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Professional Development in Mathematics

2012-2013 Algebra Academy  Exploring Student’s Mathematical Thinking  Probing the Math Needed for Algebra. Professional Development in Mathematics For District 287 & Member Districts Special Education Staff Nancy Nutting nancynutting@comcast.net

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Professional Development in Mathematics

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  1. 2012-2013 Algebra Academy  Exploring Student’s Mathematical Thinking Probing the Math Needed for Algebra Professional Development in Mathematics For District 287 & Member Districts Special Education Staff Nancy Nutting nancynutting@comcast.net Mary Peters mkpeters@district287.org Christina Shidlacachidla@district287.org Scott Swanson saswanson@district 287.org

  2. District 287 Program Facilitor for Professional Learning Jennifer Nelson jlnelson@district287.org 763-550-7241

  3. A Minnesota ProjectDeveloped by Districts 287 & 916, Metro ECSU, U of M, Hamline, Normandale CC Math Success: It’s In Our Hands Minneapolis/St. Paul Region 11 Math and Science Teacher Center www.region11mathandsciencecenter.com

  4. Surveyed 287 Special Ed Staff Math Committee Recommendations THIS is what we are doing today Algebra Academy held in 2011-12  positive feedback  share with others

  5. Today’s Session . . . Understanding Equality is Easy: True or False?

  6. In this session . . . • Identify benchmarks students reach on their way to understanding equality and being successful with algebra • Identify strategies students might use in working with equations • Practice crafting and sequencing equations that support student learning • Understand PLC structure and assessments you will use in your school between sessions

  7. We want to . . . . . . think more deeply about arithmetic in ways that are consistent with thinking in algebra. ARITHMETIC merely involves calculation. ALGEBRA involves seeing relationships. Thomas Carpenter, et. al., Thinking Mathematically: Integrating Arithmetic and Algebra, 2003

  8. This year, each session combines . . . Learning instructional strategies to increase skills & understandings Exploring the math behind what we teach When you “get” something, practice your questioning skills to help others understand Assessing our students and planning from what we learn Take care of your needs

  9. Designed for students who . . . Have 1 - 1 correspondence Can do single-digit addition and have strategies for basic subtraction Would benefit from understanding mathematics

  10. This year . . . cover + 2 school-based PLCs between sessions Full Day PD Sessions Wednesday, Sept. 19 (Room 321) EQUALITY Wednesday, Nov. 7 (Room 321) MODELING WORD PROBLEMS Wednesday, Dec. 12 (Room 321) RELATIONAL THINKING Wednesday, Jan. 16 (Room 321) OPERATIONS & BASIC FACTS Wednesday, Feb. 27(Room 321) FRACTIONS & DECIMALS Wednesday, May 1 (Room 321) EFFORT, PROBLEM SOLVING & REASONING

  11. Back at Your Table . . . 1 Talk about: How would you recognize a student who is mathematically powerful? • On chart paper, create a circle about 6” in diameter • Web the characteristics of a mathematically powerful student

  12. Checking Out Our Own Number Sense 2 “EQUAL 9” Post-it Note Posters Use +, —, x and/or ÷as many times as you like with 3, 4 or 5 of these numbers: 8 5 10 6 2to EQUAL 9 Write each equation on its own post-it note and add to your group’s poster. Be sure to write the whole equation. “equation” is a # sentence with an equal sign

  13. KRYPTO for older students 2 WARNING: It could be impossible to make the target number! Draw 5 numbers from a half deck of cards. Draw a 6th number for the target number. Students create table posters or contribute to a poster on the board. Try: http://mphgames.com

  14. http://illuminations.nctm.org 2 Deal Hint Solve Go to activities tab, search for Krypto. 2 9 7 4 5 Target 3 9 + 4 ̶ 5 + 2 ̶ 7 Primary Krypto uses just the digits 1-10.

