key advances, math practices, and translating math standards into college and career...
TRANSCRIPT
Key Advances, Math Practices, and Translating Math Standards Into
College and Career Readiness-Aligned Curriculum and Instruction
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Benefits of the CCR Standards in AE
Aligned with 2014 GED® test assessment targets and
21st century high school equivalency expectations
Based on employer, college, and community college input
Selected with adult learners in mind
Provide guidance (and continuity) from literacy level to HSE
(High School Equivalancy)/advanced level
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A Shift in Approach
• Why is it important?
- broadens the scope beyond obtaining GED credential
- give teachers the “big picture”
- meets the needs of the student at a real life level
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Sidebar about GED® Test
What about the GED® test?
• GED® test contains many CCR standards
and is still important
• Implementing CCR instruction will still be a part of
GED® instruction but will go beyond that as an end goal
• Teaching good math foundational skills vs. just teaching to
one test
• What about after your GED® credential?
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Agenda for Workshop
• Introduce the key advances for mathematics in CCRS
• Introduce math practices for effective instruction
• Discuss the resource alignment tool that may be used to
evaluate a resource – (Day 2)
• Evaluate a lesson to determine alignment to CCR
standards and math practices through the lesson study
process – (Day 2)
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Three Key Advances Prompted by the CCR Standards in Mathematics
1. Focus: Focus strongly where the CCR Standards focus
2. Coherence: Design learning around coherent
progressions from level to level
3. Rigor: Pursue conceptual understanding, procedural skill
and fluency, and application—all with equal intensity
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FOCUS
Focus means that some content is more important than other
content and receives more time and attention.
Other content supports the more important content.
The Standards for Mathematical Practice (more on this later)
become a critical focus in the CCR Standards and the
mathematics curriculum.
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Areas of Focus in the CCRS
Level A: Whole numbers - addition and subtraction concepts, skills,
and problem-solving to 20; place value and whole number
relationships to 100; and reasoning about geometric shapes and
linear measures
Level B: Whole numbers and fractions - place value, comparison,
and addition and subtraction to 1000; fluency to 100; multiplication
and division to 100; fractions concepts, skills, and problem-solving; 2-
dimensional shape concepts; standard units for measuring time,
liquid, and mass; and area measurements
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Areas of Focus in CCRS cont’d
Level C: Positive whole numbers, fractions, and decimals - fluency
with multi-digit whole number and decimal operations; decimal place
value concepts and skills to thousandths; comparing, ordering, and
operating with fractions; fluency with sums and differences of
fractions; understanding rates and ratios; early expressions and
equations; area, surface area, and volume; classification of 2-
dimensional shapes; and developing understanding of data
distributions, including creating dot plots in the coordinate plane
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1010
Areas of Focus in CCRS cont’d
Level D: Rational numbers—fluent arithmetic of positive and
negative rational numbers; applying rates, ratios, and proportions;
applying linear expressions, equations, and functions; systems of
linear equations; classification and analysis of 2- and 3-dimensional
figures; developing similarity and congruence concepts, including
problems of scale; solving right triangles using the Pythagorean
theorem; random sampling of populations to summarize, describe,
display, interpret, and draw inferences; bivariate data and a line of
good fit; and development of probability concepts
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Level E: Real numbers—extending number system to include all real
numbers; equivalent expressions involving radicals and rational
exponents; reasoning about units and levels of precision; linear,
quadratic, and exponential expressions, equations, and functions;
linear inequalities; algebraic and graphic models of functions;
applying similarity and congruence to 2-dimensional figures; and
analyzing 1- and 2-variable data sets, including using frequency
tables
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Areas of focus in CCRS cont’d
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Activity
1. Circle the topics on the worksheet for each level that are part of the major focus for that level.
2. Use the Major Work of the Levels resource to help you make your decisions. Reference a copy of the CCR Standards, if needed.
3. Discuss your selections and rationales at your table. On the following slide, there are some questions to guide your discussion.
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What are some rationales for why you did or did not circle a
particular topic as critical to the level?
Did anyone have a hard time deciding to which critical area a
topic belongs? Tell us where and why.
