Problem Solving
and Supporting
Algebraic Thinking at the
Middle and Secondary Grades
Amy Barto
LDA of Michigan
November 14, 2011
Food For Thought

“Mathematical know-how is not only one
of the most important qualities for
workers to possess in the future, it is
critical for successful functioning in
life”.
- Jo Boaler, 2008.
Prime Your Circuit
What is
mathematical
know-how?
How would you
describe your
mathematical
know-how?
How would you
describe yourself
as an instructor of
mathematics?
Best Practices Tell Us:
Abstract
Concrete
Representational
More Food For Thought
“… twenty-first-century citizens need
mathematics. But the mathematics that people
need is not the sort of math learned in most
classrooms.
People do not need to regurgitate hundreds
of standard methods.
They need to reason and problem solve, flexibly
applying new methods in new situations.
Mathematics is now so critical to American
citizens that some have labeled it the ‘new
civil right’.”
- Jo Boaler, 2008.
But WHAT IS MATH?
It is the language
and logic of our
technological
world.
Mathematics is the
science of patterns
and relationships.
Patterns
•
•
•
•
Number Sense
Place Value
Operations
Formulas
•
•
•
•
Functions
Graphs
Probability
Statistics
DO THIS AT HOME!
•
Concept A: Mathematical Representations Around Us
o 5 minute Math Hunt: Examples of Math Around Us
o Output: Make a List & Be Able to Defend Mathematical
Connection (can be related to most recent unit of
instruction)
•
Concept B: Number Sense
o 5 minute Number Hunt: Representations of
 Levels of Difficulty = Differentiation
 Whole Numbers; decimals; fractions, exponential
numbers
o Output: Make a List (must have different types of
examples; i.e. not a list of numerals)
5 Minutes – GO!
Concept:
Mathematical
Representations
Objective: Use
mathematical
language to
describe the world.
Why? Mathematics
is the language
and logic of our
technological
world.
Concept:
Number
Sense
Objective:Continue to
recognize/reinforce that
a numeral is a
representation of a
number (vs. the number
itself).
Why? A variable is
also a representation
of a number - NOT a
letter included in an
equation.
Concrete (number, set
of items) representational
(numeral, variable) abstract (using without
context in an equation)
Explore
Effectively
Solve
Problems
Conjecture
Mathematical
Power
Use a Variety
of
Mathematical
Methods
Reason
Logically
Mathematical Observations
HOMEWORK
• Bring at least # examples of addition you
observe over the next 24 hours
• Examples of a Numerical Representation
• Examples of Fractions AND explanation of
how it is representational
EXPECTATIONS
• Minimum Number of Examples
• Examples Represent Concept
• Shows Comprehension of
Concept Through Explanation
STUDENT OUTPUT
•
•
•
•
•
list
collage
photo
sketch
objects
Scaffolding Exploration and
Effective Application
Development of
Mathematical
Power
• Conjecture
• Use a variety of
methods effectively
• Reason Logically
Opportunities for
Development of
Patient Problem
Solving
Math Class Needs a
Makeover
Dan Meyer (at TED)
Speaking of……

“Children need to solve complex
problems, to ask many forms of
questions, and to use, adapt, and apply
standard methods, as well as to make
connections between methods and to
reason mathematically – and they can
engage in such methods at home and
(one hopes) at school”.
- Jo Boaler, 2008.
Say What?!?!?!?!?!
Engrade.com
(Discussion Board only
accessible to that class)
www.mrsbsmathlab.blogspot.com
Some Structures for Assessment
Math
Journaling
Partner/Group
Share
Record of
Tools Used
Mathematical
Definition(s)
• Think Green ;)
• Blogs, discussion boards, video clips, drawing, Math “Ads”
• http://mrsbmathlab.blogspot.com/2011/10/web-surfinggeometric-thinking.html
• Turn to your neighbor
• Group By Similar Strategy or Assignment
• Peer "assess" - checklist for completion/credit for assignment
• Calculator, Discussion, Number Line, Protractor, Hundreds
Chart
• Graphic Organizer/Notes
• Examples, Non-Examples, Definition
• Graphic Organizers
• Think back to vocabulary strategies!!
Some Less-Traditional
Sources for Exploring
How Many Sets?
Find the Difference
The
Grocery
Aisle
Student
Artwork
MENSA
Puzzles and
Entertainment
Sarah’s father is 60 years old.
Sarah is ¼ as old her father.
How many years older than
Sarah is her father?
At what age will Sarah be 1/10
her father’s age?
Mathematical Know-How
Effective
Strategy
Assists with
organizing
concepts and
materials
Effective
Strategy
Works for
ALL similar
situations
Good Problem Solvers
Solve Problems
Based on Common Structures
Solve It!
Read
Paraphrase
Visualize
Hypothesize
Estimate
Compute
Check

