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

2. 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”. “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.

3. 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?

4. Best Practices Tell Us: Concrete Representational Abstract

5. 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.

6. But WHAT IS MATH? Mathematics is the science of patterns and relationships. It is the language and logic of our technological world.

7. Patterns Number Sense Number Sense Place Value Place Value Operations Operations Formulas Formulas Functions Graphs Probability Statistics

8. 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 Us o Output: Make a List & Be Able to Defend Mathematical Connection (can be related to most recent unit of instruction) o Output: Make a List & Be Able to Defend Mathematical Connection (can be related to most recent unit of instruction) Concept B: Number Sense 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)

9. 5 Minutes – GO!

10. Concept: Mathematical Representations Objective: Use mathematical language to describe the world. Why? Mathematics is the language and logic of our technological world.

11. 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)

12. Mathematical Power ExploreConjecture Reason Logically Use a Variety of Mathematical Methods Effectively Solve Problems

13. 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

14. STUDENT OUTPUT list collage photo sketch objects

15. 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

16. http://blog.mrmeyer.com/ http://blog.mrmeyer.com/ blog: http://blog.mrmeyer.com/http://blog.mrmeyer.com/ Math Class Needs a Makeover Dan Meyer (at TED) Dan Meyer (at TED) http://blog.mrmeyer.com/?cat=59 http://blog.mrmeyer.com/?cat=59 his archive for algebra tags: http://blog.mrmeyer.com/?cat=59 http://blog.mrmeyer.com/?cat=59

17. 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”. “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.

18. Say What?!?!?!?!?! Engrade.com (Discussion Board only accessible to that class) www.mrsbsmathlab.blogspot.com

20. Some Structures for Assessment Think Green ;) Blogs, discussion boards, video clips, drawing, Math “Ads” http://mrsbmathlab.blogspot.com/2011/10/web-surfing- geometric-thinking.htmlhttp://mrsbmathlab.blogspot.com/2011/10/web-surfing- geometric-thinking.html Math Journaling Turn to your neighbor Group By Similar Strategy or Assignment Peer "assess" - checklist for completion/credit for assignment Partner/Group Share Calculator, Discussion, Number Line, Protractor, Hundreds Chart Graphic Organizer/Notes Record of Tools Used Examples, Non-Examples, Definition Graphic Organizers Think back to vocabulary strategies!! Mathematical Definition(s)

21. Some Less-Traditional Sources for Exploring Student Artwork How Many Sets? The Grocery Aisle Find the Difference Puzzles and Entertainment MENSA

24. 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?

25. Mathematical Know-How Effective Strategy Works for ALL similar situations Effective Strategy Assists with organizing concepts and materials

26. Good Problem Solvers Solve Problems Based on Common Structures

27. Solve It! Read Paraphrase Visualize Hypothesize Estimate Compute Check

28. READ the problem. READ the problem. Decoding the Text Decoding the Math

29. Identify the important information. Identify the important information. More than key words. More than key words. Mathematical Concepts Mathematical Concepts Put it into (your own) words. Put it into (your own) words. LANGUAGE gives meaning.

30. Diagram Graph Chart A Schematic: A MAP or math blueprint MUST RELATE TO THE PARAPHRASE.

31. Make a plan Make a plan Choose operations Choose operations Putting together the puzzle…. Putting together the puzzle…. Setting up your mathematical experiment/notation Setting up your mathematical experiment/notation Pattern and Logic in Mathematical Language

32. Create a realistic target. Create a realistic target. The Goal is the 2 nd Ring – not a bullseye. The Goal is the 2 nd Ring – not a bullseye. Relies on Comprehension of Mathematical Concept/Operations.

33. Computation Supports o Multiplication Chart o Calculator o Tactile Materials Visualization & Hypothesis o Self-Made Supports o Self-talk & routine o Developing Algorithms

34. Using the Inverse Using the Inverse “Am I right?” “I am right AND I can prove it.” “Am I right?” “I am right AND I can prove it.”Meta-cognition.

