gary s. thomas, ed.d. superintendent education support services cucamonga math presentation 1

Gary S. Thomas, Ed.D.Superintendent

Education Support Services

Cucamonga Math Presentation

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Outcomes

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• Identify the core components and organization of the Mathematics CCSS

• Understand the design of the “Next Generation Assessments”

• Implications for practice



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NEXT GENERATION ASSESSMENTS

Common Core State Standards



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• Smarter Balanced assessments will go beyond multiple-choice questions to include extended response and technology enhanced items, as well as performance tasks that allow students to demonstrate critical-thinking and problem-solving skills.



Smarter Balanced Item Formats

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• Selected Response

– Assessment items where students select from a given set

of response options

• Constructed Response/Extended Response

– Assessment items where students produce the response

instead of selecting from a list

• Performance Tasks

– Assessment items where students perform a skill or create

a product

• Technology-Enhanced / Technology-Enabled

– Assessment items that employ technology to elicit a

response from a student.



Examples of StimuliWhich of the numbers is

• A factor of 100.• A multiple of 5. A. 4B. 40C. 50D. 500

The table below shows the number of students in each third-grade class at Lincoln School.

Students in the Third-Grade

There are 105 fourth-grade students at Lincoln School. How many more fourth-grade students than third grade students are at Lincoln School? Show your work or explain how you know.

Class Number of Students

Mrs. Roy 24

Mr. Grant 21

Mr. Harrison 22

Ms. Mack 25



Selected Response (SR)

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Test Item Type Student Evidence Maximum Time Requirement

Selected Response (SR) ELA- Most selected response (SR) items will relate to a stimulus (e.g., a reading passage or audio/visual stimulus).

Mathematics- A series of options from which to choose correct response(s)

Grades 3: 1 – 3 minutes per itemGrades 4-11: 1 – 2 minutes per item

Items scored based on single or multiple answers

Traditional Selected Response Items

• Traditional Selected Response Item

• Key and Distractor Analysis

Which number is both a factor of 100 and a multiple of 5?

A. 4

B. 40

C. 50

D. 500

A. 4

B. 40

C. 50

D. 500

OPTIONS: Possible answers the students must select from

Which number is both a factor of 100 and a multiple of 5?

STEMStatement ofthe question

A. Did not consider criteria of “multiple of 5”

B. Did not consider criteria of “factor of 100”

C. Correct

D. Multiplied 100 and 5

KEY

RATIONALE

DISTRACTOR



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Sample Math Non-Selected Response (SR)

STIMULUS

STEM

OPTIONS

DOK 2: Skills and Concepts

4th grade

Rubric



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Sample Math Non-Selected Response (SR)

STEM

OPTIONS


7th grade

Rubric



Constructed Response (CR)

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Test Item Type Student Evidence Maximum time requirement

Constructed Response (CR)short and extended

Items may require test-takers to enter a single word, phrase, sentence, number, or set of numbers

2 to 10 minutes

Items scored on a rubric



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STIMULUS

STEM

RESPONSE

Sample Math Constructed Response (CR)

120


5th grade

1 point item



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STIMULUS

STEM

RESPONSE

Sample Math Constructed Response (CR)High School

STEMDOK 2: Skills and Concepts

0.4 (or equivalent)

0.2 (or equivalent)

RESPONSE

1 point item



Technology Enhanced

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Test Item Type Student Evidence Maximum Time requirement

Technology Enhanced (TE) Items will take advantage of drag and drop, hot spot, ‐ ‐drawing, graphing, gridded-response items, and simulation technologies.

