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Becoming Better Reasoners: Supporting Students to Develop as Problem-Solvers

3:30 - 5:00 Saturday, December 7
    Max Ray, Philadelphia, PA

How do we move students along the continuum from novice to expert problem solvers? We'll explore strategies such as Solve a Simpler Problem, Look at Cases, Make a Table, and Make a Model, exploring what it means to get better at each. We will solve problems together and use our own work and student work to practice recognizing novice versions of strategic thinking, as well as discuss activities and teacher moves that can help students see problem solving as a process they can get better at.

Introductions
 
Overture: Guess and Check
In which I tell a story about getting good at a problem-solving strategy.
See the whole PowerPoint I excerpted from, in context [HTML]
 
Act 1: Make a Mathematical Model
A mathematical model is a way of encoding or describing a situation about quantity or shape in a way that is "sharable, transportable, and reusable" (Lesh and Doerr, Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching)
 
Anticipating
What makes it hard for students?
What approaches have you seen?
What common "stages" have you seen?
 
Congruent Rectangles -- Hidden Constraints, Easy Numbers
In this activity, we consider multiple approaches to solving this problem and how to encourage students to consider a symbolic model.
Download the Problem [PDF]
Download some Extensions [PDF]
 
Eating Contest -- Student Work
In this activity, we look at a range of student work to see what it can unveil about how students develop as algebraic modelers.
Download the problem students solved [PDF]
Download the student work [PDF]
 
Reflecting on Making a Mathematical Model
Different ways?
What does is mean to get good at making tables?
How do we help students get better at making tables?
 
Act 2: Tables and Patterns
 
Lucky Lollipops -- Table Gallery
In this activity, we focus on the question "what needs to be organized?" and use a "Gallery Walk" to consider different ways to organize information.
Download the Problem-solving Scenario [PDF]
Download the handout [PDF] (1st page for teacher, 2nd page for students)
 
Falling Leaves -- Student Work
In this activity, we look at a range of student-generated tables and use a "Representation Station" to reflect on and compare different approaches.
Download the problem students solved [PDF]
Download the student work [PDF]
View a blog post about the student work
Download the Representation Station activity description [PDF]
 
Reflecting on Tables
Different ways?
What does is mean to get good at making tables?
How do we help students get better at making tables?
 
Discussion
What ideas might you try?
What challenges do you foresee?
What will you look for as a success?
What resources might you need? What resources do you have that you know you could use?
 
Coda: Solve a Simpler Problem
 
Don't Be Square -- What makes this problem hard? What are valid simplifications?
In this handout, you'll find a wide range of student work which all addresses what makes the problem hard. The validity question seems important!
Download the problem students solved [PDF]
Download the student work [DOC]
 
Trick or Treat Route -- How can simplifications be used to find patterns and build up solutions?
In this handout, you'll find different ways that students attempted to organize their work to tackle a problem about "how many ways...".
Download the problem students solved [PDF]
Download the student work [DOC]
 
Additional Resources
Blog: Max at the Math Forum [HTML]
 
Annie Fetter: Ever Wonder What They'd Notice? [Ignite! Talk]
Suzanne Alejandre: Unsilence Students' Voices [Ignite! Talk]
Max Ray: Why 2 > 4: A Proof by Induction [Ignite! Talk]
 
Ostrich Llama Count--Examining Solution Methods [PDF]
How to Start Problem Solving in Your Classroom [PDF]
Problem Solving-It Has to Begin with Noticing and Wondering [PDF]
Unsilence Students' Voices [PDF]
Supporting Sense-Making and Perseverance [PDF]
Think You Don't Have Time to Use the PoWs? [PDF]
Writing to Develop Understanding: The Math Forum @ Drexel's PoWs [PDF]
 
Making Effective Use of Math Forum Resources: K-2 and 3-5 [HTML] 6-8 and 9-12 [HTML]
 
Our book: Powerful Problem Solving: Activities for Sense-Making with the Mathematical Practices [HTML]
Our book's companion website: [HTML] including freely accessible classroom videos

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