35. Solve It = Math Power! Read Paraphrase Visualize Hypothesize Estimate Compute Check

36. Mathematical Power ExploreConjecture Reason Logically Use a Variety of Mathematical Methods Effectively Solve Problems

37. 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.

38. Novelty + Application can develop Fluid Thinking while Mastering Concepts Developing & Supporting Pattern Recognition, Pattern Formation and Mathematical Language Development

39. Some Math Lab Staples Shapes Dry Erase Markers/Whiteboards Toothpicks or Straws Page ProtectorsMy SmartPhoneSidewalk Chalk Straight Edges VandeWalle!!!!Random Objects

40. Other Barto Staples VandeWalle SousaNumber Devil MintonMontague

41. 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?

42. Common Core Mathematical Power

43. 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.

44. Montague, Marjorie. Math Problem Solving for Middle School Students with Disabilities http://www.k8accesscenter.org/training_resources /MathProblemSolving.asp http://www.k8accesscenter.org/training_resources /MathProblemSolving.asp Curriculum Resource available from Exceptional Innovations at http://www.exinn.net/solve-it.html http://www.exinn.net/solve-it.html

45. Barto’s Gurus Montague, M. (2003). Solve It!: A mathematical problem- solving 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 The Van de Walle Professional Mathematics Series

46. Material Resources.. ETA Cuiseniare ETA Cuiseniarehttp://www.etacuisenaire.com/ Exceptional Innovations Exceptional Innovations Solve It!: A Practical Approach to Teaching Mathematical Problem Solving Skills Solve It!: A Practical Approach to Teaching Mathematical Problem Solving Skillshttp://www.exinn.net/solve-it.html Pearson AGS Globe Pearson AGS Globehttp://www.pearsonschool.com/index.cfm?locator=PSZ1Ai Key Curriculum Press Key Curriculum Press http://www.keypress.com/ Corwin Press Corwin Presshttp://www.corwin.com/corwinSearch.nav?&_requestid=2043573

47. Some Resources Boaler, Jo. What’s Math Got to Do With It?. 2008. 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. Brown, C. A Roadmap for Math Achievement for All Students. Center for Comprehensive School Reform and Instruction.Center for Comprehensive School Reform and Instruction.Center for Comprehensive School Reform and Instruction. Enzensberger, Hans. The Number Devil. 1997 Enzensberger, Hans. The Number Devil. 1997 VanDerHeyden, A. RTI and Math Instruction. RTI Action Network. VanDerHeyden, A. RTI and Math Instruction. RTI Action Network.RTI Action Network.RTI Action Network.

48. Some On-Line Resources National Math Panel Reports, Jan. 2007. National Math Panel Reports, Jan. 2007. National Math Panel Reports National Math Panel Reports Math Laboratory Blog Math Laboratory Bloghttp://www.mrsbsmathlaboratory.blogspot.com LDA of America LDA of Americahttp://www.ldaamerica.us/ National Library of Virtual Manipulatives National Library of Virtual Manipulativeshttp://nlvm.usu.edu/ The Algebra Project The Algebra Projectwww.algebra.org Funbrain Funbrainhttp://www.funbrain.com/math/index.html National Council for Teachers of Mathematics National Council for Teachers of Mathematicshttp://illuminations.nctm.org

49. Yeah, I Can Keep Going http://www.spotthedifference.com/explorer.asp http://www.spotthedifference.com/explorer.asp http://www.spotthedifference.com/explorer.asp http://www.spotthedifference.com/explorer.asp Basher Books Basher Books Basher Books Basher Books Math Practice Guides – What Works! Math Practice Guides – What Works! Math Practice Guides – What Works! Math Practice Guides – What Works! http://www.lumosity.com/ http://www.lumosity.com/ http://www.lumosity.com/ Common Core for Math Common Core for Math Common Core for Math Common Core for Math Facebook/Mindjolt Games Facebook/Mindjolt Games

50. Contact Information Amy Barto tchrqueen1@gmail.com www.bartosworld.blogspot.com My Class: www.mrsbmathlab.blogspot.com