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Article Contents

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

• 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:

• 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.

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

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 Ⅰ 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 Ⅱ 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. Ⅲ 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. Ⅳ 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. Ⅴ 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. Ⅵ 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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