Abstract
In this chapter, the theoretical construct of guided reinvention is extended to include desirable pedagogical practices for teachers implementing RME sequences. First, we explain what a guided reinvention teaching approach looks like and how it evolved out of over 25 years of research. We then articulate the planning and teaching practices of guided reinvention teachers and describe how those practices move beyond what many call “inquiry approaches” to mathematics teaching. We end the chapter by offering a set of learning goals that professional developers might use when mentoring aspiring guided reinvention teachers.
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Notes
- 1.
It is important to emphasize that the hypothetical learning trajectory describes the learning of the class as a collective. That is, it refers to the taken-as-shared learning of the group of students as a whole. It does not refer to the learning of individual students.
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Appendices
Appendix A: Integer Hypothetical Learning Trajectory (HLT)
Phase | Tool | Imagery | Activity/taken-as-shared interests | Possible topics of mathematical discourse | Possible gesturing and metaphors |
|---|---|---|---|---|---|
One | Net worth statements | Assets and debts are quantities that have opposite effect on net worth | Learning finance terms | Conceptualizing an asset as something owned, a debt as something owed Conceptualizing a net worth as an abstract quantity (not tangible) | |
Two | Net worth statements (vertical number line) | Differences in collections of assets and collections of debts | Determining a person’s net worth Who is worth more? | Different strategies for finding net worths | Pay off |
Three | Symbols (+ and −) | + means asset and − means debt | Determining and comparing net worths | Different strategies for finding net worths Creating additive inverses as objects | Pay off |
Four | Good decisions increase net worth Bad decisions decrease net worth | Which transactions have good and bad effects on net worth? | When taking away an asset, is this good or bad? When taking away a debt, is this good or bad? Judging the results of transactions and therefore direction to move on a number line | Arms moving up and down to indicate good or bad movements | |
Five | Vertical number line (VNL) Model of to model for transition | Empty number line to express (+ and −) movements | Transactions Reasoning with number line to find a net worth after a transaction has occurred | How do various transactions affect net worth? Going through zero The effect of different transactions Different strategies for finding net worths | Arms moving up and down to indicate good or bad movements Pay off |
Six | Unknown transaction/net worth problems | Determining different possible transactions | Inventing integer rules +(+) = + −(−) = + +(−) = − −(+) = − | Pay off |
Appendix B: Sample of Cognitive-Based Interview Task
| Ellen, Jim, and Steve bought three helium-filled balloons and paid $2.00 for all three. They decided to go back to the store and get enough balloons for everyone in their class. How much did they have to pay for 24 balloons? |
Appendix C: A Resource for Transforming GR Classroom Practices
Teacher evidence | Student evidence | |
|---|---|---|
Teacher practice: social norms | ||
• T encourages Ss to explain | ||
• T encourages Ss to ask questions | ||
• T encourages Ss to ask questions to other Ss | ||
• T encourages Ss to understand other Ss solutions | ||
• T encourages Ss to use mistakes as learning opportunities | ||
• T encourages Ss to indicate agreement or disagreement | ||
• T encourages Ss to take responsibility/ownership for their learning | ||
Teacher practice: discourse | ||
• T restates Ss explanation in clearer language | ||
• T restates Ss explanation in a more advanced way | ||
• T introduces vocabulary when students have invented an idea | ||
• T asks Ss to repeat other Ss solutions | ||
• T asks questions that promote higher-level thinking (e.g., comparing, analyzing, synthesizing) | ||
• T uses Ss solutions effectively to engineer his/her summary | ||
Teacher practice: mathematical | ||
• T encourages conjecturing | ||
• T encourages proving | ||
• T encourages different solutions | ||
• T encourages efficient solutions | ||
• T encourages sophisticated solutions | ||
Teacher practice: imagery | ||
• T encourages Ss to record their thinking | ||
• T encourages Ss to model their thinking | ||
• T encourages Ss to draw on previous images when they are stuck | ||
• T “cements” Ss ideas on board or in display around the room | ||
Teacher practice: small group | ||
• T encourages Ss to ask each other for help | ||
• T collects data, not fix Ss mistakes | ||
• T asks Ss how they solved problems | ||
• T encourages Ss to draw on previous images when they are stuck | ||
Appendix D: Coaching Template
Mathematical Idea(s) of Lesson |
Launch |
Explore: Anticipated Student Thinking |
Whole-Class Discussion |
Assessment: What Evidence Shows Mathematical Ideas Are/Are Becoming Realized |
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Stephan, M., Underwood-Gregg, D., Yackel, E. (2014). Guided Reinvention: What Is It and How Do Teachers Learn This Teaching Approach?. In: Li, Y., Silver, E., Li, S. (eds) Transforming Mathematics Instruction. Advances in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-04993-9_4
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