  15. What’s does this equal? 2 3 + 4 x 5 = ? 10 – 6 + 5 = ? 35 or 23? ̶ 1 or 9? Parenthesis & Exponents PE Please Excuse Mult & Divide (in order) MDMy Dear Add & Subtract (in order) ASAunt Sally MN Benchmark 5.2.2.1 Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.

  16. Think about this problem 3 8 + 4 = + 5 What would students say belongs “in the box”? What does belong “in the box”?

  17. A Provocative Study (mid 90s) 3 Source: Carpenter, Franke and Levi

  18. What if the equations were . . . 3 Does format matter? 8 +  = 7 + 5 8 + 4 = 7 +  8 + 4 = k + 5, what is k? 8 + 4 = 7 + n, what is n?

  19. Instruction matters! Students’ Increase in Understanding of the Meaning of the Equal Sign (Number of students answering 8 + 4 = ___ + 5 correctly) NCISLA inBrief: ”Building a Foundation for Learning Algebra,” Fall 2000, http://ncisla.wceruw.org/publications/briefs/fall2000.pdf.

  20. Instruction matters! Minnesota Metro Area School in 2007 (Sept) 8 + 4 =  + 5 (Dec) 9 + 7 =  + 8 *CORRECT RESPONSE

  21. Metro Area Junior High Data Junior High Intervention Students n = 671

  22. Same Metro Area High School December 2, 2004

  23. Instruction Matters for Students with Special Needs too! Results from 2011-12 Anecdotes from Mary, Chris & Scott

  24. CGI (Cognitively Guided Instruction) Began as a research project between University of Wisconsin at Madison mathematicians and math educators and Madison area teachers. Goal: To explore how children understand mathematics and to move instruction from their understanding – what they cognitively know.

  25. Benchmarks in Student Thinking about the Equal Sign 4 • BASIC NUMBER SENTENCE SENSE • Children begin to write number sentences and describe their thinking about the equal sign. They begin to see that numbers or expressions on one side of the equal sign are the same amount asnumbers or expressions on the other side. Adapted from: Carpenter, Franke and Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Heinemann. Portsmouth, NH 2003 www.heinemann.com

  26. Begin even with young learners 2 and 3 are the same (amount) as 5 4 and 1 make the same as 5, etc. 5 is the same (amount) as 3 and 2 5 makesthe same as0 and 5, etc. 4 + 6 = 10 or 10 = 4 + 6 4 plus 6 is the same (amount) as 10 10 is the same (amount) as 4 plus 6

  27. Kevin in Kindergarten 5 Equation given to student: What does Kevin understand about the equal sign? If you were this child’s teacher, what problems would you have him work on next?

  28. Equality as Balance Number Balance Pan Balance

  29. = = = Balance 8 14 + 16 14 + 16 14 + 16 = = = 17 +  17 +  17 + 13

  30. Check balances on pages 6-12 • What’s my mathematical goal – What do I want students to notice? • What will you be doing in math during the next few weeks? On pp. 13-15 make a page or two of balance problems to bring out the mathematical ideas in a particular lesson. • Save p.15 as a master or get template from the Algebra Academy website.

  31. One caution about balances . . . Balances are a weight metaphor which causes some issues, e.g. work with subtraction or negative numbers. Watch carefully to see if students are using computation, in which case the equal sign as a balance point may work well. But if students are thinking about the concepts of balancing weights it may cause some misunderstandings.

  32. With older students,consider these two balances . . . + 5 + 5

  33. PAN BALANCES http://illuminations.nctm.org/ActivityDetail.aspx?ID=26

  34. You Tube Videos Linear Equations – Balancing the Equation http://www.youtube.com/watch?feature=fvwp&NR=1&v=DO-hzLh79uw Pan Balance with Shapes http://www.youtube.com/watch?feature=endscreen&v=vbX83p0xJ9c&NR=1