READ the problem.
Decoding the Text
Decoding the Math
•
Identify the important
information.
•
•
•
More than key words.
Mathematical Concepts
Put it into (your own) words.
LANGUAGE gives meaning.
•
•
•
•
Diagram
Graph
Chart
A Schematic: A MAP or math
blueprint
MUST RELATE TO THE
PARAPHRASE.
•
•
•
•
Make a plan
Choose operations
Putting together the puzzle….
Setting up your mathematical
experiment/notation
Pattern and Logic in
Mathematical Language
•
Create a realistic target.
•
The Goal is the 2nd Ring – not a bullseye.
Relies on Comprehension of Mathematical
Concept/Operations.
Visualization &
Hypothesis
o Self-Made Supports
o Self-talk & routine
o Developing Algorithms
Computation Supports
o
o
o
Multiplication Chart
Calculator
Tactile Materials
•
•
Using the Inverse
“Am I right?”
I can prove it.”
“I am right AND
Meta-cognition.
Solve It = Math Power!
Read
Paraphrase
Visualize
Hypothesize
Estimate
Compute
Check
Explore
Effectively
Solve
Problems
Conjecture
Mathematical
Power
Use a Variety
of
Mathematical
Methods
Reason
Logically
Teacher’s Role
providing students with sufficient practice
distributed over time
including a conceptually rich and varied mix of
problems to support their learning
encourage and support students in their efforts to
master difficult mathematics content
Students who believe that effort, not just inherent
talent, counts in learning mathematics can improve
their performance.
Novelty + Application can
develop Fluid
Thinking while Mastering
Concepts
Developing & Supporting
Pattern Recognition,
Pattern Formation and
Mathematical Language
Development
Some Math Lab Staples
Shapes
Dry Erase
Markers/Whiteboards
Toothpicks or Straws
Page Protectors
My SmartPhone
Sidewalk Chalk
Straight Edges
Random Objects 
VandeWalle!!!!
Other Barto Staples
VandeWalle
Sousa
Number Devil
Minton
Montague
Back to The Fun Stuff!!!!!
Sarah’s father is 60 years old.
Sarah is ¼ as old her father.
How many years older than
Sarah is her father?
At what age will Sarah be 1/10
her father’s age?
Mathematical
Power
Common
Core
The ultimate goal of mathematics
education is for all students to
develop mathematical power to
participate fully as a citizen and
worker in our contemporary world.
Montague, Marjorie. Math Problem
Solving for Middle School Students with
Disabilities
http://www.k8accesscenter.org/training_resources
/MathProblemSolving.asp
Curriculum Resource available from
Exceptional Innovations at
http://www.exinn.net/solve-it.html
Barto’s Gurus
Montague, M. (2003). Solve It!: A mathematical problemsolving instructional program. Reston, VA: Exceptional
Innovations.
Montague, M., Warger, C., & Morgan, H. (2000) Solve it!
Strategy instruction to improve mathematical problem solving.
Learning Disabilities Research and Practice, 15, 110-116.
Wong, B.Y, Harris, K.R., Graham, S., & Butler, D.L. (2003).
Cognitive strategies instruction research in learning disabilities.
In H.L. Swanson, K.R. Harris, & S. Graham (Eds.). Handbook
of learning disabilities (pp. 273-292). New York: Guilford.
Sousa, David. How the Brain Learns Mathematics. 2007.
The Van de Walle Professional Mathematics Series
Material Resources..
ETA
Cuiseniare
http://www.etacuisenaire.com/
Exceptional Innovations

Solve It!: A Practical Approach to Teaching Mathematical Problem
Solving Skills
http://www.exinn.net/solve-it.html
Pearson AGS Globe
http://www.pearsonschool.com/index.cfm?locator=PSZ1Ai
Key
Curriculum Press
http://www.keypress.com/
Corwin Press
http://www.corwin.com/corwinSearch.nav?&_requestid=2043573
Some Resources
•
•
•
•
Boaler, Jo. What’s Math Got to Do With It?.
2008.
Brown, C. A Roadmap for Math
Achievement for All Students. Center for
Comprehensive School Reform and
Instruction.
Enzensberger, Hans. The Number Devil.
1997
VanDerHeyden, A. RTI and Math
Instruction. RTI Action Network.
Some On-Line Resources
National
Math Panel Reports, Jan. 2007.
Math Laboratory Blog
http://www.mrsbsmathlaboratory.blogspot.com
LDA of America
http://www.ldaamerica.us/
National Library of Virtual Manipulatives
http://nlvm.usu.edu/
The Algebra Project
www.algebra.org
Funbrain
http://www.funbrain.com/math/index.html
National Council for Teachers of Mathematics
http://illuminations.nctm.org
Yeah, I Can Keep Going
http://www.spotthedifference.com/explorer
.asp
 Basher Books
 Math Practice Guides – What Works!
 http://www.lumosity.com/
 Common Core for Math
 Facebook/Mindjolt Games

Contact Information
Amy Barto
tchrqueen1@gmail.com
www.bartosworld.blogspot.com
My Class:
www.mrsbmathlab.blogspot.com