Depending on the task, time varies




Sample Math Technology Enhanced

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STIMULUS

STEM

1 point item


3rd Grade



Sample Math Technology Enhanced

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STIMULUS

STEMDOK 1: Recall and Reproduction

1 point item

Multiple Solution Path(64, 8) (1000, 10)(100, 100) (64, 4)

8th grade



Performance Task (PT)

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Test Item Type Student Evidence Maximum Time Requirement

Performance Task (PT)requires analytical thinking, evaluations, and support of students’ own responses across multiple content standards

ELA- Students will produce either a full written text or present a speech

Mathematics- Integrate knowledge and skills to formulate a problem, create model, evaluate data, and interpret results

Grade 3-8: 105 minutes (over 2-3 days)

High School: 120 minutes (over 2-3 days)




Sample Math Performance Task

DOK 4: Extended Thinking

18 point item Rubric scoring for each section

5th Grade

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Sample Math Performance Task

DOK 3: Strategic Thinking and Reasoning16 point item

Rubric scoring for each section

High School

ç



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CCSS MATHEMATICSCommon Core State Standards



Math Standards

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• Recurring practices grades K-12

Mathematical Practice

• Different at each grade level focus on skill set

Mathematical Content



Content Standards

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Grade level Focus or Priority• Identified in CCSS document not in the

California Common Core document

Domains• Content topics

Cluster Heading • Identifies standard

Cluster Standards• Group of related standards• CA added 15% are in italics and bold

Progressions across grade levels



Grade Level Focus or Priority

For each grade level from kindergarten through grade 8, the Critical Areas outline the essential mathematical ideas for each grade level.

d

d



Reading of K-8 Standards

Domain

Domain

Domain

Domain

Domain



Reading of K-8 Standards

Stem: if standard isfollowed by a,b,c, etc.

Cluster

Domain

Cluster Heading

Standards

http://www.clipartillustration.com/clipart-illustration-orange-people-talking-in-chit-chat-conversation-or-speech.php

http://www.clipartillustration.com/clipart-illustration-orange-people-talking-in-chit-chat-conversation-or-speech.php



Examples of Additional 15%:

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• Added language to existing standard

– Grade 2-Measurement and Data

– Grade 4-Geometry



Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the understanding of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

28 CISC Mathematics Toolkit


Locate the CCSS for Mathematics and read the first three words for each mathematical practice and notice the similarities.

What do they begin with?Mathematically proficient students…

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Briars & Mitchell (2010)Getting Started with the Common Core State Standards




“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.”

(CCSS, 2010)

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The Standards for [Student] Mathematical Practice

What are the verbs that illustrate the student actions for your assigned

mathematical practice?

Circle, highlight or underline them for all eight practices…

Discuss with a partner:

What stands out?



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CLASSROOM STRATEGY: POSE MEANINGFUL TASKS

Common Core State Standards



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Not only this

But also….

7 x 5 = ___



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Amber didn’t know what 7 x 5 equals, but she know 5 x 5 = 25 and 2 x 5 = 10.

Use drawings, words and/or equations to explain why Amber can add 25 and 10 to find out what 7 x 5 equals.

Source: PARCC Item Department ITN, Appendix F



Multiplication Problem

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• Which mathematical practices are needed to complete the task?

• What mathematics content is needed to complete the task?



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Not only this

Find 20% of 350,000.

But also….



Claim

A fast food chain reports that 8% of people in the U.S. eat at its restaurants each day. The chain has 12,800 stores. According to

the 2010 Census Bureau report approximately 310 million people live in the

U.S..

Make a conjecture as to whether or not you believe the report from the fast food chain

to be accurate. Create a mathematical argument that validates your conclusion.



Claim Problem

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Not only this

A rectangle has side lengths of 7 m and 9 m. A square with side length 5 m is cut out of the rectangle. Find the area of that rectangle after the square is cut out.

But also….



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Here is a satellite photo of Geoff’s yard. He is interested in re-sodding his entire yard. Help him out by providing a cost estimate. Develop a list of questions you would need answered before you can solve the problem. Bristol Central Geometry Teachers



Claim Problem

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MATHEMATICAL DISCOURSEDaily Instruction

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Mathematical Discourse

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The ability to read, write, listen, ask questions, think, and communicate about math will develop and deepen students’ understanding of mathematical concepts. Students should read text, data, tables, and graphs with comprehension and understanding. Their writing should be detailed and coherent, and they should use correct mathematical vocabulary. Students should write to explain answers, justify mathematical reasoning, and describe problem-solving strategies.

