Mathematics for ELL Students (Workshop 2) focuses on the ways in which middle grades educators can support the specific needs of English Language Learners in the math classroom. This presentation is part of a broader workshop for educators. More information at http://middlegradesmath.org

1. Teaching High-Level Mathematics to English Language Learners in the Middle Grades 1135 Tremont Street, Suite 490 Boston MA 02120 © Copyright 2009 Center for Collaborative Education/Turning Points Workshop 2

2. Teaching High-Level Mathematics to English Language Learners in the Middle Grades was developed by Turning Points, a project at the Center for Collaborative Education in Boston, MA. This tool is part of the Mathematics Improvement Toolkit , a project of the National Forum to Accelerate Middle Grades Reform, and was supported by the U.S. Department of Education’s Comprehensive School Reform Initiative, grant #S332B060005. Opinions expressed are those of the authors and are not necessarily those of the Department. Developed by Turning Points, a project of the Center for Collaborative Education

4. Workshop 2 Agenda Warm-up/Checking In Big Ideas of Workshop 2 – Mathematical Tasks and Cognitive Demand Comparing the Cognitive Demands of Two Tasks: Linguistic, mathematical and cultural Doing Mathematics Classroom Video #1 Doing Mathematics: Finding the Best Box Reflecting on Doing Mathematics Keeping Classroom Activities at a High Level of Cognitive Demand Action Plans: Next Steps in Your Own Classroom Closing Circle 2-1

6. Warm-Up / Checking In

7. Read HANDOUT 2-3: Best Practices Modeled in Workshop 1 WRITE on HANDOUT 2-4: o ne practice from the list that you have added to your personal toolkit. Explain how it has changed your teaching and how it has helped the English language learners in your classroom to learn high-level mathematics. Warm-up/Checking In 2-3, 2-4

9. Warm-up/Checking In (continued) Reflecting on your experience Take some notes for yourself. Why did you choose this particular strategy to try out? What did you learn through sharing with a partner? 2-6 Share

10. Introducing the “Big Ideas” of Workshop 2

12. Quasar Project findings about teaching high-level math to urban middle school students Research-based strategies that support English language learners to develop critical thinking, academic language, communication skills A coherent approach to teaching high-level math to English language learners High-Level Mathematics for English Language Learners Research on Learning High-Level Math Research on English Language Learners This Project

13. “ Big Idea” of the Quasar Project: High Levels of Cognitive Demand Lead to Substantial Learning Gains “… students who performed best … were in classrooms in which tasks were … set up and implemented at high levels of cognitive demand … For these students, having the opportunity to work on challenging tasks in a supportive classroom environment translated into substantial learning gains on an instrument designed to measure student thinking, reasoning, problem solving and communication. ”* * A Quote from: Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development . Teachers College Press, 2000, by M. K. Stein, M. S. Smith, M.A. Henningson and E. A. Silver. 2-7

15. The cognitive demand of a task is the sum total of what a student needs to know, understand and be able to do in order to solve a problem or complete a task successfully. What is Cognitive Demand? The level of cognitive demand depends not only on the task, but also on the prior knowledge of the students. 2-7

17. Comparing the Cognitive Demands of Two Tasks

18. Consider the cognitive demands of the following two tasks for English language learners (HANDOUT 2-9) Task 1. Find the surface area and volume of a rectangular prism that measures 2” x 4” x 24”. 2-9 Task 2. Out of This World Candies plans to sell Starburst candies in a new package containing 24 individually wrapped Starbursts. Your challenge is to find the dimensions of the least expensive box that can hold exactly 24 Starbursts. Each wrapped Starburst has a square shape that measures 2 cm on a side and 1 cm high. Write a report including the dimensions for all the possible boxes we can use to package 24 Starbursts, the dimensions of the least expensive box, and explain how you know your answers are correct. 2 cm 2 cm 1 cm

21. Jigsaw Activity for Task 1 (continued) Reassemble into groups of three people, with one linguistic expert, one math expert, and one cultural expert in each group. Each person needs to share your expert group’s responses with the other two. Ask questions and make comments. Make sure that everyone becomes an expert on all three questions! 2-13

