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.netnancynutting@comcast.net Mary Peters mkpeters@district287.orgmkpeters@district287.org Christina Shidla cachidla@district287.orgcachidla@district287.org Scott Swansonsaswanson@district 287.org

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

3. A Minnesota Project Developed by Districts 287 & 916, Metro ECSU, U of M, Hamline, Normandale CC Minneapolis/St. Paul Region 11 Math and Science Teacher Center Math Success: It’s In Our Hands 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... Exploring the math behind what we teach Learning instructional strategies to increase skills & understandings Assessing our students and planning from what we learn When you “get” something, practice your questioning skills to help others understand 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... 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 + 2 school-based PLCs between sessions cover

11. Back at Your Table... 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 1

12. Checking Out Our Own Number Sense “EQUAL 9” Post-it  Note Posters Use +, —, x and/or ÷ as many times as you like with 3, 4 or 5 of these numbers: 8510 6 2 to 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 2

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

14. http://illuminations.nctm.org Deal Hint Solve 2

15. What’s does this equal? 3 + 4 x 5 = ? 10 – 6 + 5 = ? 35 or 23? ̶ 1 or 9? Parenthesis & Exponents PE Please Excuse Mult & Divide (in order) MD My Dear Add & Subtract (in order) AS Aunt 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. 2

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

17. A Provocative Study (mid 90s) Source: Carpenter, Franke and Levi Percent Responding responses 7*1217 12&17 other Gr. 3 & 4 94925107 Gr. 5 & 6 2762120 Gr. 1 & 2 55813 8 16 3

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

19. Instruction matters! Students’ Increase in Understanding of the Meaning of the Equal Sign (Number of students answering 8 + 4 = ___ + 5 correctly) Before adjusting instruction After adjusting instruction Gr. 1 & 25%66% Gr. 3 & 49%72% Gr. 5 & 62%84% NCISLA inBrief: ”Building a Foundation for Learning Algebra,” Fall 2000, http://ncisla.wceruw.org/publications/briefs/fall2000.pdf. http://ncisla.wceruw.org/publications/briefs/fall2000.pdf

20. Instruction matters! 7 * (8*)12 (16)17 (24)other School W Sept. 07 Dec. 07 Sept. 07 Dec. 07 Sept. 07 Dec. 07 Sept. 07 Dec. 07 Grade 2 6%33%53%40%17%13%25%15% Grade 3 34%65%44%28%13%6%9%1% Grade 4 26%75%59%17%15%8%0%3% Grade 5 65%88%25%5%7%6%3%1% Minnesota Metro Area School in 2007 (Sept) 8 + 4 = + 5 (Dec) 9 + 7 = + 8 *CORRECT RESPONSE

21. Metro Area Junior High Data Responses n % 726840 1227441 177611 12 or 17315 other223 Junior High Intervention Students n = 671

22. Same Metro Area High School December 2, 2004 Geo1Geo2Geo3BCGeo Responses n = 29 n = 31n = 34n = 11 65.52%61.29%85.29%72.73%7 20.69%22.58%8.82%9.09%12 0.00%6.45%5.88%0.00%17 6.90%3.23%0.00% 9 0.000.00%12 or 17 6.90%6.45%0.00%18.18%other

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 1. 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 as numbers 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.comwww.heinemann.com 4

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 makes the same as 0 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 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? 5

28. Equality as Balance Pan Balance Number Balance

29. Balance = 14 + 1617 +  = = 14 + 1617 +  = = 14 + 1617 + 13= 8

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... Consider what numbers create balance in the following situation? Consider what students might say about how this equation works on a balance: What numbers do the circle and the triangle represent if this balance DOES balance? 3 – 5 + 6 = 5 – 1 + 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=fvw p&NR=1&v=DO-hzLh79uw Pan Balance with Shapes http://www.youtube.com/watch?feature=end screen&v=vbX83p0xJ9c&NR=1

35. Equality Benchmarks 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 4

36. Equality Benchmarks 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 4

37. Equality Benchmarks 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.” 4

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 a) The arrow points to a symbol. What is the name of the symbol? b) What does the symbol mean? c) 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 Best Definition Percent of Students (n=375) RelationalOperationalOther No response don’t know Gr. 7 365293 Gr. 8 464581 Gr. 6 29587 6

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

42. 4 th Grade Bilingual Classroom 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? 16

43. Usually avoid equal sign in strings 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 17

44. Equation Chains 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 17

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 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 27 = Whole Numbers +

