1. Alabama Explorations Guide:
Grade 3 Mathematics
©2010 Wisconsin Cooperative Educational Service Agencies (CESAs) School Improvement Services Permission is granted to the Alabama Department of Education for dissemination and use in any whole or part in any form within the Alabama Department of Education region.

2. Today’s Agenda Foundations of the Standards
Exploring Grade Level Intent
Exploring the Content Standards’ Structure
Exploring Standards for Mathematical Practice
Exploring Mathematical Understanding
Exploring the Expectations of Understanding
Exploring Two Standards
Exploring Vertical Connections
Determining Implications and Next Steps

3. Outcomes
1. To understand the foundation of the Alabama College-and Career-Ready Standards 2. To explore the critical focus areas by grade level 3. To explore the grade level standards 4. To explore mathematical understanding 5. To reflect on implications to your practice

4. The Message Cannot/should not be rushed–a marathon, not a race.
Your LEAs teacher leaders are needed.
Our focus–to learn HOW to explore these standards.
We aren’t exploring all standards today. You will be given a process that can be duplicated in your school.
We won’t be aligning today–because alignment cannot be done effectively without careful exploration.

5. To explore, you will need …
Copy of the 2010 Alabama Course of Study: Mathematics
The Explorations Guide
Highlighters
Pen or pencil
Tables for group work
Timer/timekeeper

6. Ground Rules for Today Information-Giving Group Work & Recording
Attentive listening
Open mindset to receive new ideas and information
Note-taking
Open mindset
Professional conversations
Careful note-taking (for taking back)
Deep thinking
Record questions–to be addressed later

7. Now … for some background information

8. Development of Common Core State Standards
Joint initiative of:
Supported by:

9. The College-and Career-Ready Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time to recognize that standards are not just promises to our children, but promises we intend to keep.

10. What Is College and Career Readiness?
College readiness means students are prepared for credit-bearing academic courses at 2-year or 4-year postsecondary schools.
Career readiness means students are prepared to succeed in workforce training programs in careers that:
1) Offer competitive, livable salaries above the
poverty line
2) Offer opportunities for career advancement
3) Are in a growing or sustainable industry

11. Why are college- and career-ready standards good for students?
College & Career Focus
Consistent
Mobility
Student Ownership

12. How can you ensure that the work in your classroom is preparing students to be ready for essential workforce and college responsibilities?

13. Which Standards Are In The Course Of Study?+ =
Alabama Added Content

14. Alabama Added Content + +
Grades K- 12
Grades K-8
Grades 9-12

15. FOCUS
Identifies key ideas, understandings and skills for each grade or course
Stresses deep learning (addresses mile-wide, inch-deep issue)
Connects topics and standards within grade or course
Requires applying concepts and skills within same grade or course

16. FOCUS: Increased Clarity and Specificity
“It is important to recognize that “fewer standards” are no substitute for focused standards. Achieving “fewer standards” would be easy to do by resorting to broad, general statements. Instead, these Standards aim for clarity and specificity.” CCSS page 3.

17. COHERENCE
Provides the opportunity to make connections between mathematical ideas
Occurs both within a grade and across grades
Is necessary because mathematics instruction is not just a checklist of topics to cover, but a set of interrelated, powerful ideas

18. LEARNING PROGRESSIONS
Learning Trajectories – sometimes called learning progressions – are sequences of learning experiences hypothesized and designed to build a deep and increasingly sophisticated understanding of core concepts and practices within various disciplines. The trajectories are based on empirical evidence of how students’ understanding actually develops in response to instruction and where it might break down.
Daro, Mosher, & Corcoran, 2011
Insert hyperlink from Hunt Institute on Learning Progressions

19. Learning Progression Framework
Ending Point
Ending Point
Starting Point
Starting Point K 1 2 3 4 5 6 7 8 HS
Counting and Cardinality
Number and Operations in Base Ten
Ratios and Proportional Relationships
Number and Quantity
Number and Operations – Fractions
The Number System
Operations and Algebraic Thinking
Expressions and Equations
Algebra
Functions
Geometry
Measurement and Data
Statistics and Probability

20. Value of Learning Progressions/Trajectories to Teachers
Know what to expect about students’ preparation.
Manage more readily the range of preparation of students in your class.
Know what teachers in the next grade expect of your students.
Identify clusters of related concepts at grade level.
Provide clarity about the student thinking and discourse to focus on conceptual development.
Engage in rich uses of classroom assessment.

