Fall 2011 Mathematics SOL Institutes

1. GRADE BAND: 3-5 Fall 2011 Mathematics SOL Institutes Grade Band Team Members Vickie Inge, University of Virginia Patricia Robertson, Arlington Public Schools, retired Beth Williams, Bedford County Public Schools Vandi Hodges, Hanover County Public Schools, retired

2. Today’s Institute Objective To improve mathematics instruction by providing district-level trainers with professional development resources focused on facilitating students' mathematical understanding through mathematical problem solving, communication, reasoning, connections, and representation.

3. Comparing and Ordering Fractions 1/15, 1/20, 10/11, 8/9, 5/8, 5/7, 3/5, 4/9, 6/12, 13/11, 3/4 • Work alone for 5 minutes to put these fractions in order from least to greatest. • Do not use a common denominator or convert to a decimal.

4. Comparing and Ordering Fractions 1/15, 1/20, 10/11, 8/9, 5/8, 5/7, 3/5, 4/9, 6/12, 13/11, 3/4 • Work with a shoulder partner to justify that the fractions are in the correct order. • Which ones were easier to determine? • How did you think about completing this task?

5. Comparing and Ordering Fractions 1/15, 1/20, 10/11, 8/9, 5/8, 5/7, 3/5, 4/9, 6/12, 13/11, 3/4 • What are the big ideas about fractions brought forward by this task? • Which fractions were the easiest to compare? • Which fractions were the more challenging?

6. Mathematics Standards of Learning for Virginia Public Schools, 2009 Handout Virginia Process Goals • Goals Introductory Paragraph • Mathematical Problem Solving Goal • Mathematical Communication Goal • Mathematical Reasoning Goal • Mathematical Connections Goal • Mathematical Representations Goal

8. Mathematical Process Skills Student Look-fors • Read your assigned part of the handout. • Underline the key ideas. • Share findings between table partners.

9. Fraction Track Activity You need: • Fraction Cards • Fraction Track Game Board • 20 beans /counters • Play with 1 or 2 players or with 2 pairs. Playing to 1 • Use only the fraction cards that are equal to 1 whole or less than 1. Mix the cards, and place the deck of cards facedown. • Use one Completed Fraction Track Game Board that goes to 1 for each pair of players. • Place seven beans on the game board, one on each track, at zero. • Players take turns drawing the top card and moving a bean (or beans) to total the amount on the card. • The goal is to use a series of moves to have beans land on exactly the number 1. When you land on 1 you win the bean. When a bean is won, place a new bean at 0 on the same track so that the next play has a bean on every track for the next player’s turn. • If you cannot move the total amount of your Fraction Card you lose that turn. Adapted from NCTM Illuminations

10. Fraction Track Activity Closing Reflection Questions: • What were some of the strategies you used to play the game? • What mathematical concepts are connected to the strategies?

11. Preparing to View the Video Use the Mathematical Process Skills -“Student Look-fors” Recording Form Handout while viewing the video. Review the indicators under the Skill Area your table was assigned. In the note’s section record specific evidence to support the indicator(s) observed in the assigned Skill Area.

12. Mathematics Process Skills “Student Look-fors” Table Group Debrief At the top of the paper, write the Skill Area the table group addressed. As a group select one or two the indicators in your assigned Skill Area that seemed to be most visible. On the chart paper record specific evidence for one or two indicators identified in the Skill Area. Post your chart on the wall. Return to your table and discuss charts for other Skill Areas.

13. VADOE Curriculum Framework Grade 4

15. analyze the unpacked standard and identify the Process Goals and where possible specific indicators that are implicit and/or explicit in the Standard. Record ideas in the Classroom Instruction column. • Analyze the unpacked standard. • Identify the Mathematical Process Skills and specific indicators that are implicit and/or explicit in the Standard. • Record ideas in the Classroom Instruction column.

16. Encourage Mathematical Communication with Talk Moves • Teacher revoices a student’s reasoning for the purpose of clarification and advancing student thinking. • Student revoices another student’s reasoning to make sense themselves and advance the other students’ reasoning and understanding. • Asking students to justify or prove someone else’s reasoning. Using justification or proof to allow for respectful discussion of ideas. • Asking student to build on the group’s reasoning by connecting and extending another student’s idea. • Wait time (means to make the other things happen).

17. Task Sort Divide up the tasks so that at least 2 people will complete each task in the set. Write on the task whether you consider the task low or high cognitively demanding. Work independently before discussing with your partner whether you thought it was a low or high cognitively demanding task. Discuss the tasks that you and your partner solved. Read the remainder of the tasks and without working these tasks indicate if you think the task is a low or high cognitively demanding tasks.

18. Characteristics of Rich Mathematical Tasks High cognitive demand (Stein et. al, 1996; Boaler & Staples, 2008) Significant content(Heibert et. al, 1997) Require Justification or explanation (Boaler & Staples, in press) Make connections between two or more representations (Lesh, Post & Behr, 1988) Open-ended (Lotan, 2003; Borasi &Fonzi, 2002) Allow entry to students with a range of skills and abilities Multiple ways to show competence (Lotan, 2003)

19. Task Analysis Guide – Lower-level Demands Involve recall or memory of facts, rules, formulae, or definitions Involve exact reproduction of previously seen-material No connection of facts, rules, formulae, or definitions to concepts or underlying understandings Require limited cognitive demand Focused on producing correct answers rather than developing mathematical understandings Require no explanations or explanations that focus only on describing the procedure used to solve Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

20. Task Analysis Guide – Higher-level Demands • Focus on use of procedures for developing deeper levels of understanding of concepts and ideas • Suggest broad general procedures with connections to conceptual ideas (not narrow algorithms) • Provide multiple representations to develop understanding and connections Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

21. Task Analysis Guide – Higher-level Demands DOING Mathematics Require complex, non-algorithmic thinking and considerable cognitive effort Require exploration and understanding of concepts, processes, or relationships Require accessing and applying prior knowledge and relevant experiences to facilitate connections Require task analysis and identification of limits to solutions Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

22. Factors Associated with Impeding Higher-level Demands Shifting emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient or too much time to wrestle with the mathematical task Letting classroom management problems interfere with engagement in mathematical tasks Providing inappropriate tasks to a given group of students Failing to hold students accountable for high-level products or processes Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

23. Consolidating Today’s Work Today we have looked at: • Using rich task to engage students • Using the Cognitive Demand Framework to analyze mathematical tasks • Identifying the Mathematical Process Goals called for in the standards in the Virginia SOL Curriculum Framework • Using the Process Goals and “Student Look-fors” in the classroom setting • Encouraging mathematical conversation using Talk Moves in the classroom setting

24. Consolidating Today’s Work Exit Card: As you reflect on our work together today respond to the following question: When a classroom teacher pays attention to each of these areas how is student engagement and ultimately student learning impacted?