22. What are the cognitive demands of Task 2 for English language learners? Think about the cognitive demands of Task 2 individually. Write your answers for Linguistic, Mathematical and Cultural demands in the second columns of HANDOUTS 2-10, 2-11 and 2-12. Task 2. Out of This World Candies plans to sell Starburst candies in a new package containing 24 individually wrapped Starbursts. Your challenge is to find the dimensions of the least expensive box that can hold exactly 24 Starbursts. Each wrapped Starburst has a square shape, that measures 2 cm on a side and 1 cm high. Write a report including the dimensions for all the possible boxes we can use to package 24 Starbursts, the dimensions of the least expensive box for us to make, and explain how you know your answers are correct. 2-10, 2-11, 2-12 2 cm 2 cm 1 cm

24. Jigsaw Activity for Task 2 (continued) Reassemble into groups of three people, with one linguistic expert, one math expert, and one cultural expert in each group. Each person needs to share your expert group’s responses with the other two. Ask questions and make comments. Make sure that everyone becomes an expert on all three questions! 2-14

25. Recall the Key Finding of the Quasar Project Answer this question individually using Handout 2-15 . “… students who performed best … were in classrooms in which tasks were … set up and implemented at high levels of cognitive demand … For these students, having the opportunity to work on challenging tasks in a supportive classroom environment translated into substantial learning gains on an instrument designed to measure student thinking, reasoning, problem solving and communication. ” 2-15 If you want English language learners to engage in high-level mathematics and become successful mathematical thinkers, reasoners, problem solvers and communicators, would you use Task 1 or Task 2? Explain why. Now let’s take a poll. How many chose Task 1? _________ Task 2? __________

26. Doing Mathematics!

28. Doing Mathematics When students are engaged in tasks at the highest level, we call this “Doing Mathematics.” This will be the focus of the rest of this workshop. We will begin with a thorough exploration of Task 2

29. Doing Mathematics What kind of package does it come in? How do you think candy makers decide what kinds of packages to use? Turn and talk to a partner. Take turns answering these questions. Share your responses to the second question. Think about your favorite kind of candy.

30. DOING MATHEMATICS “ FINDING THE BEST BOX” Read the complete instructions in Handout 2-16: Thinking Geometrically. Our company, Out of This World Candies , plans to sell our Starburst candies in a new package containing 24 individually wrapped Starbursts. Your challenge is to find the dimensions of the least expensive box that can hold exactly 24 Starbursts. Write a report including the dimensions for all the possible boxes we can use to package 24 Starbursts, the dimensions of the least expensive box, and explain how you know your answers are correct. You and your teammates represent the Best Solutions Consulting Company . Out of This World Candies wants you to solve the following problem: 2-16 Each wrapped Starburst has a square shape that measures 2 cm on a side and 1 cm high. 2 cm 2 cm 1 cm

33. After viewing the video and writing your observations and questions Turn and Talk with a partner. Discuss what you have both written on HANDOUT 2-17 . SHARE A Classroom Example of Doing Mathematics Finding the Best Box (continued)

34. DOING MATHEMATICS “ FINDING THE BEST BOX” Read the complete instructions in Handout 2-16: Thinking Geometrically. Our company, Out of This World Candies , plans to sell our Starburst candies in a new package containing 24 individually wrapped Starbursts. Your challenge is to find the dimensions of the least expensive box that can hold exactly 24 Starbursts. Write a report including the dimensions for all the possible boxes we can use to package 24 Starbursts, the dimensions of the least expensive box, and explain how you know your answers are correct. You and your teammates represent the Best Solutions Consulting Company . Out of This World Candies wants you to solve the following problem: 2-16 Each wrapped Starburst has a square shape that measures 2 cm on a side and 1 cm high. 2 cm 2 cm 1 cm

36. 1st step: Unpacking the Problem Take a few minutes to re-read the problem on HANDOUT 2-16 . Think individually about each topic listed below. What SPECIFIC INFORMATION is given in the problem? What PRIOR KNOWLEDGE can we use to solve the problem? What do we need to FIND OUT that will help solve the problem? 2-16, 2-19, 2-20, 2-21

37. 1st step: Unpacking the Problem (continued) Use HANDOUT 2-19 . Individually write your answer to the question: What SPECIFIC INFORMATION is given in the problem? Then take turns using Speak, Listen, Question and Respond to talk with a partner about the specific information given. SHARE with the large group. 2-16, 2-19 Speak Listen Question Respond The problem tells us that there are 24 Starbursts in a box. Hmm ... She talked about how many Starbursts are in a box. Why is that important? Because that will tell us how big the boxes have to be.