21. A Vision for Implementation
Administrator and teacher awareness of new standards, grade-specific training in the critical areas of focus, structure of the standards, verbs and vocabulary, student practice standards, etc…
District curricula, assessments, and instruction (the exterior and interior)
Alabama College- and Career-Ready Standards
(2010 Alabama Course of Study: Mathematics)

22. Exploring the Standards:
Grade 3 Mathematics

23. Activity #1
Focus Area Narratives
Important descriptions at the beginning of each grade level.
Provide the intent of the mathematics at each grade.
Provide 3-4 critical focus areas for the grade level .
Provide a sense of …
The sophistication for mathematical
understanding at the grade level.
The learning progressions for the grade.
Extensions from prior standards.
What’s important at the grade level.
Focus area narratives are important descriptions at the beginning of each grade level. They express the intent of the standards for the grade and specify 3 to 4 critical mathematical areas of focus.

24. Turn to page 27 in your ACOS
Activity #1
Grade-Level Intent
Grade 3 Narrative
Turn to page 27 in your ACOS
for Grade 3.

25. Activity #1: Exploring Grade Three Intent

26. Components of the Mathematics Course of Study
Activity #2
Components of the Mathematics Course of Study
Number & Quantity
Algebra
Functions
Modeling
Geometry
Statistics & Probability
Mathematical Practice Standards
Mathematical
Content Standards

27. Structure of the Standards
Activity #2
Structure of the Standards
Standards for Mathematical Practice
Carry across all grade levels
Describe habits of mind of a mathematically expert student
ACOS –
pages 6-8
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments & critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning

28. The Practices are also listed on the Grade 3 Overview.
Activity #2
Grade 3 Overview
Standards for Mathematical Practice are provided in detail on pages 6-8 of the ACOS.
The Practices are also listed on the Grade 3 Overview.

29. K-12 Standards for Mathematical Content
Activity #2
K-12 Standards for Mathematical Content
Refer to the ACOS
K-8 standards presented by grade level
Organized into domains that progress over several grades
Grades K-8 introductions give 2 to 4 focal points at each grade level
High school standards presented by conceptual theme (Number & Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability)

30. Structure of the Standards
Activity #2
Structure of the Standards
Content standards define what students should understand and be able to do
Clusters are groups of related standards
Domains are larger groups that progress across grades
Content Standard Identifiers
Domain
Cluster Statement
The actual standards are written in a format that will be new to Alabama teachers. As we mentioned a few moments ago, you will see domains in each grade level. These are akin to the content strands that most math teachers have been planning their instruction with for years. All of the standards that relate to that domain are organized by clusters. The individual standards are akin to learning objectives.
Within each grade level you find, numbered content “standards,” preceded by the phrase “students will” that define what students should know and be able to do.
Related standards are grouped into “clusters,” which are housed within “domains.” Domains are larger groups which progress across grade levels.
Direct participants to look at page 10 in ACOS 2010.
Point out the content standard identifier at the end of each standard the correlates with CCSS numbers.
Cluster
Standards

31. Grade-Level Standards
Activity #2
Grade-Level Standards
“…grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators, researchers and mathematicians.”
(2010 Alabama Course of Study, p.1)

32. Activity #2: Exploring Structure of Content Standards
Grade 3

33. Mathematical Practice
Activity #3
Standards for
Mathematical Practice
“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.”
(2010 Alabama Course of Study, p. 6)

34. Underlying Frameworks
Activity #3
Underlying Frameworks
National Council of Teachers of Mathematics
5 PROCESS Standards
Problem Solving
Reasoning and Proof
Communication
Connections
Representations
NCTM (2000M). Principles and Standards for School Mathematics. Reston, VA: Author.

35. Underlying Frameworks
Activity #3
Underlying Frameworks
National Research Council
Strands of Mathematical Proficiency
Conceptual Understanding
Procedural Fluency
Strategic Competence
Adaptive Reasoning
Productive Disposition
NRC (2001). Adding It Up. Washington, D.C.: National Academies Press.

