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David C. Webb University of Colorado Boulder
Freudenthal Institute US
NCTM 2014 Annual Meeting April 10, 2014
ALGEBRA ON MY MIND
TOOLS TO PROMOTE ALGEBRAIC REASONING
PROPOSED LEARNING GOALS • Getting to know a bit more about…
• What is Realistic Mathematics Education? • What is progressive formalization? • Ways to promote algebraic reasoning
• Develop a deeper understanding of the design of questions,
tasks, & opportunities to develop algebraic reasoning • Use tools and representations that support “seeing
structure” “making connections” and other math practices
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REALISTIC MATHEMATICS EDUCATION • Hans Freudenthal
• “mathematics as a human activity” • realiseren – to imagine • Guided reinvention
• Role of the teacher • Student-centered modeling
• Eliciting students’ representations & strategies • Mathematics should make sense to students
• Progressive formalization • Purposeful sequence of questions and activities
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FIUS, University of Colorado Boulder, USA FIsme, Utrecht University, the Netherlands
The Development of Mathema0cal and Scien0fic Reasoning through
Contexts and Representa0ons in STEM
5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity
© F.M.- N.B.
informal, situational
preformal, structured
The Iceberg Metaphor
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5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity5+ 2= 7
5 25 2 37fo rm a l no ta tio ns to p o f the ic e b e rgflo a tingc a p a c ity
© F.M.- N.B.
informal, situational
preformal, structured
What are models and tools in “icebergs for algebra”?
Patterns & Generalization
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“The language of arithmetic focuses on answers while the language of algebra focuses on relationships.”
-- MacGregor, M. & Stacey, K. (1999). A flying start to algebra. Teaching Children Mathematics, 6(2), 78-86.the ways we have students’ demonstrate algebraic reasoning include …. ?
What are the strategies and tools that we can use to support student algebraic reasoning?
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AS YOU WORK THROUGH THE TASKS, CONSIDER THE FOLLOWING…
• How do the features of task (context, representation, information) support making connections and seeing the mathematical structure of the problem?
• How does the problem • help students see mathematical structure? • Support relational reasoning, making connections?
• What do students organize? Reason about?
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15 …. 10
Add the circles to find the square
24 46
Add the circles to find the square
….
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…. 46
Add the circles to find the square
….
…. 46
Same Rule Applies: Add the circles to find the square
….
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46
Same Rule Applies: Add the circles to find the square
….
FEATURES OF ALGEBRAIC REASONING… • Noticing, extending, and generalizing patterns
• Numerical, visual, pre-formal rules, etc
• Relating numbers and representations that support algebraic reasoning
• Understanding equality and operations
• Other ideas?
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ACTIVITY: REFLECT ON TASKS ENCOUNTERED
• As we work through these tasks, consider the • Use of number • Use of representation • Access to less formal solution strategies
• How does the model/tool lead to other relationships?
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ACTIVITY: TASK DESIGN
• How does the representation promote algebraic reasoning? Can this be a basis for other questions?
• What are other questions that come to mind?
FOR MORE INFORMATION • WWW.FIUS.ORG • Contact David Webb – [email protected]
• Britannica Mathematics in Context – Exhibit Hall Booth • Fosnot, Dolk, Jacob et al – Math in the City (K-8) • Webb, Boswinkel & Dekker (2008). Beneath the Tip of the Iceberg.
Mathematics Teaching in the Middle School
• Webb, van der Kooij & Geist (2011). Design Research in the Netherlands: Introducing Logarithms Using Realistic Mathematics Education. Journal of Math Ed Teachers College.
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Questions?
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