Gibbs, Promoting modelling and covariational reasoning among secondary school students in the context of big data. Statistics Education Research Journal, 2017. 16(2): 163-190.

[52]

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Figure 1.  Connections between the STEM Fields' Education, K-12 Curricula, and Teacher Education
Figure 2.  Visualization of the process of mathematical modelling in the new Ontario 1-8 mathematics curriculum [29]
Figure 3.  Modelling is at a centre of STEM teacher education
Table 1.  Aligning the existing frameworks into the educational framework for modelling in STEM
Stage Revised Kolb's Learning Cycle [56] MBI learning cycle [13] Epistemological foundations [16]: Stages are cumulative Teaching modelling [17]
0 Setting the parameters: Select a phenomenon that is within students' reach and interest. Preparing students for epistemic introspection: Discuss, how is the epistemology reflected in modelling. Getting ready: Develop activities, anticipate student difficulties, questions, potential challenges, etc.
1 Immersing in contextually rich concrete experiences: Students engage through both mind and body, while working in groups in authentic contexts. Organizing what students know and what they want to know: Students are given resources; initial questions emerge. Gaining epistemic awareness: Understand what and how members of one's discipline come to know and how each member of the group can contribute. Enacting: Organize students, guide and scaffold modelling activities, keep them relevant and focused, thus opening opportunities for deep learning to occur.
2 Conducting critical reflective observations: In an investigator-like manner, students weigh what they know and what knowledge the situation requires. Developing epistemic humility: Students recognize the limitations of their knowledge and assess how the present situation challenges and extends what they know. Enacting: Monitor students' work; provide adaptive interventions when needed (Blum, 2015) to help students formulate their own questions and seek answers.
3 Conducting contextual-specific abstract conceptualization: Students propose work-ing hypotheses; under-stand that all knowledge is provisional & needs testing in context. Generating testable, revisable, explanatory, conjectural, & generative hypotheses: Students propose patterns, models, theories that might explain the relationships between observed phenomena. Acquiring and practising epistemic empathy: Use different perspectives of the group members to interpret and understand the phenomenon. What new insights does this process bring?
4 Pragmatic active experimentation: Testing if and how abstract conceptualizations agree with new concrete experiences. Seeking evidence to test suggested hypotheses: Collect new evidence; use proposed models to generate new data. Constructing an argument: Explain the phenomenon, allow for alternative explanations. Exercising epistemic control: "Think like a …" The group critically examines their model and tests it in view of the ill-structured context-based conditions. If it fits, consider the work done or start a new cycle of inquiry. Enacting: Teacher monitors students' work, asks questions, and regroups students when required.
1* Returning to stage 1 with enhanced understanding of the phenomenon. Returning to stage 1 with a set of new questions as a motivation for a new cycle. Returning to stage 1: Start a new cycle of inquiry at a deeper epistemological level. Reflecting, modifying, revising: Teacher consolidates or revisits activity, with modifications/follow-up.
Stage Revised Kolb's Learning Cycle [56] MBI learning cycle [13] Epistemological foundations [16]: Stages are cumulative Teaching modelling [17]
0 Setting the parameters: Select a phenomenon that is within students' reach and interest. Preparing students for epistemic introspection: Discuss, how is the epistemology reflected in modelling. Getting ready: Develop activities, anticipate student difficulties, questions, potential challenges, etc.
1 Immersing in contextually rich concrete experiences: Students engage through both mind and body, while working in groups in authentic contexts. Organizing what students know and what they want to know: Students are given resources; initial questions emerge. Gaining epistemic awareness: Understand what and how members of one's discipline come to know and how each member of the group can contribute. Enacting: Organize students, guide and scaffold modelling activities, keep them relevant and focused, thus opening opportunities for deep learning to occur.
2 Conducting critical reflective observations: In an investigator-like manner, students weigh what they know and what knowledge the situation requires. Developing epistemic humility: Students recognize the limitations of their knowledge and assess how the present situation challenges and extends what they know. Enacting: Monitor students' work; provide adaptive interventions when needed (Blum, 2015) to help students formulate their own questions and seek answers.
3 Conducting contextual-specific abstract conceptualization: Students propose work-ing hypotheses; under-stand that all knowledge is provisional & needs testing in context. Generating testable, revisable, explanatory, conjectural, & generative hypotheses: Students propose patterns, models, theories that might explain the relationships between observed phenomena. Acquiring and practising epistemic empathy: Use different perspectives of the group members to interpret and understand the phenomenon. What new insights does this process bring?
4 Pragmatic active experimentation: Testing if and how abstract conceptualizations agree with new concrete experiences. Seeking evidence to test suggested hypotheses: Collect new evidence; use proposed models to generate new data. Constructing an argument: Explain the phenomenon, allow for alternative explanations. Exercising epistemic control: "Think like a …" The group critically examines their model and tests it in view of the ill-structured context-based conditions. If it fits, consider the work done or start a new cycle of inquiry. Enacting: Teacher monitors students' work, asks questions, and regroups students when required.
1* Returning to stage 1 with enhanced understanding of the phenomenon. Returning to stage 1 with a set of new questions as a motivation for a new cycle. Returning to stage 1: Start a new cycle of inquiry at a deeper epistemological level. Reflecting, modifying, revising: Teacher consolidates or revisits activity, with modifications/follow-up.
Table 2.  Elaboration of modelling in the new British Columbia Computer Science Curriculum [27, https://curriculum.gov.bc.ca/curriculum/search Keyword "model"]
(emphasis and capitalizations in the original)
Model with mathematics in situational contexts Computer Science 11 Reasoning and modelling Keyword: Model Elaboration: Use Mathematical Concepts And Tools To Solve Problems And Make Decisions (E.G., In Real-Life And/Or Abstract Scenarios) Take A Complex, Essentially Non-Mathematical Scenario And Figure Out What Mathematical Concepts And Tools Are Needed To Make Sense Of It
Keyword: Situational Contexts Elaboration: Including Real-Life Scenarios And Open-Ended Challenges That Connect Mathematics With Everyday Life
Ways to model mathematical problems Computer Science 11 No CCG Keyword: Mathematical Problems Elaboration: Estimate Theoretical Probability Through Simulation represent Finite Sequences And Series solve A System Of Linear Equations, Exponential Growth/Decay solve a Polynomial Equation calculate Statistical Values Such As Frequency, Central Tendencies, Standard Deviation Of Large Data Set compute Greatest Common Factor/Least Common Multiples
Model with mathematics in situational contexts Computer Science 11 Reasoning and modelling Keyword: Model Elaboration: Use Mathematical Concepts And Tools To Solve Problems And Make Decisions (E.G., In Real-Life And/Or Abstract Scenarios) Take A Complex, Essentially Non-Mathematical Scenario And Figure Out What Mathematical Concepts And Tools Are Needed To Make Sense Of It
Keyword: Situational Contexts Elaboration: Including Real-Life Scenarios And Open-Ended Challenges That Connect Mathematics With Everyday Life
Ways to model mathematical problems Computer Science 11 No CCG Keyword: Mathematical Problems Elaboration: Estimate Theoretical Probability Through Simulation represent Finite Sequences And Series solve A System Of Linear Equations, Exponential Growth/Decay solve a Polynomial Equation calculate Statistical Values Such As Frequency, Central Tendencies, Standard Deviation Of Large Data Set compute Greatest Common Factor/Least Common Multiples
Table 3.  The refined version of the Educational Framework for Modelling in STEM. The Stages I-VI correspond to Stages 0-1* in Table 1
Stage Teacher's and students' roles during modelling* Release of control
Teacher prepares students : Discusses how the epistemology is reflected in modelling.
Teacher prepares a lesson: Selects a phenomenon, develops activities, anticipates student difficulties, questions, potential challenges, etc.
Teacher releases control of student learning as students advance from Stage Ⅰ to Ⅵ

