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Module 6A for Elementary
Teachers
Florida Standards for Mathematics:
Focus on Content Standards
2
Professional Development Session Alignment
Set 1Governing Board
School Leaders
Module 6 Florida Standards Math Module 7
ELA & Data Use
Teachers Math
Leadership Teams Session 2
Session1
ELAData Use
Data Use ELA Math
Data Use
3
Professional Development Session Alignment
Set 2Governing Board
School Leaders
Module 5 Florida Standards ELA
Module 6 Florida Standards Math Module 7
ELA & Data Use
Module 8 Math & Data Use
Teachers Math
Leadership Teams
Session 4
Session3
ELAData Use
AssessmentsData
AnalysisVAM
Florida Standards
Data &ELA
Data &Math
Session 5
Session 6
4
Module 5 ELA
Module 6 Math
Module 7 ELA & Data
Use
Module 8Math &
Data Use
You Are Here
Module 2ELA
Module 1 Data Use
Module 4 Data Use
Module 3Math
6
8 Components of Full Florida Standards Implementation
Learn more about the Practice Standards
Examine the language and structure of the Florida Standards for Math Content
Create and solve standards-based tasks
Observe Florida Standards for Math-aligned instruction
Share implementation successes and challenges and plan next steps
Focus on Content Standards Outcomes
7
Welcome and Introductions• Pre-Assessment• Sharing Implementation Experiences• The Language of the Content Standards• The Progression of Mathematical ConceptsLunch• Meeting the Expectations of the Content Standards
by Teaching with High Level Tasks• Teaching the Content Standards Through Problem
Solving• Next Steps• Post-AssessmentWrap Up
Today’s Agenda
8
Pre-Assessment
Introductory Activity
9
GuidePage
3
Sharing Implementation Experiences
Section 1
10
Instructional Shifts for Mathematics
11
• The Standards for Mathematical Content• The Standards for Mathematical Practice
Focus Coherence Rigor
Two Areas
Fewer standards allow for focusing on the major work for each grade
Focus
12
The Standards are designed around coherent progressions and conceptual connections.
Coherence
Grade 1 Grade 2 Grade 3
Use place value understanding and
properties of operations to add and subtract
Use place value understanding and
properties of operations to add and subtract
fluently
Use place value understanding and
properties of operations to perform multi-digit arithmetic
13
The Florida Standards for Math are designed around coherent progressions and conceptual connections.
Coherence
Math Concept Progression K-12
All Roads Lead to Algebra……
14
The major topics at each grade level focus equally on:
Rigor
CONCEPTUAL UNDERSTANDING
• More than getting answers
• Not just procedures
• Accessing concepts to solve problems
PROCEDURAL SKILL AND FLUENCY
• Speed and accuracy
• Used in solving more complex problems
• Comes after conceptual understanding
APPLICATION OF MATHEMATICS
• Using math in real-world scenarios
• Choosing concepts without prompting
15
The Standards for Mathematical Practice
Developing Mathematical Expertise
16
1. Make sense of problems and persevere in solving them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of
others 4. Model with mathematics 5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning
Activity 1: Sharing Experiences Implementing Math Practice Standards
17
Sharing Implementation Experiences
1. Each participant will discuss with their table group one positive highlight, one challenge, and one lesson learned from their personal implementation of the Practice Standards thus far.
2. Each table group will then determine two positive highlights, one common challenge, and one common lesson learned that they will present to the larger group.
3. Participants will record notes and “New Ideas” generated from the discussion.
Guide Pages
8-9
Positive Highlights
Challenges
Lessons Learned
Guide Pages
5-6
The Language of the Mathematical
Content Standards
Section 2
18
Activity 2: What Do These Students Understand? – Part 1
19
What Do These Students Understand? – Part 1
1. Read and analyze the “Who Knows Math” handout on page 8 in the Participant Guide. Record your observations on what these students know and what they can do on page 9 in the Participant Guide.
2. Would you be comfortable with his/her understanding if s/he continued to approach division in his/her particular way? Explain your reasoning.
Guide Pages
8-9
Guide Pages
8-9
2 412
12
X 412
The major topics at each grade level focus equally on:
Rigor
CONCEPTUAL UNDERSTANDING
• More than getting answers
• Not just procedures
• Accessing concepts to solve problems
PROCEDURAL SKILL AND FLUENCY
• Speed and accuracy
• Used in solving more complex problems
• Comes after conceptual understanding
APPLICATION OF MATHEMATICS
• Using math in real-world scenarios
• Choosing concepts without prompting
20
21
Guide Pages10-11
“Conceptual understanding refers to an integrated and functional grasp of mathematical ideas.”
(Adding it Up: Helping Children Learn Mathematics. 2001)
Conceptual Understanding
22
Example
Question: What is 20 + 70?
Student Response: 20 is 2 tens and 70 is 7 tens. So, 2 tens and 7 tens is 9 tens. 9 tens is the same as 90.