  35. Equality Benchmarks 4 2. EXPERIENCE WITH A VARIETY OF EQUATION TYPES • Children accept as true number sentences that go beyond the form a + b = c. They understand that equations in these forms might be true: 7 = 3 + 4 2 + 8 = 5 + 5 356 + 42 = 354 + 44 y = 3x + 7 Use: Equal 9 Posters or Krypto

  36. Equality Benchmarks 4 3. CALCULATING TO DETERMINE TRUTH (Operational Thinking) • Children recognize that the equal sign separates two equal values. They carry out calculations to determine that the two sides of an equation are equal or not equal. 8 + 4 = ___ + 5 12 12

  37. Equality Benchmarks 4 4. RELATIONAL THINKING • Children compare the expressions on each side of the equation and check for truth by identifying relationships among numbers and reasoning instead of actually carrying out the calculations. 8 + 4 = ___ + 5 • “7 is the missing number because 5 is one more than 4, so I need a number that is one less than 8.”

  38. Why does algebra make sense? 3m + 4 = 13 ̶̶ 4 ̶̶ 4 3m = 9 3m 3 = 9  3 m = 3 What if you only think about the equal sign as a signal to compute rather than separating two sides that have the same value? Terry Wyberg, U of M, Region 11 Grant Developer

  39. Middle School Study by Knuth 3 + 4 = 7 The arrow points to a symbol. What is the name of the symbol? What does the symbol mean? Can the symbol mean anything else? If yes, please explain. Source: Knuth, “The Importance of Equal Sign Understanding for the Middle Grades,” MTMS, May 2008

  40. Study by Knuth 3 + 4 = 7 Source: Knuth, “The Importance of Equal Sign Understanding”, MTMS, May 2008

  41. What if students were asked . . . 3 +  = 7 or 3 + m = 6 The arrow points to a symbol. What does this symbol mean?

  42. 4th Grade Bilingual Classroom 16 Note equations given to students (LH column). Why is each number sentence useful in developing students thinking about equality? If you were these children’s teacher, what equations might you use with them next?

  43. Usually avoid equal sign in strings 17 8 + 4 = 12 + 5 = 17 what’s wrong? but 8 + 4 ≠ 12 + 5 and 8 + 4 ≠ 17 Use a “goes to” arrow to track ongoing thinking 8 + 4 12 + 5 17

  44. Equation Chains 17 • An “equation chain”can use multiple equal signs if all the terms surrounding any equal sign are equal to each other. For example, children might generate many ways to make 10 and write the following “equation chain”: 10 = 6 + 4 = 7 + 3 = 20 – 10 = 100 – 90 = 7 + 2 + 1 

  45. How long can you go? • Consider having students create chains on adding machine tape to encourage flexible thinking about a given quantity and expressions that represent that amount. 75 = 3 x 25 = 100 – 25 = 7 x 10 + 5 Thanks to teachers at Willow Lane, White Bear Lake Area Schools, MN for the adding machine tape idea

  46. Work equation chains with a partner Whole Numbers + 27 = Start with the day of one of your birthdates. Take a strip of adding machine paper equal to your height. Add one or two expressions equal to that number. Move to next strip and add to it. Use with Calendar Math or Morning Meeting or Number of Day

  47. Misconceptions about Equality 18 • It is difficult to sort out exactly why misconceptions about the meaning of the equal sign are so pervasive and so persistent. A good guess is that many children see only examples of number sentences with an operation to the left of the equal sign and the answer on the right and they over generalize from those limited examples. Carpenter, Franke and Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Heinemann. Portsmouth, NH 2003 www.heinemann.com p. 22

  48. The Clothespin Card 18 Benchmarks 1, 2, 3  4 6 + 4

  49. Generate Equations 18 Algebrafy basic fact work and work with equality misconceptions. Why do you think you have found all the ways to make 10 with 2 numbers? Introduce TRUE OR FALSE as a way to look at an equation in its entirety.

  50. True or False Sentences Write the number sentence and then show whether it is TRUE or FALSE? 5 + 7 = 10 Benchmarks 1, 2, 3  4

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