47. Misconceptions about Equality 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 www.heinemann.com 18

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

49. Generate Equations 6 + 4 = 100 + 10 = 10 4 + 6 = 101 + 9 = 10 2 + 8 = 10 7 + 3 = 103 + 7 = 10 4 + 6 = 10 5 + 5 = 10 6 + 4 = 10 7 + 3 = 10 etc. 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. 18

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

51. “Play” True or False The “combinations that make 10 activity” was only the context for another goal – introducing the true/false strategy for evaluating equations and checking out their understanding of equality. 10 = 6 + 4 T or F? Math people can do things forwards and backwards as long as they tell the truth! 18

52. Vocabulary Issues Establish the words true or false connected to students. e.g. Today everyone is wearing the color red. True or False? ¿Es verdad o falso? “Grown-up” vs “Kid” Hmong Tseeb Tsi Tseeb truenot true YogTsi Yog correctnot correct

53. Order from Easiest to Hardest select one quadrant If you wanted to expose your students to equations beyond a + b = c which equations would be easiest and which would be hardest for your students? Use label: T or F? Benchmarks 1, 2, 3 19

54. Open Number Sentences-Make it True Write the number sentence with a box or a variable representing some missing information. What goes in the box (or what does the letter have to be) to make the sentence (equation) true? 10 = 6 + n  + 7 = 10 Benchmarks 1, 2, 3  4

55. Using True-False Sentences Using Open Number Sentences On p. 20 create some true/false sentences or open number sentences you could use next week with your students 20

56. One Side is the Same as the Other Which symbol makes the sentence true?   Benchmarks 1 & 2 21

57. One Side is the Same as the Other Insert the  or  into each equation – circle or highlight all the true equations To what mathematical ideas was I trying to draw attention? What do you notice? What mathematical conversations do I want my students to have? Benchmarks 1, 2, 3, 4 22-27

58. One Side is the Same as the Other What is your mathematical focus for your students (or adults with whom you work) next week or in the near future? What theme will you use for crafting  or  equations on pp. 28-29? (save p. 30 as a master or get template at Algebra Academy website) What kind of conversation do you want to occur? Benchmarks 1, 2, 3  4 28-29

59. Fitting in True/False, Open Number and =/≠ Work True or False? Why? Daily board work as students come in to class Math Circles/Morning Meeting/Daily Oral Math Do a few before/after lessons using white boards Quick transition after specialist or lunch

60. Exploring Student Thinking Between our Professional Development sessions, engage in 2 PLC sessions, facilitated by a staff member at your school or form a professional partnership. PLC #1 centers on the Baseline Assessment PLC #2 centers on Instructional Strategies and Interviews with Students Putting PD into practice with YOUR students

61. 2 PLC sessions + Next PD Session PLC 1 PLC 2 Bring to Next PD Session Nov. 7

62. Throughout our ten-year study, whenever we found an effective school or an effective department within a school, without exception that school or department has been a part of a collaborative professional learning community. - Milbury McLaughlin Need for a Collaborative Culture

63. based on the work of Nancy Love, ICTM Pre Session Conference, Oct. 17, 2002 State & District Evidence Building Evidence Classroom specific evidence such as student work samples, observations, surveys, interviews Improved Student Learning Improvement in Curriculum, Instruction and Assessment Drilling Down through Data to Improve Student Learning

64. Baseline Assessment/PLC#1 How well do your students understand equality? Give the Baseline Assessment, preferably THIS WEEK, to your students without any direct instruction. You may read items to students. Manipulative may be available. We are looking for a baseline – what do they actually understand right now? Summarize your results on the recording sheets and bring students’ work and recording sheets to your 1 ST PLC meeting. (Also save for Nov. 7 th )

65. Baseline Assessment Form B 5 problems to give to students: 1) True or false? 2) 3) 4) (grades PK-3?) 5) Two teams have the same number of players. There are 8 girls and 4 boys on one team; if 5 boys are on team two, how many are girls? Check out recording sheets

66. Baseline Assessment Form A 5 easier problems to give to students: 1) 2 + 3 = 2) 10 = 6 + 4 True or false? 3) 8 + 4 = + 5 4) 3 + = 5 + 2 5) Two teams have the same number of players. There are 8 girls and 4 boys on one team; if 5 boys are on team two, how many are girls? Check out recording sheets

67. If your students are less verbal... read problems to them, as needed Then collect papers  Discuss some of the problems with a student or a group of students  Record their ideas and bring to PLC/next session

68. What do you think? In one Minnesota District in 2006… 59% of grade 4-5-6 students correctly put “7” in the equation in response to: 2. What number can you put in the box to make this true? 8 + 4 = + 5

69. Consider this similar problem… 4. The Cardinals and Blue Jays soccer teams have the same number of players on each team. There are 8 girls and 4 boys on the Cardinals. If there are 5 boys on the Blue Jays, how many girls are there on the team? Explain you thinking. Answer __________ Write a number sentence to show how you solved this problem. of the same grade 4-5-6 students got this problem right. 92% Context matters!