Conceptual Understanding
23
Example
Question: What is 5 + 6?
Student Response: I know that 5 + 5 = 10; since 6 is 1 more than 5, then 5 + 6 much be 1 more than 10. 1 more than 10 is 11.
Conceptual Understanding
24
Example
Question: Why is 7 an even or odd number? Explain how you know.
Student Response: 7 is odd because I cannot make pairs with all of the cubes like I can with 8 cubes. When I can make pairs with all of the cubes it is an even number.
Conceptual Understanding
25
“Procedural skill and fluency is demonstrated when students can perform calculations with speed and accuracy.”
(Achieve the Core)“Fluency promotes automaticity, a critical capacity that allows students to reserve their cognitive resources for higher-level thinking.”
(Engage NY)
Procedural Skill and Fluency
Check all the equations that are true.� 8 x 9 = 81� 54 ÷ 9 = 24 ÷ 6� 7 x 5 = 25� 8 x 3 = 4 x 6� 49 ÷ 7 = 56 ÷ 8
26
Mariana is learning about fractions. Show how she can divide this hexagon into 6 equal pieces. Write a fraction that shows how much of the hexagon each piece represents.
Adding / subtracting with tens[Ask orally](a) Add 10 to 17(b) Add 10 to 367(c) Take 10 away from 75(d) Take 10 away from 654
Procedural Skill and Fluency
27
• The Standards call for students to use math flexibly for applications.
• Teachers provide opportunities for students to apply math in context.
• Teachers in content areas outside of math, particularly science, ensure that students are using math to make meaning of and access content.
(Frieda & Parker, 2012)(Achieve the Core, 2012)
Application of Mathematics
28
There are 9 cookies left in the pan. Five students want to share the cookies equally. How many cookies will each student get?
(Investigations Grade 3 Unit 7, Session 1.5)
Application of Mathematics
Example
29
5.MD – Minutes and Days
What time was it 2011 minutes after the beginning of January 1, 2011?
(Illustrative Mathematics)
Example
Application of Mathematics
Activity 2: What Do These Students Understand? – Part 2
30
What Do These Students Understand? – Part 2
Return to the “Who Knows Math” handout on pages 8-9 in the Participant Guide. Which students have shown conceptual understanding, which have shown procedural skill and fluency, which have shown both, and which pieces of work would you need to know more to make the determination?
Guide Pages
8-9
2 412
12
X 412
How does the approach of the Florida Standards for Math Content differ from previous approaches to mathematics teaching and learning?
Think About It…
32
Let’s Take A Break…
33
Be back in 15 minutes…
The Progression of Mathematical Concepts
Section 3
34
35
The Organization of the Standards Domain
Cluster
Standards
Domain Distribution
36
Domain Progression
37
Number and Operations in Base Ten
Number and Operations in Fractions
Counting and Cardinality & Operations and Algebraic Thinking
Measurement and Data
Geometry
Exploring the Content Standards
39
Activity 3 (part 1): Explore a Progression
40
Part 1
Examine your assigned domain and determine which standards focus on:
• Conceptual Understanding (CU)• Procedural Skill and Fluency (PSF)• Application (A)
2.NBT.1Understand that three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g.
706 equals 7 hundreds, 0 tens, and 6 ones.
K.NBT.1
Compose and decompose
numbers from 11to19 into ten
ones and some further ones
e.g. by using objects,
drawings, and record each
composition or decomposition
by a drawing or equation.
4.NBT.4
Fluently add and subtract multi-digit whole numbers to any place.
Guide Page
13
Activity 3 (part 2): Explore a Progression
41
Part 2
Examine your assigned domain across the K-5 grade band and record five general observations about the progression and two observations about the relationship between the Content and Practice Standards.
2.NBT.1Understand that three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g.
706 equals 7 hundreds, 0 tens, and 6 ones.
K.NBT.1
Compose and decompose
numbers from 11to19 into ten
ones and some further ones
e.g. by using objects,
drawings, and record each
composition or decomposition
by a drawing or equation.
1.NBT.2a
Understand that numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
Guide Pages14-15
Guide Pages 13-14
Activity 3 (part 3): ‘Chunking’ the Standards
42
Part 3
Examine all of the Content Standards for your assigned grade level. Make connections across domains and create clusters that can be taught as part of a lesson or unit.
3.NBT.2Fluently add and subtract within 1000 using strategies
and algorithms based on place value, properties of operations, and/or the relationship between addition
and subtraction.
3.OA.8
Solve two-step word problems
using the four operations.
Represent these problems
using equations with a letter
standing for the unknown
quantity. Assess the
reasonableness of answers
using mental computation and
estimation strategies including
rounding.
3.MD.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. Guide
Page15
1. What domains of the Florida Standards for Math do you think will be the most exciting and/or productive for you to teach to your students?
