Algebra ABBC

1
Algebra AB-BC

• Pittsburgh Public Schools Mathematics In-service,
  August 31- September 1, 2009

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Engaging in Lessons
3
Engaging in lesson 1.1

• Read p. 8.
• What do you expect from this investigation based
  on reading the investigation introduction?
• Examine the arithmetic tables
• What are the missing numbers in each table?
• Find and explain several patterns in each table.
• Continue with investigation on p. 10
• Why is it helpful to understand the patterns in
  the multiplication and addition tables?

4
Engage in Lesson 1.2

• Quickly engage in 1-3 on p. 12.
• Read Subtraction Results on the top of p. 13.
  Explain in your own words how the book is
  describing the subtraction of 15-17.
• Read Opposite of a Number on p. 13.
• In what way is 0 special in addition/subtraction?
• Explain in your own words how you can use the
  idea of opposites to solve a subtraction problem.
• Do problems 2, 3, and 6 on pp. 14-15

5
Engage in Lesson 1.3

• Examine the addition table
• How can we use the addition table to make sense
  of why 17-15 2?
• How can we connect 15 17 -2 to what takes
  place in the addition table?
• Read Minds in Action, p. 17-18.
• Complete the extended addition table
• Think-Pair-Share Do all the patterns found for
  positive numbers work? Explain.

6
Consistency in Mathematics

• Sometimes the way things work in mathematics
  seems like magic.
• What seems like magic is purposeful choice.
• What would happen if we filled in the table
  differently?
• Are we willing to accept these consequence?
• To help students understand that anytime
  something seems like magic, it is often the
  result of some purposeful choice. The rules were
  chosen in order to make things work the way
  people wanted them to.
• Consistency is critical in the discipline of
  mathematics to the extent that it is possible.

7
What Can We Learn from the History of Numbers?

• Counting Numbers (found in everyday experience)
  are closed under addition and multiplication.
• 0, negative numbers and rational numbers emerged
  because counting numbers not otherwise closed.
• Rational numbers are closed under squaring
• Irrational and imaginary numbers emerged because
  rational numbers are not closed under
  square-rooting
• With counting numbers and positive rationals,
  math moved from real world to abstract world.
  However other numbers moved in opposite
  direction.

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Curriculum Approach
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CME Project-Philsophy

• Designed around a Habits of Mind approach
• Embraced a disciplinary approach to the doing of
  mathematics, hence much time spend focused on
  doing explorations, problems or in discussion
• Focused on building experience with ideas before
  students are asked to formalize them, as iss done
  in the discipline by mathematicians as they
  learn.
• Designed to privilege effort low-threshold, high
  ceiling problems
• Designed as a formative assessment curriculum

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Cuoco et al Habits of Mind

• Take 30 minutes to read the article and take
  notes on the following 
• What is problematic about the focus of
  traditional school mathematics?
• Why do the authors hold that habits of mind are
  actually a critical focus for school mathematics,
  more so even then the particular products?
• What habits of mind feel familiar? Explain. Which
  ones feel more unusual? Explain.
• How does the habits of mind approach connect to
  the mathematics discipline?

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Engage in More Lessons
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Engage in Lesson 1.4

• Examine the multiplication table How does the
  table change when you move
• Up 1 row
• Down 1 row
• Right 1 column
• Left on column
• How does this compare to how these moves change
  the addition table? Explain.
• Look at row number 2. As you move right, you add
  2 each time. As you move left, you subtract 2
  each time. Why?
• If you wanted this pattern to continue throughout
  the extended multiplication table, what would you
  do as you move left?
• Would extended the pattern another way make
  sense? Explain.

13
Engage in Lesson 1.4

• Fill in the extended multiplication table
  purposefully in a way that enables you to test
  conjectures that you made.
• Do all your conjectures hold when you include the
  negative numbers?
• Discuss 6, p. 24.
• Complete 3, 6, 7, 8 and 14 on pp. 24-26.

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Engage in 1.5

• Think-Pair-Share
• Using the arithmetic tables, can you show what
  each property means?
• Explain why each property is true.
• Can you come up with an example showing when the
  property can make a calculation easier?
• Rephrase the any order, any grouping properties
  in your own words.
• Work on 1-3 on pp. 30-31

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Making Sense of the Curriculum Philosophy
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Experience before Formalization

• How did the flow of lessons from 1.1 1.5 help
  students gain experience working with numbers and
  operations before formalizing the this work?
• Did we move from concrete ideas to abstract one,
  or from abstract ideas to concrete ones?

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Wilensky Abstract and Concrete

• Take 20 minutes to read the article and take
  notes around the following 
• What are the main ideas of the article?
• How would you describe the difference between the
  authors understanding of abstract and concrete,
  and the typical assumptions of meaning?
• How does something become concrete?
• What does it mean to have a relationship with
  mathematics?

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Set-Up of Book

• Chapters divided into investigations
• Investigations divided into sections that build
  from Getting Started.
• NEVER skip Getting Started- activates prior
  knowledge and allows assessment of where students
  are
• Do not pre-teach do not confirm/deny ideas.
• Generally useful to allow students to read the
  investigation intro and have very brief
  discussion
• Critical to return to reflections at end of each
  investigations
• Some elements of a traditional book

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Elements of an Investigation

• For You To Do
• For Discussion
• For you to Explore
• Minds in Action
• Developing Habits of Mind
• Check Your Understanding
• On Your Own
• Maintain Your Skills

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Examining the Core Curriculum
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Elements of Curriculum

• Examine Unit Opening in Curriculum.
• What information is available to you?
• What is the role of the keys and overarching
  questions?
• Examine individual lessons.
• How are the lessons set-up?
• What supports are there for you as a teacher?
• What is the role of the Generalizing Question?
• Time is set up in class for students to work
  together on problems.

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Connecting to District Goals

• How does this new course and curriculum support
  an Growth mindset necessary in an effort-based
  system.

23
Connecting to District Goals

• How does this new curriculum support formative
  assessment, particularly in relation to the 5 key
  strategies (Wiliam, D)?
• Clarifying and sharing learning intentions and
  criteria for success
• Engineering effective classroom discussions,
  questions, and learning tasks that elicit
  evidence of learning
• Providing feedback that moves learners forward
• Activating students as instructional resources
  for one another (distributed disciplinary
  authority, learning communities)
• Activating students as the owners of their own
  learning (meta-cognition, interest, motivation,
  attribution).

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Curriculum Planning
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Planning Investigation 2A

• Count off by 5s and then form groups.
• Review the five investigations of Investigation
  2A, including the investigation opener.
• Engage in your particular section and analyze the
  curriculum around the lesson
• How are habits of mind developed?
• How does this lesson build towards the key idea
  of 2A?
• What is the role of the generalizing question?
• Where are students likely to struggle? What are
  ways you can advance them without removing the
  struggle?
• How can you make use of formative assessment in
  the lesson?
• How can you support a growth-mindset?