Students gain control of their learning as they advance from Stage Ⅰ to Ⅵ
Students get immersed in contextually rich concrete experiences; discuss how each member of the group can contribute.
Teacher provides students with resources; records initial questions; organizes what students know and what they want to know; organize them into groups; discusses what and how members of one's discipline come to know; guides and scaffolds modelling activities.
Students contribute what they know and what knowledge the situation requires; assess how the present situation challenges and extends what they know.
Teacher organizes the activity, provides resources, organizes students' initial questions, discusses limitations of each individual knowledge, monitors their work.
Students propose working hypotheses; propose patterns, models, theories, etc. that might explain the relationships between observed phenomena. They discuss and use different perspectives of the group members to interpret and understand the phenomenon.
Teacher monitors students' work.
Students test if and how the results of Stage IV agree with new concrete experiences; collect new evidence; use proposed models to generate new data, explain the phenomenon, allow for alternative explanations; test the models. Decide if the work is done and could be reported or start a new cycle of inquiry.
Teacher monitors students' work, asks questions, and regroups students when required.
Students return to Stage Ⅱ with enhanced understanding of the phenomenon, with a set of new questions as a motivation for a new cycle.
Teacher consolidates or revisits the modelling activity, suggests modifications for the follow-up.
Stage Teacher's and students' roles during modelling* Release of control
Teacher prepares students : Discusses how the epistemology is reflected in modelling.
Teacher prepares a lesson: Selects a phenomenon, develops activities, anticipates student difficulties, questions, potential challenges, etc.
Teacher releases control of student learning as students advance from Stage Ⅰ to Ⅵ

Students gain control of their learning as they advance from Stage Ⅰ to Ⅵ
Students get immersed in contextually rich concrete experiences; discuss how each member of the group can contribute.
Teacher provides students with resources; records initial questions; organizes what students know and what they want to know; organize them into groups; discusses what and how members of one's discipline come to know; guides and scaffolds modelling activities.
Students contribute what they know and what knowledge the situation requires; assess how the present situation challenges and extends what they know.
Teacher organizes the activity, provides resources, organizes students' initial questions, discusses limitations of each individual knowledge, monitors their work.
Students propose working hypotheses; propose patterns, models, theories, etc. that might explain the relationships between observed phenomena. They discuss and use different perspectives of the group members to interpret and understand the phenomenon.
Teacher monitors students' work.
Students test if and how the results of Stage IV agree with new concrete experiences; collect new evidence; use proposed models to generate new data, explain the phenomenon, allow for alternative explanations; test the models. Decide if the work is done and could be reported or start a new cycle of inquiry.
Teacher monitors students' work, asks questions, and regroups students when required.
Students return to Stage Ⅱ with enhanced understanding of the phenomenon, with a set of new questions as a motivation for a new cycle.
Teacher consolidates or revisits the modelling activity, suggests modifications for the follow-up.
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