Are any of the lesson objectives that you did not select for the
given level critical to a different level?
Do you think that any of the topics you did not select are
important to teach? If so, how might you relate them to one of
the critical areas for the level?
Discussion Questions
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Reflections
Focusing on fewer topics allows students to develop a deeper
understanding of the content that matters most
There’s so much mathematics that students could be learning,
why limit what students are taught?
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Coherence
Understand the research(next slide) base that explains the importance of coherence in standards and curricula.
Extend understanding about the focus of content in each level to include coherence within and across the levels.
Develop an understanding of the progressions of critical
concepts across the CCR levels as a foundation for developing
a coherent and rigorous mathematics curriculum.
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Rationale for Coherence
Relevance and Importance Based on the Research
Research emanating from TIMSS and the ACT National Curriculum Survey support the premise that coherent standards and curricula are important for college and career readiness:
Coherence allows students to demonstrate new understanding built on foundations from previous study.
Coherence prevents standards from being a list of isolated topics.
Coherence means that each standard is not a new event, but rather an extension of previous learning.
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Implications for Coherence
Content unfolds meaningfully.
Connections between concepts are made both within and
across the levels.
Students and teachers expect knowledge and skills to build
and grow.
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1. Identify the progression topic to which each color-coded group of standards belongs: fluency with operations, expressions and equations, or real-world applications.
2. Begin with the fluency (blue) cards. Use knowledge of how concepts and skills build on one another to organize the color-coded cards in a logical order of progression from the lowest to the highest level.
3. Use knowledge of the CCR Standards and the Unit 1 resource, Major Work of the Levels, to help identify the level (A, B, C, D, or E) for each standard on a fluency card.
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Activity
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4. Share results with others at your table, and discuss any points of agreement and disagreement.
5. Repeat steps 2 through 4 for expressions and equations (yellow) and real-world applications (green) cards.
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Directions (Continued)
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Reflections
How has participating in this activity changed your thinking about the CCR standards?
How will you use the information and understanding you have acquired to improve your teaching practice and student learning?
What additional training and tools would strengthen your ability to do so?
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Rigor
Rigor in lessons relates to the depth at which the major work
of each level should be addressed:
Conceptual Understanding: Comprehending key concepts
behind the procedures
Fluency: Gaining speed and accuracy in applying procedures
Application: Supporting problem-solving and deeper
mathematical thinking
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Rationale for Rigor
Relevance and Importance Based on the Research
Surveys of employers and professors of first-year credit-bearing mathematics courses validate the importance of rigor in instruction.
• Students with solid conceptual understanding can generalize and apply concepts from several perspectives.
• Students who can perform calculations with speed and accuracy (fluency) are able to access more complex concepts and procedures.
• When students have the ability to use math flexibly, they are able to apply their knowledge to solve problems.
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Activity
1. Check the component(s) of rigor likely to be required in a
lesson, activity, or task that targets each CCR standard on
the Engaging the Three Components of Rigor worksheet.
Make notes about your rationales.
2. Discuss your reasoning at your table, using the four
questions listed on the following slide.
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Discussion Questions
1. What makes you think a particular component of rigor applies?
2. Are there certain words or phrases in the standard that provide
clues?
3. Which components of rigor might appear together in a single
standard? Explain.
4. Which components of rigor are not likely to appear together in a
single standard? Explain.
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Reflections
How do you define “real-world” in application problems?
Is it enough to give very difficult problems to students to
advance rigor in a lesson?
How can we separate difficulty of technique from
rigorous thinking?
In what way(s) did your understanding of the meaning of
“rigor” change as a result of this activity?
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Math Practices
• What are math practices?
• - math practices describe a variety of expertise at all levels that educators should seek to develop in their students
• Why are they important?
