5 Pillars of Mathematics

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5 Pillars of Mathematics

• Training 2 Mathematical Discourse and Rigorous
  Mathematical Tasks
• Dawn Perks

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Todays Agenda

• Review TAF, DPS 3 focus practices, and 5 pillars
  of mathematics
• Review main focus of the mathematics blueprint
• Review what makes a good task from session 1
• Introduce Stein and Smiths Task Analysis Guide
• Categorizing Mathematical Tasks
• Define open-ended questions
• 2 methods for creating open-ended questions
• Transforming Mathematical Task Activity
• Introduce Bay-Williams and Van de Walles
  worthwhile task evaluation
• Corey the Camel
• Debrief if tasks fufilled the evaluation
• Review 5 pillars and blueprint focus

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Teaching and Assessment Framework
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• 5 Pillars of Mathematics
• Reasoning to make sense of mathematics
• Productive use of discourse when explaining and
  justifying mathematical thinking
• Procedural fluency
• Flexible and appropriate use of mathematical
  representations
• Confidence and perseverance in solving

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Mathematics Lesson

• Rigorous Task/Problem
  Classroom Discourse
• In order for a problem/task to be rigorous
  Meaningful classroom discourse is
• it must meet the following criteria
  imperative to extend students
  thinking
• The problem/task has important, useful
  and connect mathematical ideas.
• mathematics embedded in it. i.e. Where is
  Discourse includes ways of representing,
• it in the standard course of study?
  thinking, talking, agreeing, and
  disagreeing
• The problem/task requires higher-level
  the way ideas are exchanged and what the
  
• thinking and problem solving.
  ideas entail and as being shaped
  by the tasks
• The problem/task contributes to the
  in which students engage as well as by
  the
• conceptual development of students.
  nature of the learning environment.
• The problem/task creates an opportunity
  -NCTM
• for the teacher to assess what his or her
• students are learning and where they are
• experiencing difficulty.

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• What Are Good Tasks?

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What Are Good Tasks?

• They help students make sense of the mathematics.
• They are open-ended, whether in answer or
  approach.
• They empower students to unravel their
  misconceptions.
• They not only require the application of facts
  and procedures but encourage students to make
  connections and generalizations.
• They are accessible to all students in their
  language and offer an entry point for all
  students.
• Their answers lead students to wonder more about
  a topic and to construct new questions as they
  investigate on their own.

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Selecting Rigorous Tasks
Memorization Tasks Procedures with Connections Tasks
Involve reproducing previously learned facts, rules, formulae, or definition to memory. Are not ambiguous-such tasks involve exact reproduction of previously seen material and what is to be reproduced is clearly and directly stated. Focus students attention on the use of procedure the purpose of developing deeper levels of understanding of mathematical concepts and ideas. Suggest pathways to follow (explicitly or implicitly) that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.
Procedures without Connections Tasks Doing Mathematics Tasks
Are algorithmic. Use of the procedure is either specifically called for or its use is evident based on prior instruction, experience, or placement of the task. Require limited cognition demand for successful completion. There is little ambiguity about what needs to be done and how to do it. Require complex and no algorithmic thinking (i.e., there is not a predictable, well-rehearsed approach or pathway explicitly suggested by the task, task instructions, or worked-out example). Require students to explore and understand the nature of mathematical concepts, processes, or relationships.
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Defining Levels of Cognitive Demand of
Mathematical Tasks

• Lower Level Demands
• Memorization
• Procedures without connections
• Higher Level Demands
• Procedures with Connections
• Doing Mathematics

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Categorizing Mathematical Task Activity
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Open-ended Questions

• Open-ended questions have more than one
  acceptable answer and/ or can be approached by
  more than one way of thinking.

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Open-ended Questions

• Well designed open-ended problems provide most
  students with an obtainable yet challenging task.
  
• Open-ended tasks allow for differentiation of
  product.
• Products vary in quantity and complexity
  depending on the students understanding.

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Open-ended Questions Method 1 Working Backward

• Identify a topic.
• Think of a closed question and write down the
  answer.
• Make up an open question that includes (or
  addresses) the answer.
• Example
• Multiplication
• 40 x 9 360
• Two whole numbers multiply to make 360. What
  might the two numbers be?

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Open-ended Questions Method 2 Adjusting an
Existing Question

• Identify a topic.
• Think of a typical question.
• Adjust it to make an open question.
• Example
• Money
• How much change would you get back from 5 if you
  buy a Caesar salad and juice?
• 3. I bought lunch at the cafeteria and got 35
  change back. How much did I start with and what
  did I buy?

• Todays Specials
• Green Salad 1.15
• Caesar Salad 1.20
• Veggies and Dip 1.10
• Fruit Plate 1.15
• Macaroni 1.35
• Muffin 65
• Milk 45
• Juice 45
• Water 55

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Transforming Mathematical Tasks Activity
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How do we determine if a task is worthwhile?
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Corey the Camel
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• 5 Pillars of Mathematics
• Reasoning to make sense of mathematics
• Productive use of discourse when explaining and
  justifying mathematical thinking
• Procedural fluency
• Flexible and appropriate use of mathematical
  representations
• Confidence and perseverance in solving

22
Mathematics Lesson

• Rigorous Task/Problem
  Classroom Discourse
• In order for a problem/task to be rigorous
  Meaningful classroom discourse is
• it must meet the following criteria
  imperative to extend students
  thinking
• The problem/task has important, useful
  and connect mathematical ideas.
• mathematics embedded in it. i.e. Where is
  Discourse includes ways of representing,
• it in the standard course of study?
  thinking, talking, agreeing, and
  disagreeing
• The problem/task requires higher-level
  the way ideas are exchanged and what the
  
• thinking and problem solving.
  ideas entail and as being shaped
  by the tasks
• The problem/task contributes to the
  in which students engage as well as by
  the
• conceptual development of students.
  nature of the learning environment.
• The problem/task creates an opportunity
  -NCTM
• for the teacher to assess what his or her
• students are learning and where they are
• experiencing difficulty.

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THANK YOUWe look forward to seeing you again at
training 3