2. What domains of the Florida Standards for Math do you think will be the most challenging for your students to learn?
Reflect
43
Guide Page
16
44
Lunch
Meeting the Expectations of the Content
Standards by Teaching with High Level Tasks
Section 4
45
• Don’t have a predictable, well-rehearsed approach or pathway to the solution.
• Require students to explore and understand the nature of mathematical concepts, processes, or relationships.
• Demand self-monitoring or self-regulation of one’s own cognitive processes.
• Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task.
High Level Tasks
46
Guide Page
18
• Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.
• Require considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required.
Adapted from (Stein, Smith, et al (2000). Implementing Standard-Based Mathematics Instruction)
High Level Tasks
47
Math Class Needs a MakeoverDan Meyer
48
Watch Video
Take a Look…
49
How can I incorporate high-level mathematics tasks that will benefit ALL of my students?
The Big Question
50
Open Questions
Strategies for Differentiating High Level Tasks
51
Parallel Tasks
• An open question is framed in such as way that multiple responses and approaches can correctly answer the question.
• An open question allows students at varying developmental and readiness levels to equally participate in and grow from thought provoking tasks.
• An open question provides multiple pathways into the mathematics.
52
What are Open Questions?
Question 1: To which fact family does the fact 3 x 4 = 12 belong?
Question 2: Describe the picture below by using a mathematical equation.
X X X XX X X XX X X X
53
(Small.2012, 7)
Example of an Open Question
• An open question should be mathematically meaningful by focusing on the expectations of the content standards.
• An open question needs just the right amount of ambiguity.
• An open question is most effectively used when followed up by a whole class discussion in which the teacher strategically calls on students, conveys the message that multiple answers are welcomed, and builds connections between students’ answers.
54
Guidelines for Using Open Questions
• Turning Around a Question• Asking for Similarities and Differences
• Replacing a Number with a Blank• Creating a Sentence• Using “Soft” Words
• Changing the Question
55
Guide Pages19-21
Strategies for Writing Open Questions
56
Parallel Tasks are sets of tasks, usually two or three, that get at the same big idea and are close enough in context that they can be discussed simultaneously.
(Small. 2012, 10)
Parallel Tasks
57
Choice 1: Create a word problem that could be solved by multiplying two one-digit numbers.
Choice 2: Create a word problem that could be solved by multiplying two numbers between 10 and 100.
(Small. 2012, 10)
Example of Parallel Tasks
58
Consider the following:
• Different developmental levels
• What operations students can use
• What size numbers students can handle
• Conceptual understandings that students have developed
Creating Parallel Tasks
59
Things to Do:
• Create the parallel tasks• Create parallel questions• Manage task choice• Hold a follow-up discussion
Guide Pages22-24
Implementing Parallel Tasks
Activity 4: Teaching with High Level Tasks
60
Teaching with High Level tasks
1. Choose a grade level.2. Choose which content and practice
standards you want to work with.3. Create Open Questions and one set of
Parallel Tasks that can be used in your instruction of the Content Standards.
4. Write your final version of both the Open Questions and the Parallel Tasks on a piece of chart paper. Be prepared to present your ideas to the full group.
Guide Pages25-26
Principles to Keep in Mind
61
1. All open questions must allow for correct responses at a variety of levels.
2. Parallel tasks need to be created with variations that allow struggling students to be successful and proficient students to be challenged.
3. Questions and tasks should be constructed in such a way that will allow all students to participate together in follow-up discussions.
1. How does teaching with high level tasks relate to conceptual understanding, procedural skill and fluency, and application of mathematics?
2. How does teaching with high level tasks support Florida’s ‘New Way of Work’?
Reflect
62
Guide Page
27
Teaching the Content Standards Through
Section 5
Problem Solving
63
64
Guide Pages29-31
Watch Video
Activity 5: A New Spin on Old Strategies
66
Center 1: Sample Lesson Plans
Center 2: Math Journals
Center 3: Concept Cards
Center 4: Developing Fluency
You will have 12 minutes at each work station.
Next Steps
Section 6
67
1. What do we think should happen at school to promote implementation of the Florida Standards for Math?
2. What can we do now in our classrooms and in the school to promote implementation of the Florida Standards for Math?
3. What are some expected challenges?
4. How can we work around the challenges?
What's Your Plan?
68
Guide Page
33
Closing Activities
69
Learned more about the Practice Standards
Examined the language and structure of the Florida Standards for Math Practice
Created and solved standards-based tasks
Observed Florida Standards for Math-aligned instruction
Shared implementation successes and challenges and planned next steps
Focus on Content Standards Outcomes
70
Click to edit Master title style
Where Are You Now?
Assessing Your Learning
71
Post-Assessment and Session Evaluation
Guide Page
35
Module 2 ELA
Module 1 Data Use
Module 3 Math
Module 4 Data Use
Module 5 ELA
Module 6 Math
Module 7 ELA & Data
Use
Module 8Math &
Data Use
What’s Next?
Module 7 ELA &
Data Use
Module 4 Data Use