Indiana Department of Education, 2006



Language in CCSS Mathematics

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• 505 math standards K-12– 89 math standards – 11 cluster headings – Standards for Mathematical Practice

Require Language in Math



Math Verbs

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’97 Standards Verbs

• Solve, demonstrate, understand

CCSS Verbs

• Explain, describe, justify, summarize



Mathematically Speaking

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• a = math teacher• b = guides students towards use of

specialized oral and written language to communicate content

• c = English teacher then

a = b and b = c

therefore a = c Krajicek, Stephanie Writing in Mathematics: Common Objections & FAQ’sVolume 2, Number 2 issue of the Write Connections




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• Where to begin….• Start with why and because

– Teacher: Start asking why?• Why did you choose that step?

– Student: Explain their thinking.• ______is correct because_____.




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• Think Alouds• Number Talks• Think-Pair-Share• Response Frames• Language Objectives• Writing in Math• Support for EL Learners



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COGNITIVE RIGORCommon Core State Standards



Cognitive Rigor Matrix

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Depth of Knowledge

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DOK 1Recall &

Reproduction•Working with facts•One–step problems•Verbs•Solve•Complete•Recall•Identify•Evaluate

DOK 2Basic Skills &

Concepts•Explain concept•Select procedure•Verbs•Explain•Model•Interpret•Organize data•Extend a pattern



Depth of Knowledge

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DOK 3Strategic Thinking &

Reasoning•Predictable outcomes•Support Conclusion•Verbs•Plan•Prove•Draw Conclusions•Compare/Contrast•Logical argument

DOK 4Extended Thinking•Conduct Investigations•Real-world problems•Verbs•Relate•Develop•Design•Apply•Justify



DOK Levels

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4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

Kentucky Department of Education

DOK 1Recall & Reproduction

DOK 2Basic Skills & Concepts

Find the next three terms in the following pattern: 2/7, 4/7, 6/7, 8/7…

Identify

Draw the next figure in the following pattern:

Use models to explain concept



DOK Levels

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ReasoningDOK 4

Extended Thinking

Find the next three terms in the pattern and determine the rule for the following pattern of numbers:1, 4, 8, 11, 15, 18, 22, 25, 29, …

Explain ideas using supporting evidence

Find the next three terms in the pattern, determine the rule for finding the next number in the pattern, and make or find a model for the pattern:1, 1, 2, 3, 5, 8, 13, 21, 34, … Design a mathematical model

4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.




DOK Levels

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DOK 1Recall & Reproduction

DOK 2Basic Skills & Concepts

The price of gasoline was $2.159 per gallon last week. This week the new price is $2.319 per gallon. Calculate the percent of increase.

On a trip across the country, Justin determined that he would have to drive about 2,763 miles. What speed would he have to average to complete the trip in no more than 50 hours of driving time? How many days would he have to drive if he drove 8 hours per day?

Multiple Steps

7.RP.3 Use proportional relationships to solve multistep ratio and percent problems.




DOK Levels

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ReasoningDOK 4

Extended Thinking

A sweater that you really been want has just been placed on sale. The original cost was $63.99. The sale price is $47.99. What is the percent of decrease from the original price You still do not have enough money saved up to purchase the sweater, so you wait just a little longer and the store now has an ad that states that all items currently on sale have been reduced by 1/3 of the sale price. What is the new sale price? What is the overall percent of decrease from the original price? Compare/contrast data

Students will visit three local grocery stores and find the prices of three different sizes of the same product at the three stores. Students will then determine the unit price for each size item at each store and make a decision as to which is the best buy. Students will then write a report chronicling their work and reporting which is the best buy, justifying their decision with their mathematical work.

Analyze multiple sources of evidence.

7.RP.3 Use proportional relationships to solve multistep ratio and percent problems.




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Connecting Practices to Instructional Materials

Instructional Materials



Gap Analysis

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• With your table group compare a sample text book item to the CCSS.

• Determine how the standard is aligned with the curriculum?

• EVERYONE MUST AGREE




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• Extending a text problem



Grade level teams

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• Within grade level teams of 4– Identify a team leader– Select 4 standards

• one for each participant

– Align materials with CCSS standards• Identify student activities from each DOK

level

– Team leader will collect and distribute copies to the team. ????

gary s. thomas, ed.d. superintendent education support services cucamonga math presentation 1

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