Key Advances, Math Practices, and Translating Math Standards Into College and Career...

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

Technology Services

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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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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


• What CCR content standards are targeted?

• Do they represent the major work of the level?

(Use the CCR Content Progressions [#2] to help you.)


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• 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.)


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• 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.)



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• 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?



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• What explanations, representations, and/or examples are

included in the lesson to make its mathematics concepts clear?



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• 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?


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?


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

Key Advances, Math Practices, and Translating Math Standards Into College and Career...

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