38. 1st step: Unpacking the Problem (continued) Use HANDOUT 2-20. Individually write your answer to the question: What PRIOR KNOWLEDGE can we use to solve the problem? Then take turns using Speak, Listen, Question and Respond to talk with a partner about the prior knowledge in the problem. SHARE with the large group. 2-16, 2-20

39. 1st step: Unpacking the Problem (continued) Use HANDOUT 2-21. Individually write your answer to the question: What do we need to FIND OUT that will help us solve the problem? Then take turns using Speak, Listen, Question and Respond to talk with a partner about what we need to find out to solve the problem. SHARE with the large group. 2-16, 2-21

40. 2nd step: Partial Solutions 2-22 CHOOSE a STRATEGY to solve a problem Use Easier Numbers 1, 2, 3 10, 20, 30 Write an equation a 2 + b 2 = c 2 Draw a Picture Create a Model Make a list, table or chart Look for a Pattern Work backwards

41. 2nd step: Partial Solutions Work by with your partner to find the dimensions in centimeters of one box that can hold exactly 24 Starbursts. First Choose a Strategy ( HANDOUT 2-22 ) you will use to find the dimensions. When you have found the dimensions of one box draw a large diagram of the box on a sheet of blank paper, showing its length, width and height in centimeters. Record your results HANDOUT 2-23 and post them on a chart where the entire group can see them. 2-22, 2-23 DIMENSIONS DIAGRAMS 8 cm x 6 cm x 2 cm 8 cm 6 cm 2 cm

42. 2nd step: Partial Solutions (continued) Look at all the dimensions of the boxes posted so far. Think and write (HANDOUT 2-24) Do you think we have found all the possible boxes? Explain: give reasons for your answer. How many more do you think there might be: hundreds of possibilities or just a few? Explain: give reasons for your answer. 2-24

43. 2nd step: Partial Solutions (continued) Listeners Choose a scaffolding question from the bottom of the handout. Next Speaker Try to answer each person’s question. Take turns speaking and asking questions until the group has reached consensus. GET INTO TEAMS OF FOUR. First speaker Share what you have written on HANDOUT 2-24 using 2-23, 2-24 “ I think there are … no a few many … more possible boxes because …”

44. Using Scaffolding Questions I think there are many more possible boxes because we’ve got only three different ones so far. Could we make a stack even taller? I can visualize a tall stack of 24 Starbursts. The stack will fill a box that’s 2 cm long, 2 cm wide and 24 cm high Can someone give me an example? #1 #2 #3 #4 2-23, 2-24 When you have reached consensus on this one question, share with the larger group.

45. 3nd step: Completing the Solution FINDING ALL THE BOXES TEAM: Work with your teammates to make a plan for finding all the possible boxes that can hold exactly 24 Starbursts. As you make your plan, think about the following questions: How will we organize our work to keep track of all the boxes we have found? How will we know for sure that we have found all the possible boxes? SHARE YOUR PLAN 2-23, 2-24

47. 3rd step: Completing the Solution (continued) HANDOUT 2-25 shows some of the data gathered by teachers in an earlier workshop. TEAM What pattern or patterns do you notice in the data? Can these patterns help you determine whether you have found all the possible boxes that can hold exactly 24 Starbursts? 2-25 FINDING ALL THE BOXES

48. TEAM Talk with your team. Review the list of boxes and dimensions posted so far. Are you all convinced that you have found all the possible boxes that will hold exactly 24 Starbursts? Can you make a convincing argument about this within your team? Try to reach consensus on this point. 3rd step: Completing the Solution (continued) Finding all the boxes

49. SHARE Does the entire workshop group agree that all possible boxes have been found? 3rd step: Completing the Solution (continued) Finding all the boxes

