Problem Solving and Supporting Algebraic Thinking at the Middle and Secondary Grades Amy Barto LDA of Michigan November 14, 2011.

Problem Solving Problem Solving and Supporting and Supporting

Algebraic Thinking at the Algebraic Thinking at the Middle and Secondary GradesMiddle and Secondary Grades

Amy BartoAmy BartoLDA of MichiganLDA of Michigan

November 14, 2011November 14, 2011

Food For ThoughtFood For Thought

““Mathematical know-how is not only Mathematical know-how is not only one of the most important qualities one of the most important qualities for workers to possess in the future, for workers to possess in the future, it is critical for successful functioning it is critical for successful functioning in life”.in life”.

- - Jo Boaler, 2008.Jo Boaler, 2008.

Prime Your CircuitPrime Your Circuit

Best Practices Tell Us:Best Practices Tell Us:

More Food For ThoughtMore Food For Thought“… “… twenty-first-century citizens need twenty-first-century citizens need mathematics. But the mathematics that people mathematics. But the mathematics that people need is not the sort of math learned in most need is not the sort of math learned in most classrooms. 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?But WHAT IS MATH?

PatternsPatterns• Number Number

SenseSense• Place ValuePlace Value• OperationsOperations• Formulas Formulas

• Functions• Graphs• Probability• Statistics

DO THIS AT HOME!DO THIS AT HOME!• Concept A: Mathematical Representations Around Us Concept A: Mathematical Representations Around Us

o 5 minute Math Hunt: Examples of Math Around Us5 minute Math Hunt: Examples of Math Around Uso Output: Make a List & Be Able to Defend Mathematical Output: Make a List & Be Able to Defend Mathematical

Connection (can be related to most recent unit of Connection (can be related to most recent unit of instruction) instruction) 

• Concept B: Number Sense Concept B: Number Sense o   5 minute Number Hunt: Representations of 5 minute Number Hunt: Representations of

Levels of Difficulty = DifferentiationLevels of Difficulty = Differentiation Whole Numbers; decimals; fractions, exponential Whole Numbers; decimals; fractions, exponential

numbersnumberso Output: Make a List (must have different types of Output: Make a List (must have different types of

examples; i.e. not a list of numerals)examples; i.e. not a list of numerals)  

5 Minutes – GO! 5 Minutes – GO!

Mathematical ObservationsMathematical Observations

ScaffoldingScaffolding Exploration and Exploration and Effective ApplicationEffective Application

Speaking of……Speaking of……

““Children need to solve complex Children need to solve complex problems, to ask many forms of problems, to ask many forms of questions, and to use, adapt, and apply questions, and to use, adapt, and apply standard methods, as well as to make standard methods, as well as to make connections between methods and to connections between methods and to reason mathematically – and they can reason mathematically – and they can engage in such methods at home and engage in such methods at home and (one hopes) at school”.(one hopes) at school”.

- - Jo Boaler, 2008.Jo Boaler, 2008.

Say What?!?!?!?!?!Say What?!?!?!?!?!

Some Structures for Some Structures for AssessmentAssessment

Some Less-Traditional Some Less-Traditional Sources for ExploringSources for Exploring

Sarah’s father is 60 years old. Sarah is Sarah’s father is 60 years old. Sarah is ¼ as old her father. ¼ as old her father.

How many years older than Sarah is How many years older than Sarah is her father?her father?

At what age will Sarah be 1/10 her At what age will Sarah be 1/10 her father’s age?father’s age?

Mathematical Know-HowMathematical Know-How

Good Problem SolversGood Problem Solvers

Solve Problems Solve Problems

Based on Common StructuresBased on Common Structures

Solve It!Solve It!

READ the problem.READ the problem.

Decoding the TextDecoding the Text

Decoding the MathDecoding the Math

• IdentifyIdentify the important the important information.information.• More than key words.More than key words.• Mathematical ConceptsMathematical Concepts

• Put it into (your own) words. Put it into (your own) words.

LANGUAGE gives meaning.LANGUAGE gives meaning.

• DiagramDiagram• GraphGraph• ChartChart• A Schematic: A MAP or math A Schematic: A MAP or math

blueprintblueprint

MUST RELATE TO THE MUST RELATE TO THE PARAPHRASE.PARAPHRASE.

• Make a planMake a plan• Choose operationsChoose operations• Putting together the puzzle….Putting together the puzzle….• Setting up your mathematical Setting up your mathematical

experiment/notationexperiment/notation

Pattern and Logic in Pattern and Logic in Mathematical LanguageMathematical Language

• Create a realistic target.Create a realistic target.• The Goal is the 2The Goal is the 2ndnd Ring – not a bullseye. Ring – not a bullseye.

Relies on Comprehension of Mathematical Relies on Comprehension of Mathematical Concept/Operations.Concept/Operations.

Computation SupportsComputation Supportso Multiplication ChartMultiplication Charto CalculatorCalculatoro Tactile MaterialsTactile Materials

Visualization & Visualization & HypothesisHypothesis

o Self-Made SupportsSelf-Made Supportso Self-talk & routineSelf-talk & routineo Developing AlgorithmsDeveloping Algorithms

• Using the InverseUsing the Inverse• ““Am I right?” Am I right?” “I am right AND I “I am right AND I

can prove it.”can prove it.”

Meta-cognition. Meta-cognition.

Solve It = Math Power!Solve It = Math Power!

Teacher’s RoleTeacher’s Role

Some Math Lab StaplesSome Math Lab Staples

Other Barto StaplesOther Barto Staples

Back to The Fun Stuff!!!!!Back to The Fun Stuff!!!!!