• - when math practices are linked to content, deeper understanding can occur enabling students to extend their learning to new situations
• - emphasis shift to “learning how to learn” vs. “how to get the answer”
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A argument for the standards
• Quote from Fabio Milner (CCRS math coach)
• “The frequently maligned Standards for Mathematical Practices are really habits of mind that refer to higher-order thinking skills empowering individuals able to use them to present better arguments, to understand and criticize the logical flaws of statements and arguments presented by the media, politicians, friends, and others, to become better problem-solvers, to make good conjectures and generalizations, to express themselves more clearly and accurately. And note that I am not talking about mathematics, but rather in any arena. I think this constitutes a powerful argument in favor of the CCR standards.
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Math Practice 1
• Make sense of problems and persevere in solving them
• “Understand the problem, find a way to attack it, and work until it is done”
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Math Practice 2
• Reason abstractly and quantitatively
• “Students should be able to break a problem apart and show it symbolically, with pictures, or in any way other than the standard algorithms”
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Math Practice 3
• Construct viable arguments and critique the reasoning of others
• “I can talk about math, using mathematical language, to support or
oppose the work of others”
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Math Practice 4
• Model with mathematics
• “I can use math to solve real-world problems, organize data, and understand the world around me”
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Math Practice 5
• Use appropriate tools strategically
• “I can select appropriate math tools and use them correctly to solve problems”
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Math Practice 6
• Attend to precision
• “Students speak and solve mathematics with exactness and
meticulousness”
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Math Practice 7
• Look for and make use of structure
• “I can find patterns and repeated reasoning that can help solve more
complex problems”
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Math Practice 8
• Look for and express regularity in repeated reasoning
• “Students keep an eye on the big picture while working out the details
of a problem”
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Implications of the Standards for Mathematical Practice on Instruction
The Standards for Mathematical Practice are applied across all
levels.
Not all Standards for Mathematical Practice are appropriate for
every lesson.
Students need opportunities to experience all of the Standards
for Mathematical Practice over the course of the unit or the
level of study.
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Activity
Consider the following problem:
A pencil and eraser together cost $1.10. The pencil cost $1 more than the eraser. How much does the eraser cost?
1. Which Standards for Mathematical Practice are central to answering this question correctly?
2. How could you enhance this question to better include standards for Mathematical Practices that you identified?
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Activity Directions cont’d
3. Working independently, come up with a problem or lesson that will involve using one or more Standards for Mathematical Practices when a student is given the problem.
4. Discuss individual decisions and rationales at your table, including how the different types of problems/lessons imagined would affect the relevance of a particular Standard for Mathematical Practice.
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Reflections
What are some of the Standards for Mathematical Practice that you think are central to the problem/lesson?
What are some of the Standards for Mathematical Practice that you think might serve in a supporting capacity to the problem?
Why is it important to emphasize the Standards for Mathematical Practice at all levels of adult learning?
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Questions and Comments
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Day 2: Agenda
• Questions and Overview of Day 1
• Part 1: Discussion about Revising a Resource
to Improve Alignment to CCR Standards
• Part 2: Creating CCR-Aligned Lessons
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A Note about Resources
• No one resource will be perfect; some resources are better than others
• How can we make sure our current resources better align with the CCRS?
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Key Advances That Frame the Resource Evaluation in Mathematics
1. Focus: Does the resource focus strongly where the
standards focus, including the relevant Standards for
Mathematical Practice?
2. Coherence: Does the resource design learning around
coherent progressions between levels and within the level?
3. Rigor: Does the resource pursue conceptual understanding,
procedural skill and fluency, and application with equal
intensity?
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Purpose of a Lesson Study
• Creates CCR-aligned lessons.
• Reinforces the key instructional advances and level-specific
demands of CCR standards.
• Provides opportunities to share, test, and hone lessons with
colleagues.
• Supports cooperative learning through observing and being
observed.
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Math Lessons and Content
• Math practices should be integrated into lessons
• Not all math practices can be present in all lessons/problems
• All math practice opportunities should be experienced over the course of the unit/content
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Sample CCR-Aligned Lesson
Selected lesson: – hand out
This lesson includes essential CCR content, including:
• Clearly identified CCR content standards for level D.
• At least one Standard for Mathematical Practice, linked to the
lesson activities.
• A clear, essential question, related to the lesson objectives.