70. Teaching is the constant in classrooms; learning is the variable. Learning should be the constant; teaching the variable. Lee Jenkins, The Ten Root Causes of Educational Frustration, DataNotGuesswork Seminar, Sept. 19, 2001

71. Instructional Strategies/PLC#2 Try some of the instructional strategies or games suggested today: Krypto/Equal 9 posters/Equal # Increasing the variety of equation formats Using “is the same as” language with “equals” Using = and ≠ or balances Using true/false or open number sentences Using equation chains or clothespin cards Come to the 2 nd PLC meeting ready to share what is happening with your students & how you use these strategies with your students.

72. Why is “Equality” so important? When students are limited in the kinds of equations with which they work, the conversation can only be about the right answer. When students can think about the equal sign differently, the conversations can focus on understanding important math concepts and skills. Listening to students’ math conversations makes us smarter teachers--we learn what students know and what they don’t know. “Math Talk” is essential!

73. PLCs and Teacher Knowledge Matter In the 7-county Metro Area, those schools which had not made AYP and whose staff members did attend the gr. 3-5 Math Academies... had greater MCA scores than schools who did not attend Region 11 Evaluation Report and Metro ECSU communication, 2010

74. Assessing Student Learning We need multiple indicators No one kind of information tells the whole story. You need to match the evidence to the audience What evidence will they trust? Thomas R. Guskey, conference address, MSDC, May 15, 2002

75. Interview Protocol/PLC #2 Select up to 3 students for a teaching interview with each of the students. Seeing (hearing & watching) is believing! Check out your Interview Protocol. Bring your recording sheets and student work to the 2 nd PLC meeting.

76. Summative Assessment/SD Day Parallel to the Baseline Give to your students before 11/7 Bring Baseline AND Summative recording sheets and student work to next session on 11/7 Get all assessments online at the Academy Website

78. MN Mathematics Standards Numbering System 78 3.2.1.4 GRADE LEVEL STRANDSTANDARDBENCHMARK K-8, 9-111 Number 2 Algebra 3 Geom/Meas 4 Data May have several per strand May have several per standard; includes examples

79. Supporting Standards & IEP Goals Using the Standards by Progression Document, identify a standard and some benchmarks that could be supported with one of today’s activities ALGEBRA NUMBER DATA/PROBABILITY GEOMETRY/MEAS.

80. MATHEMATICS & SCIENCE FRAMEWORKS FOR MN STANDARDS STEM TEACHER RESOURCE CENTER www.scimathmn.org/stemtc A New Resource for Teachers to Identify Resources to Help Students Achieve Minnesota Standards in Mathematics or Science Launched 6/30/2011 – work in progress – your feedback is appreciated!

83. OVERVIEW  Summarizes Big Ideas and Essential Understandings of the Standard/Benchmarks  Correlations to NCTM Standards  Correlations to national Common Core Standards MISCONCEPTIONS  Points out frequent misconceptions and common errors students have about these math ideas VIGNETTE  A snapshot into a classroom featuring an acitivity and teacher/student dialogue  Gives a picture of what standard/benchmarks mean RESOURCES  Teaching notes and ideas for learning  Resource lists  Key vocabulary and ideas for working with vocab.  Professional development/PLC resources  References

84. ASSESSMENT  Sample tasks to assess this standard/benchmarks  May be used formatively or summatively DIFFERENTIATION  Ideas for struggling learners and English Learners  Ideas to extend the learning for kids who “get it”  Resource lists PARENTS/ADMINISTRATORS  Checklists and resources for parents, coaches, administrators

85. Scavenger Hunt Select a mathematics standard and cluster of benchmarks on which to focus Explore the 7 tabs Scavenger Hunt

86. Next time (Wednesday, Nov. 7) Bring both Baseline & Summative Assessment recording sheets/student work. We’ll review your experiences with equality, equations and talking about the equal sign We’ll focus on Modeling Word Problems Is every word problem a unique experience for students? How can graphic organizers help students understand, categorize and solve word problems?

87. Feedback please... Please fill out the sheet with shapes to help us plan for next time Check in with your school/district team about PLC meetings Talk about what you can try out from today’s session with your students Identify an article to read