50. 3rd step: Completing the Solution (continued) WRITE: Individually answer the question at the top of HANDOUT 2-26. FINDING THE LEAST EXPENSIVE BOX How can we determine which box is least expensive to produce? 2-26

51. 3rd step: Completing the Solution (continued) TEAM First speaker. Use this pattern language to share what you have written on HANDOUT 2-26: To find the least expensive box I suggest that we _________________ because ____________________. Listeners Choose a scaffolding question from HANDOUT 2-26 Next Speaker Answer each person’s question. Take turns speaking and asking questions until the group has reached consensus on a plan. SHARE FINDING THE LEAST EXPENSIVE BOX 2-26

53. 3rd step: Completing the Solution (continued) THINK AND WRITE: Individually, use HANDOUT 2-27a and 2-27b to plan your group’s report to Out of This World Candies . Re-read HANDOUT 2-16 to remember what the company asked for. TEAM Take turns sharing items from your lists and deciding whether to include each item in the report. Together, write a list of what you think the report should contain. SHARE Planning a Final Report 2-27

58. Essential Questions for Lesson Planning (continued) 2. How does this lesson insure that all English Language Learners are engaged at all times throughout the learning process? 2-29 Write: Individually write your responses. Pair: Discuss your responses with your partner. Share with the whole group. Think: Take about a minute to think about the second question as it applies to the activity we just completed . Think Write Pair Share

60. Keeping Classroom Activities at a High Level of Cognitive Demand

64. Identifying Levels of Cognitive Demand 2-30, 2-31 TEAM: Form groups of three. Use the Final Word protocol to discuss this with your team mates. Each of you will be the first speaker for one of the three tasks.

65. Identifying Levels of Cognitive Demand 2-30, 2-31 TEAM: The first speaker says which level they chose for Task A and explains why. The other speakers respond in order to agree, disagree and comment. The first speaker has the Final Word for Task A. Repeat the process starting with a different first speaker for Tasks B and C. SHARE I think task A is at level … Agree or disagree and comment Agree or disagree and comment Final Word #1 #2 #3 #1

66. Modifying Levels of Cognitive Demand 2-32 The cognitive demand of a task can be changed. It can be raised to make a task more challenging. It can be lowered to make a task simpler and more routine. Consider Task B ( a Level 1 task): 3/5 of the students in Ms. Jones’ class of 30 students are boys. How many of the students are boys and how many are girls? TEAM Brainstorm at least three different ways that Task B can be modified to make it more challenging. Write your responses on HANDOUT 2-32 . SHARE

67. Modifying Levels of Cognitive Demand 2-32 Consider Task A ( a Level 2 task): Students at Grayson Middle School are ordering school T-shirts. They come in two colors. A survey of 25 students shows that 10 students preferred blue T-shirts and 15 students preferred red. Explain how you can estimate how many red and blue T-shirts to order, if a total of 180 students order shirts. Use diagrams, tables and mathematical expressions to make your explanation easier to understand. TEAM Brainstorm at least three different ways that Task A can be modified to make it more challenging. Write your responses on HANDOUT 2-32 . SHARE

68. Modifying Levels of Cognitive Demand 2-32 Consider Task C ( a Level 3 task): The student council has budgeted $300 to buy drinks for the graduation dance. Your job is to order drinks that most students will like. A survey of 40 students shows that 15 prefer cola, 5 prefer ginger ale and the rest prefer lemonade. Sodas cost $2.00 for a 2-liter bottle and lemonade costs $3.00 for a 2-liter container. To have enough drinks for everyone, you want to spend as much of the $300 as possible without going over budget. Decide how much of each drink to order and write a report to the student council justifying your decision. TEAM Task C is written at a high-level of cognitive demand. But it is possible to simplify a task and make it less challenging. Often this happens inadvertently as teachers try to help their students. Brainstorm at least three different ways that Task C can be simplified to make it less challenging. Write your responses on HANDOUT 2-32 . SHARE

69. How Tasks Change When They are Implemented in the Classroom We have seen that teachers can intentionally modify curriculum tasks to make them more challenging. However, research has shown that teachers and students often lower the cognitive demand level of a mathematical task, often without being aware that they are doing so. 2-33