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Practice at Your Table
• Complete items 5 through 7of the Lesson Revision
Template [#5]:
o Specify the learning goals of the lesson.
o Identify the key CCR standards of the lesson.
o Identify math practices that are evident in the lesson
• Use the CCR Content Progressions [#2] to help you.
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Group Debrief
1. Did the lesson focus on a major work of the level?
2. What CCR standards did you select?
3. What key advances(Focus, Coherence and Rigor) are
evident or not evident in the lesson and or resource?
4. What math practices are evident in the lesson?
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Materials: What You Need
• CCR Content Progressions (#2)
• Standards for Mathematical Practice (#3)
• Checklist to Guide Mathematics Lesson Development (#6)
• Lesson Study Protocol (#7)
• Lesson plan template (provided)
• Sample lesson aligned to CCR standards
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Materials to Support Your Work
Checklist to Guide Mathematics Lesson Development [#6]:
• Is designed to accompany your development or revision of a
lesson.
• Reflects much of the content in the Lesson Revision Template
[#5].
• Serves as a final quality check of a lesson for the lesson study.
Lesson Study Protocol [#7]:
• Offers step-by-step guidance for developing, teaching and
observing, critiquing, and improving a lesson.
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Now, using the checklist (#6) and your notes, over the
next hour, work to strengthen this lesson.
Work to Strengthen the Lesson
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Apply the Checklist to Guide Mathematics Lesson Development
Fill out section 1 of the checklist:
• What are the learning goals for the students in this lesson?
Are they defined?
• How long do you think this lesson would take to complete?
Does the lesson define this?
Add notes about any refinements you might make to the lesson.
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Apply the Checklist - Section 2
Fill out section 2 of the checklist:
• What CCR content standards are targeted?
• Do they represent the major work of the level?
(Use the CCR Content Progressions [#2] to help you.)
Add notes about any refinements you might make to the lesson.
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Apply the Checklist - Section 3
Fill out section 3 of the checklist:
• Which specific Standards for Mathematical Practice are already included in the lesson?
• Are there other practices that are central to the lesson goals?
• Is there information about how the practices should be observed and assessed?
(Use the Standards for Mathematical Practice [#3] to help you.)
Add notes about any refinements you might make to the lesson.
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Fill out section 4 of the checklist:
• What foundational knowledge is needed to succeed in this
lesson? Does the lesson define this?
• How do concepts acquired in this lesson support future
learning? Does the lesson define this?
(Use the CCR Content Progressions [#2] to help you identify
concepts for foundational and future learning.)
Add notes about any refinements you might make to the lesson.
Apply the Checklist - Section 4
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Fill out section 5 of the checklist:
• Which aspect(s) of rigor do the targeted standards require?
• Do the lesson activities address those aspects?
• Does the lesson include thought-provoking tasks?
• Would additional tasks, activities, and problems strengthen the
lesson?
Add notes about any refinements you might make to the lesson.
Apply the Checklist - Section 5
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Fill out section 6 of the checklist:
• What explanations, representations, and/or examples are
included in the lesson to make its mathematics concepts clear?
Add notes about any refinements you might make to the lesson.
Apply the Checklist - Section 6
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Fill out section 7 of the checklist:
• When does student sharing happen in the lesson? Does the
lesson identify this?
• What are the expected responses to the discussion questions?
• Does the lesson tell us how to judge student understanding
based on the discussion?
Add notes about any refinements you might make to the lesson.
Apply the Checklist, Section 7
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Fill out Section 8 of the checklist:
• What strategies and opportunities are used to check for student
understanding throughout the lesson?
Add notes about any refinements you might make to the lesson.
Apply the Checklist, Section 8
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Group Debrief: Recap of Steps to Follow in a Lesson Study
What kinds of additions did you make to the lesson to
strengthen it?
• What challenges did you encounter if any?
Recap of the steps in a lesson study:
1. Choose a goal for the lesson study: What goal would you set
for the lesson study of the sample lesson?
2. Situate the goal in a sequence of learning: Where would the
sample lesson best be positioned in a sequence of learning?
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Questions and Comments