fall 2011 mathematics sol institutes debbie delozier, stafford county public schools fanya morton,...
TRANSCRIPT
FALL 2011 MATHEMATICS SOL INSTITUTES
Debbie Delozier, Stafford County Public SchoolsFanya Morton, Stafford County Public SchoolsKathryn Munson, Chesterfield County Public SchoolsKaren Watkins, Chesterfield County Public SchoolsLouAnn Lovin, Ph. D., James Madison University
GRADE BAND: K-2
Icebreaker – MAKING Connections!• Please introduce yourself to those at your table.
• Discuss a possible pattern that exists in your group. Examples of possible patterns might be (brown hair, blonde hair, brown hair, blonde hair) or (glasses, no glasses, glasses, no glasses).
• When it is your table’s turn, please stand up so that the pattern you have selected is visible to the rest of the group. The rest of the group will make conjectures about what the pattern could be.
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Overview of Day• Communication and Reasoning through
Number Talks• The Teacher’s Role in Mathematical Discourse• Mathematics through Problem Solving• Looking at Student Communication and
Reasoning – The Frog Problem• Tasks to Promote Communication and
Reasoning• Processing and Summarizing
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What is the power in having more than one strategy?
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Think-Pair-Share
NUMBER TALK VIDEO: Grade 2
• Watch video• Table discussion
–What strategies are the students using?
–What mathematics do students understand that allows them to be flexible with numbers in this way?
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Number Talks
Quick Jot
What is the teacher’s role during a number talk?
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NUMBER TALK VIDEO: Kindergarten
• Watch video• Table discussion
–What student understanding is being assessed through this number talk?
–What do you notice about the way the teacher orchestrates the discourse?
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Number Talks
What is the value in implementing number talks?
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Think-Pair-Share
Key Components of Number Talks• Classroom environment and community• Classroom discussions• Teacher as facilitator, questioner, listener,
learner• Mental math• Purposeful computation problems
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Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies. Sausalito, CA: Math Solutions. 10-15
Creating the Environment
Table Discussion:
What do you think a teacher may need to do to create an environment that allows for this kind of math talk?
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Productive Talk Moves
• Revoicing - “You used the 100s chart and counted on?”
• Rephrasing - “Who can share what Ricardo just said, but using your own words?”
• Reasoning - “Do you agree or disagree with Johanna? Why?”
• Elaborating - “Can you give an example?”
• Waiting - “This question is important. Let’s take some time to think about it.”
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Adapted from: Chapin, S., O’Connor, C. & Anderson, N. (2003). Classroom Discussions Using Math Talks to Help Students Learn, Grades 1-6. Sausalito, CA: Math Solutions Publications. 11-16.
NUMBER TALK VIDEO – 16 + 15
• Watch video• Watch video again and record talk moves you notice • What kind of math talk currently happens in your
classroom? Which talk moves are you already comfortable with? Which ones might you wish to incorporate in future lessons?
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Mathematics Through Problem Solving
“Children need to experience mathematics as problem solving…”
Kathy Richardson, 1999
Think-Pair-Share – How do the process standards described in this
quote match what you have observed in math classrooms? (yours or others)
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Mathematics SOL Institutes – FocusFive goals for students:
– become mathematical problem solvers that– communicate mathematically; – reason mathematically;– make mathematical connections; and– use mathematical representations to model
and interpret practical situations
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Frog Problem• Work with a small group (3 – 5 people).• Record your solution on chart paper.• Try to solve the problem in at least two
different ways.• Post your chart paper for a
Gallery Walk.
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Pay attention to the mathematical processes you are using as you solve this problem.
There were 57 frogs in the pond. Some were swimming and some were sunning. There are about twice as many frogs swimming as were sunning. How many frogs were swimming and how many frogs were sunning? Use pictures, numbers, and/or words to prove that your answer makes sense.
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Frog Problem – First Grade Work• Work with a partner• Examine the student work samples• For each student, discuss and record:
– How does this student represent his/her thinking?– What does this student appear to understand (or
not understand)?– What would I do next with this student?
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Frog Problem – First Grade
How did this problem provide opportunities for students to reason and communicate?
If you were going to choose students to share their solutions, which students would you choose and why?
Does the order that you have students share matter?
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“The level and kind of thinking in which students engage determines what they will learn.”
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997
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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)
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“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.”
Stein, Smith, Henningsen, & Silver, 2000
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Task Sort• With your small group, use the collection of
tasks provided and discuss whether you think the task requires lower-level thinking or higher-level thinking.
• Record your group’s sort on the T-chart.• Be ready to share the criteria you used to
categorize the task.
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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
25Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press
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
26Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press
Task Analysis Guide – Higher-level Demands• 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
27Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press
DOING
Mathematics
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
28Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press
Factors Associated with Promoting Higher-level Demands• Scaffolding of student thinking and reasoning• Providing ways/means by which students can
monitor/guide their own progress• Modeling high-level performance• Requiring justification and explanation through
questioning and feedback• Selecting tasks that build on students’ prior knowledge
and provide multiple access points• Providing sufficient time to explore tasks
29Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press
“There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.”
Lappan & Briars, 1995
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Bumping up the Cognitive Demand• Work with a small group• Choose task #5 or #11• Think of a way to make this task more
cognitively demanding- provide context- make it open-ended- ask for an explanation/justification
• Rewrite your new task and be ready to share
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Why use cognitively demanding tasks?• More engaging for students• Allows students to wrestle with important
mathematical ideas and make sense of them• Engages students in mathematical processes
of reasoning, representation, communication, connections, and problem solving
• Allows teachers to assess students’ level of mathematical understanding
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“If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks.”
Stein & Lane, 1996
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Laying the Foundation
• Vertical Articulation documents• As students study higher levels of
mathematics, they must draw on the previous understandings they’ve constructed as well as their capacity to reason and problem solve while maintaining the attitude that they can persevere and figure it out!
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Assessments – Then and Now
35Grade 3
OLD
NEW
Why are the process standards important?Each person writes a few words or short phrases on sticky notes and places them in the middle of the table.As a group, synthesize your ideas to create a bumper sticker to address this question.
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I MATH
Standards of Learning and Mathematical Processes
“The content of the mathematics standards is intended to support the following five goals for students: becoming mathematical problem solvers, communicating mathematically, reasoning mathematically, making mathematical connections, and using mathematical representations to model and interpret practical situations.”
VDOE 2009 Mathematics Standards of Learning (page iv)
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