Redesigning Mathematical Curriculum for Blended Learning
Abstract
:1. Introduction: Student Pods during the Pandemic
2. Designing for Virtual Math Teams
- First, it generated and collected data on small online groups of public-school students collaborating on problem solving.
- Second, it provided computer support, including a shared whiteboard and a dynamic-geometry app.
- Third, it analyzed the group interaction that unfolded in the team discourse.
- Fourth, it elaborated aspects of a theory of “group cognition” [19]. Several papers published during this period and contributing to the broad vision of CSCL have now been reprinted and reflected upon in Theoretical Investigations: Philosophic Foundations of Group Cognition [11]. Several chapters in this volume analyze aspects of group cognition based on excerpts of student discourses during VMT sessions.
3. Redesigning for Pandemic Pods with GeoGebra Classes
4. Findings from VMT Trials
5. Supporting Group Practices in Blended Learning
6. Broadening the Model for Blended Learning
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Group Collaboration Practices: |
1. Discursive turn taking (responding to each other and eliciting responses). |
2. Coordinating activity (deciding who should take each step). |
3. Constituting a collectivity (e.g., using “we” rather than “I” as agent). |
4. Sequentiality (establishing meaning by temporal context). |
5. Co-presence (being situated together in a shared world of concerns). |
6. Joint attention (focus on the same, shared images, words and actions). |
7. Opening and closing topics (changing discourse topics together). |
8. Interpersonal temporality (recognizing the same sequence of topics, etc.). |
9. Shared understanding (common ground). |
10. Repair of understanding problems (explicitly fixing misunderstandings). |
11. Indexicality (referencing the same things with their discourse). |
12. Use of new terminology (adopting new shared words). |
13. Group agency (deciding what to do as a group). |
14. Sociality (maintaining friendly relations). |
15. Intersubjectivity (sharing perspectives). |
Group Dragging Practices: |
1. Do not drag lines to visually coincide with existing points, but use the points to construct lines between or through them. |
2. Observe visible feedback from the software to guide dragging and construction. |
3. Drag points to test if geometric relationships are maintained. |
4. Drag geometric objects to observe invariances. |
5. Drag geometric objects to vary the figures and see if relationships are always maintained. |
6. Some points cannot be dragged or only dragged to a limited extent; they are constrained. |
Group Construction Practices: |
1. Reproduce a figure by following instruction steps. |
2. Draw a figure by dragging objects to appear right. |
3. Draw a figure by dragging objects and then measure to check. |
4. Draw a figure by dragging objects to align with a standard. |
5. Construct equal lengths using radii of circles. |
6. Use previous construction practices to solve new problems. |
7. Construct an object using existing points to define the object by those points. |
8. Discuss geometric relationships as results of the construction process. |
9. Check a construction by dragging its points to test if relationships remain invariant. |
Group Tool-Usage Practices: |
1. Use two points to define a line or segment. |
2. Use special GeoGebra tools to construct perpendicular lines. |
3. Use custom tools to reproduce constructed figures. |
4. Use the drag test to check constructions for invariants resulting from custom tools. |
Group Dependency-Related Practices: |
1. Drag the vertices of a figure to explore its invariants and their dependencies. |
2. Construct an equilateral triangle with two sides having lengths dependent on the length of the base, by using circles to define the dependency. |
3. Circles that define dependencies can be hidden from view, but not deleted, and still maintain the dependencies. |
4. Construct a point confined to a segment by creating a point on the segment. |
5. Construct dependencies by identifying relationships among objects, such as segments that must be the same length. |
6. Construct an inscribed triangle using the compass tool to make distances to the three vertices dependent on each other. |
7. Use the drag test to check constructions for invariants. |
8. Discuss relationships among a figure’s objects to identify the need for construction of dependencies. |
9. Points in GeoGebra are colored differently if they are free, restricted or dependent. |
10. Indications of dependency imply the existence of constructions (such as regular circles or compass circles) that maintain the dependencies, even if the construction objects are hidden. |
11. Construct a square with two perpendiculars to the base with lengths dependent on the length of the base. |
12. Construct an inscribed square using the compass tool to make distances on the four sides dependent on each other. |
13. Use the drag test routinely to check constructions for invariants. |
Group Practices Using Chat and GeoGebra Actions: |
1. Identify a specific figure for analysis. |
2. Reference a geometric object by the letters labeling its vertices or defining points. |
3. Vary a figure to expand the generality of observations to a range of variations |
4. Drag vertices to explore what relationships are invariant when objects are moved, rotated, extended. |
5. Drag vertices to explore what objects are dependent upon the positions of other objects. |
6. Notice interesting behaviors of mathematical objects |
7. Use precise mathematical terminology to describe objects and their behaviors. |
8. Discuss observations, conjectures and proposals to clarify and examine them |
9. Discuss the design of dependencies needed to construct figures with specific invariants. |
10. Use discourse to focus joint attention and to point to visual details. |
11. Bridge to past related experiences and situate them in the present context. |
12. Wonder, conjecture, propose. Use these to guide exploration. |
13. Display geometric relationships by dragging to reveal and communicate complex behaviors. |
14. Design a sequence of construction steps that would result in desired dependencies. |
15. Drag to test conjectures. |
16. Construct a designed figure to test the design of dependencies. |
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Stahl, G. Redesigning Mathematical Curriculum for Blended Learning. Educ. Sci. 2021, 11, 165. https://doi.org/10.3390/educsci11040165
Stahl G. Redesigning Mathematical Curriculum for Blended Learning. Education Sciences. 2021; 11(4):165. https://doi.org/10.3390/educsci11040165
Chicago/Turabian StyleStahl, Gerry. 2021. "Redesigning Mathematical Curriculum for Blended Learning" Education Sciences 11, no. 4: 165. https://doi.org/10.3390/educsci11040165