36. Refer to pages 6-8 in the ACOS
Activity #3
Standards for
Mathematical Practice
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments & critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Refer to pages 6-8 in the ACOS

37. Make sense of problems and persevere in solving them#1
When presented with a problem, I can make a plan, carry out my plan, and evaluate its success.
AFTER…
CHECK
Is my answer correct?
How do my representations connect to my algorithms?
EVALUATE
What worked?
What didn’t work?
What other strategies were used?
How was my solution similar to or different from my classmates?
BEFORE…
EXPLAIN the problem to myself.
Have I solved a problem like this before?
ORGANIZE information.
What is the question I need
to answer?
What is given?
What is not given?
What tools will I use?
What prior knowledge do I
have to help me?
DURING…
PERSEVERE
MONITOR my work
CHANGE my plan if
it isn’t working out
ASK myself, “Does this
make sense? “

38. Reason abstractly and quantitatively
I can use reasoning habits to help me contextualize and decontextualize problems. #2
CONTEXTUALIZE
I can take numbers and put them in a real-world context.
For example, if given
3 x 2.5 = 7.5,
I can create a context:
I walked 2.5 miles per day for 3 days. I walked a total of 7.5 miles.
DECONTEXTUALIZE
I can take numbers out of context and work mathematically with them.
For example, if given
I walked 2.5 miles per day for 3 days, How far did I walk?
I can write and solve
3 x 2.5 = 7.5.

39. Construct viable arguments and critique the reasoning of others#3
I can make conjectures and
critique the mathematical
thinking of others.
I can critique the reasoning
of others by…
listening
comparing arguments
identifying flawed logic
asking questions to
clarify or improve
arguments
I can construct, justify, and
communicate arguments by…
considering context
using examples and non-
examples
using objects, drawings,
diagrams and actions

40. Model with mathematics#4
I can recognize math in everyday life and use math I know to solve everyday problems.
concrete
models
I can…
make assumptions and
estimate to make
complex problems
easier.
identify important
quantities and use tools
to show their relation-
ships.
evaluate my answer and
make changes if needed.
pictures
symbols
Represent
Math
oral
language
real-world situations

41. Use appropriate tools strategically#5
I know when to use certain tools to help me explore and deepen my math understanding.
I have a math toolbox.
I know HOW to use math
tools.
I know WHEN to use math
I can reason: “Did the tool I
used give me an answer
that makes sense?”

42. I can use precision when solving problems and communicating my ideas.#6
Attend to precision
I can use precision when solving problems and communicating my ideas.
Problem Solving
I can calculate accurately.
I can calculate efficiently.
My answer matches what the problem asked me to do–estimate or find an exact answer.
Communicating
I can SPEAK, READ, WRITE, and LISTEN mathematically.
I can correctly use…
math symbols.
math vocabulary.
units of measure.

43. Look for and make use of structure.
I can see and understand how numbers and spaces are organized and put together as parts and wholes. #7
SHAPES
For example:
Dimension
Location
Attributes
Transformation
NUMBERS
For example:
Base 10 structure
Operations and properties
Terms, coefficients, exponents

44. Look for and express regularity in repeated reasoning
I can notice when calculations are repeated. Then, I can find more efficient methods and short cuts. #8
Patterns:
1/9 = ….
2/9 = …
3/9 = …
4/9 = ….
5/9 = ….
I notice the pattern which leads to an efficient shortcut!!!

45. The Mathematical Practices
Activity #3
The Mathematical Practices
These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections.
Lets look at mathematical practices in action in a classroom.
Video Clip
What Mathematical Practices did you see the students use?

46. Activity #3: Exploring Standards for Mathematical Practice
Grade 3

47. Activity #4
Exploring Domains
Domains are common learning progressions that can progress across grade levels.
Domains do not dictate curriculum or teaching methods.
Topics within domains are not meant to be taught in the order presented.
Teachers must present the standards in a manner that is consistent with decisions that are made in collaboration with their K-12 mathematics team.

48. Mathematical Language
Activity #4
Mathematical Language
Mathematical language may be different than everyday language and other disciplinary area language.
Questions may arise about the meaning of the mathematical language used. This is a good opportunity for discussions and sense making in the ACOS.
Questions about mathematical language can be answered by investigating the progression of the concepts in the standards throughout other grades.

49. Activity #4: Exploring Domains
Grade 3

50. Outline of Grade 3 Math Standards
Domain
Clusters
Standards
Operations & Algebraic Thinking 4 9
Number and Operations in Base Ten 1 3
Number and Operations--Fractions
Measurement and Data 8
Geometry 2
TOTAL
25 Total

51. Mathematics Understanding
Activity #5
Mathematics Understanding
The Alabama College- and Career-Ready Standards for mathematics provide a major focus on UNDERSTANDING.
Questions to think about …
What is meant by understanding?
How do we see it in students?
How do we teach it?

52. Activity #5: Exploring Understanding

53. Mathematical Understanding Reflected in the Standards
Activity #6
Mathematical Understanding Reflected in the Standards
Interpretation
Explanation
Application
Mathematics Procedural Skills
From Kindergarten through to Grade 12, there is a strong emphasis and specificity on ways that students will be expected to show their understanding.