70. How Tasks Change When They are Implemented in the Classroom Consider this schematic diagram*: 2-33 * Adapted from: Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development . Teachers College Press, 2000, by M. K. Stein, M. S. Smith, M.A. Henningson and E. A. Silver. Student Learning Tasks as they appear in curriculum materials Tasks as set up by teachers Tasks as enacted by teachers and Students

73. How Tasks Change in the Classroom Teachers commonly simplify challenging mathematical tasks for English language learners! Why do you think this is true? Turn and Talk with a partner about this. 2-33 SHARE

75. How Tasks Change in the Classroom Most teachers simplify problems for their students. Some do it more often than others. Many teachers simplify problems more often for English language learners than for native English speakers. Why do you think this is so? 2-33 Student Learning Tasks as they appear in curriculum materials Tasks as set up by teachers Tasks as enacted by teachers and Students

76. How Tasks Change in the Classroom Think of a time when you simplified a problem for your students by telling students how to approach a problem or giving them hints that led them to the procedure you wanted them to follow, or to the answers? 2-33 Student Learning Tasks as they appear in curriculum materials Tasks as set up by teachers Tasks as enacted by teachers and Students Think Write Pair Share

77. How Tasks Change in the Classroom The types of changes we’ve been discussing can have the net effect of reducing the cognitive demand of a task that was designed to be at the level of Doing Mathematics, to the lowest level, following step-by-step procedures and plugging numbers into formulas. Teachers who make such changes mean well. They want to help students “succeed” in solving a particular problem with less frustration, but they deprive them of opportunities to learn high-level mathematics. 2-33 To summarize:

78. How Tasks Change in the Classroom Teachers of English language learners must make sure their students have access to high level mathematics in the middle grades. To do this they must guard against the cognitive demands of mathematical tasks the set for their students. To Summarize (continued):

79. Supporting High-Level Learning of Mathematics for English Language Learners If we don’t help our students by simplifying a problem, showing them which steps to take, or giving them hints, how can we help them avoid frustration when they are stuck?

81. Scaffolding High-Level Learning of Mathematics for English Language Learners Work with a partner. Make a list of at least 3 ways to scaffold “Finding the Best Box” for English language learners. Write your suggestions on Handout 2-34 Scaffolds for “Finding the Best Box” 2-34 Think Write Pair Share

82. Scaffolding High-Level Learning of Mathematics for English Language Learners 2-35 Read HANDOUT 2-35 . Compare the suggestions in the handout with the suggestions our group came up with. Scaffolding Suggestions for “Finding the Best Box”

84. A Few Final Thoughts High-level mathematics in middle school is challenging, engaging, and enjoyable. High-level mathematics instruction helps English language learners build confidence, develop problem-solving strategies, learn and practice basic skills and develop an understanding of the language, methods and purposes of mathematics. 2-36

85. A Few Final Thoughts To keep mathematics instruction at a high level teachers must assign challenging tasks, allow students enough time to work on them, establish classroom procedures and expectations so that students work together comfortably, build on each other’s ideas and take responsibility for their own and each other’s learning. It is challenging for teachers as well as students to maintain a high level of cognitive demand. Scaffolding by teachers is essential in fostering the learning of high-level mathematics. This takes professional insight, preparation, and time. 2-36

86. A Few Final Thoughts High level mathematics is within the reach of all middle grades students, of all English language learners. We hope this challenge will become one that sustains, guides and focuses your work as a teacher. 2-36

87. Action Plans: Next Steps in your own classroom

88. WRITE on HANDOUT 2-37 one best practice from this workshop for supporting high-level mathematical learning with English language learners that you plan to try out with your own students. PAIR with a partner to discuss the practice and how you plan to implement it. Plan how you will collect evidence (student work, teacher lesson plans, video, audio, etc.) of how well it worked. Use HANDOUT 2-38 “Changes I have made in my practice” to document what happens. 2-37, 2-38

89. Workshop 1 Teaching High-Level Mathematics to English Language Learners in the Middle Grades © Copyright 2009 Center for Collaborative Education/Turning Points Thank you for attending the workshop 2-39 Please help us out by completing the Workshop Evaluation, Handout 2-39 before you leave.