Sarah’s father is 60 years old. Sarah is Sarah’s father is 60 years old. Sarah is ¼ as old her father. ¼ as old her father.

How many years older than Sarah is How many years older than Sarah is her father?her father?

At what age will Sarah be 1/10 her At what age will Sarah be 1/10 her father’s age?father’s age?

The ultimate goal of mathematics The ultimate goal of mathematics education is for education is for all studentsall students to to

develop mathematical power to develop mathematical power to participate fully as a citizen and participate fully as a citizen and

worker in our contemporary world.worker in our contemporary world.

Montague, Marjorie.Montague, Marjorie. Math Problem Math Problem Solving for Middle School Students with Solving for Middle School Students with

DisabilitiesDisabilitieshttp://www.k8accesscenter.org/training_resources/MathProblemSolving.asp

Curriculum Resource available from Curriculum Resource available from Exceptional Innovations at Exceptional Innovations at

http://www.exinn.net/solve-it.htmlhttp://www.exinn.net/solve-it.html

Barto’s GurusBarto’s Gurus

Montague, M. (2003). Solve It!Montague, M. (2003). Solve It!: A mathematical problem-: A mathematical problem-solving instructional program.solving instructional program. Reston, VA: Exceptional Reston, VA: Exceptional Innovations.Innovations.

Montague, MMontague, M., ., Warger, C., & Morgan, H. (2000) Solve it! Warger, C., & Morgan, H. (2000) Solve it! Strategy instruction to improve mathematical problem solving. Strategy instruction to improve mathematical problem solving. Learning Disabilities Research and PracticeLearning Disabilities Research and Practice, , 1515, 110-116., 110-116.

Wong, B.Y, Harris, K.R., Graham, S., & Butler, D.L. (2003). Wong, B.Y, Harris, K.R., Graham, S., & Butler, D.L. (2003). Cognitive strategies instruction research in learning disabilities. Cognitive strategies instruction research in learning disabilities. In H.L. Swanson, K.R. Harris, & S. Graham (Eds.). In H.L. Swanson, K.R. Harris, & S. Graham (Eds.). Handbook Handbook of learning disabilities of learning disabilities (pp. 273-292). New York: Guilford. (pp. 273-292). New York: Guilford.

Sousa, David. Sousa, David. How the Brain Learns MathematicsHow the Brain Learns Mathematics. 2007.. 2007.

The Van de Walle Professional Mathematics SeriesThe Van de Walle Professional Mathematics Series

Material Resources..Material Resources..ETA CuiseniareETA Cuiseniare

http://www.etacuisenaire.com/http://www.etacuisenaire.com/Exceptional InnovationsExceptional Innovations Solve It!: A Practical Approach to Teaching Mathematical Problem Solve It!: A Practical Approach to Teaching Mathematical Problem

Solving SkillsSolving Skillshttp://www.exinn.net/solve-it.htmlhttp://www.exinn.net/solve-it.html

Pearson AGS GlobePearson AGS Globehttp://www.pearsonschool.com/index.cfm?locator=PSZ1Aihttp://www.pearsonschool.com/index.cfm?locator=PSZ1Ai

Key Curriculum PressKey Curriculum Presshttp://www.keypress.com/http://www.keypress.com/

Corwin PressCorwin Presshttp://www.corwin.com/corwinSearch.nav?&_requestid=2043573http://www.corwin.com/corwinSearch.nav?&_requestid=2043573

Some ResourcesSome Resources• Boaler, Jo. Boaler, Jo. What’s Math Got to Do With It?.What’s Math Got to Do With It?.

2008.2008.• Brown, C. Brown, C. A Roadmap for Math A Roadmap for Math

Achievement for All StudentsAchievement for All Students. . Center for Center for Comprehensive School Reform and Comprehensive School Reform and Instruction.Instruction.

• Enzensberger, Hans. Enzensberger, Hans. The Number DevilThe Number Devil. . 19971997

• VanDerHeyden, A. VanDerHeyden, A. RTI and Math RTI and Math InstructionInstruction. . RTI Action Network.RTI Action Network.

Some On-Line ResourcesSome On-Line ResourcesNational Math Panel Reports, Jan. 2007.National Math Panel Reports, Jan. 2007.Math Laboratory BlogMath Laboratory Bloghttp://www.mrsbsmathlaboratory.blogspot.comhttp://www.mrsbsmathlaboratory.blogspot.comLDA of AmericaLDA of America

http://www.ldaamerica.us/http://www.ldaamerica.us/National Library of Virtual ManipulativesNational Library of Virtual Manipulatives

http://nlvm.usu.edu/http://nlvm.usu.edu/The Algebra ProjectThe Algebra Project

www.algebra.orgwww.algebra.orgFunbrainFunbrain

http://www.funbrain.com/math/index.htmlhttp://www.funbrain.com/math/index.htmlNational Council for Teachers of MathematicsNational Council for Teachers of Mathematics

http://illuminations.nctm.org http://illuminations.nctm.org

Yeah, I Can Keep GoingYeah, I Can Keep Going

http://www.spotthedifference.com/http://www.spotthedifference.com/explorer.aspexplorer.asp

Basher Books Basher Books Math Practice Guides – What Works!Math Practice Guides – What Works!http://www.lumosity.com/http://www.lumosity.com/Common Core for MathCommon Core for MathFacebook/Mindjolt GamesFacebook/Mindjolt Games

Contact InformationContact Information

Amy BartoAmy [email protected]@gmail.com

www.bartosworld.blogspot.comwww.bartosworld.blogspot.com

My Class:My Class:

www.mrsbmathlab.blogspot.comwww.mrsbmathlab.blogspot.com