54. explain … interpret …apply
Activity #6
Students who understand
a concept can:
explain … interpret …apply
For example, they can …
use it to make sense of and explain quantitative situations (see "Model with Mathematics" in Practices)
incorporate it into their own arguments and use it to evaluate the arguments of others (see " Construct viable arguments and critique the reasoning of others" in Practices)
bring it to bear on the solutions to problems (see "Make sense of problems and persevere in solving them")
make connections between it and related concepts

55. Activity #6: Exploring the Expectations of Understanding
Grade 3

56. Three Important Lenses to Explore the Standards
Activity #7
Three Important Lenses to Explore the Standards
Lens #1: Student-Friendly Language
Lens #2: Key Vocabulary
Lens #3: Mathematical Practices 56 56

57. Lens #1: Student-Friendly Language
Activity #7
Lens #1: Student-Friendly Language
Explaining the intended learning in student-friendly terms at the outset of a lesson is the critical first step in helping students know where they are going...Students cannot assess their own learning or set goals to work toward without a clear vision of the intended learning.  When they do try to assess their own achievement without understanding the learning targets they have been working toward, their conclusions are vague and unhelpful.
-Stiggins, Arter, Chappuis & Chappuis,
2004, pp

58. Activity #7
Lens #2: Key Vocabulary
Why identify key vocabulary in the standards for instruction?
To clarify the teacher’s understanding
To activate prior vocabulary in context
To make connections to the prior learning and experiences of students
To observe how vocabulary is developed in the learning progressions of the standards
What implications does the vocabulary of the standards hold for teacher professional development?

59. Lens #3: Mathematical Practices
Activity #7
Lens #3: Mathematical Practices
“…those content standards which set an expectation of understanding are potential ‘points of intersection’ between the Standards for Mathematical Content and the Standards for Mathematical Practice.”
“…attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.”
(2010 Alabama Course of Study: Mathematics, page 9)

60. Activity #7: Exploring the Content Standards
Grade 3

61. Activity #8
Vertical Connections
All Standards in mathematics have a connection to early and subsequent concepts and skills.
The flow of those connections is documented by how a student develops the concepts.
Prior Standards
Prior Standards
Current Standard
Future Standards
Future Standards

62. Big Ideas that Carry Across the Document (K-12)
Activity # 8
Big Ideas that Carry Across the Document (K-12)
(from Phil Daro, one of three lead writers on
the Common Core Standards for Mathematics)
Properties of operations: their role in arithmetic and algebra
Mental math and algebra vs. algorithms (Inspection)
Units and unitizing
Operations and the problems they solve
Quantities Variables Functions Modeling (As a sequence across grades)
Number Operations Expressions Equations (As a sequence across grades)
Modeling
Practices

63. Fractions Progression
Activity #8 K
Understanding that arithmetic of fractions draws upon four prior progressions that informed the CCSS
Number Line in Quantity and Measurement
Equal Partitioning
Fractions
Rational numbers
Properties of Operations
Rational Expressions
Unitizing in Base 10 and in Measurement
Rates, proportional and Linear Relationships

64. Vertical Connections (example) Fractions, Grades 3–6
Activity #8
Vertical Connections (example) Fractions, Grades 3–6
Gr. 3. Develop an understanding of fractions as numbers.
Gr. 4. Extend understanding of fraction equivalence and ordering.
Gr. 4. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Gr. 4. Understand decimal notation for fractions, and compare decimal fractions.
Gr. 5. Use equivalent fractions as a strategy to add and subtract fractions.
Gr. 5. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Gr. 6. Apply and extend previous understandings of multiplication and division to divide by fractions.

65. Functions and Equation Progression
Activity #8
Functions and Equation Progression K
Quantity and Measurement
Operations and Algebraic Thinking
Ratio and Proportional Relationships
Expressions and Equations
Functions
Modeling Practices
Modeling (with Functions)

66. Activity #8: Exploring Vertical Connections
Grade 3

67. Determining Implications and Next Steps
Activity #9
Determining Implications and Next Steps
We’ve been exploring the standards–now, what do we do?

68. Activity #9: Determining Implications

69. Activity #10: Determining Next Steps

71. Exit Ticket
Please complete the exit ticket as instructed. 3. Write three significant things that you learned today. 2. Write about two areas that are still confusing to you. 1. Write one immediate step you will take to implement the 2010 Alabama Course of Study: Mathematics when you return to your LEA.