what difference will they make?

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WHAT DIFFERENCE WILL THEY MAKE?

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K-12 Mathematics Common Core State Standards

Jaime Aquino, Ph.D.General Manager

North Region

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Objectives

• Develop a understanding of the Common Core Sate Standards in Mathematics by relating their implementation to the past, current and future work of Networks.

• Identify the implications of the CCSS Math Standards to instruction, assessment, leadership and professional development.

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Why do we study mathematics in school?

• Because it’s hard and we have to learn hard things at

school. We can learn easy stuff at home like manners.

Carrine, K

• Because it always comes after reading. Roger, 1

• Because all the calculators might run out of batteries or

something. Thomas, 1

• Because it’s important. It’s the law from President Bush

and it says so in the Bible on the first page. Jolene, 2

• Because you can drown if you don’t. Amy, K

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Why do we study mathematics in school?

• Because what would you do with your check from work when you

grow up? Brad, 1

• Because you have to count if you want to be an astronaut. Like

10…9…8…blast off. Michael, 1

• Because you could never find the right page. Mary, 1

• Because when you grow up you couldn’t tell if you are rich or not.

Raji, 2

• Because my teacher could get sued if we don’t. That’s what she

said. Any subject we don’t know – Wham! She gets sued and

she’s already poor. Corky, 3

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What are standards?

• Standards define what students should understand and be able to do.

• Standards must be a promise to students of the mathematics they can take with them.

• We haven’t kept our old promise and now we make a new one.

• What difference will it make?

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Lessons Learned

After two decades of standards based accountability:

• Too many standards

• Lack of student motivation

• “Cover” at “pace” is a failure

– Tells teachers to ignore students

– Turn the page regardless

– Shrug your shoulders and do what “they” say

– Mathematics is not a list of topics to cover

• Singapore: “Teach less, learn more”

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Lessons Learned

• TIMSS: math performance in the US is being compromised by a lack of focus and coherence in the “mile wide, inch deep” curriculum

• Hong Kong students outscore U.S. students on the grade 4 TIMSS, even though Hong Kong only teaches about half of the tested topics. U.S. covers over 80% of the tested topics.

• High-performing countries spend more time on mathematically central concepts: greater depth and coherence.

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Answer Getting vs. Learning Mathematics

United States

How can I teach my kids to get the answer to this problem? Use mathematics they already know. Easy,

reliable, works with bottom half, good for classroom management.

Japan

How can I use this problem to teach mathematics they don’t already know?

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Math Standards

1. Mathematical Performance: what kids should be able to do

2. Mathematical Understanding: standards for what kids need to understand

3. Mathematical Practices: behaviors students need to exhibit in mathematics

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Performance

• Performance: what kids should be able to

do• multiply and divide within 100 • 3rd grade sample

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Understanding

• Understanding: what kids should understand about mathematics

• 3rd grade sample– Understand properties of multiplication and

the relationship between multiplication and division.

• Table Talk:– Why is that our kids do not perform as well as

students in other countries do?

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215 + 31

2.15

+ 3.1

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24 x 5 = 120

4 x 1/2 = 2

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Impact of MisconceptionsImpact of Misconceptions

We use ideas we already have (BLUE DOTS)

to construct a new idea (RED DOT)

John Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 2004, page 23.

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Correcting Misconceptions versus Typical Remedial Correcting Misconceptions versus Typical Remedial LearningLearningCorrecting Misconceptions versus Typical Remedial Correcting Misconceptions versus Typical Remedial LearningLearning

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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 reasoning 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.

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There are 125 sheep

and

5 dogs in a flock.

How old is the shepherd?

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A Student’s Response

There are 125 sheep and 5 dogs in a flock.

How old is the shepherd?

125 x 5 = 625 extremely big

125 + 5 = 130 too big

125 - 5 = 120 still big

125 5 = 25 That works!

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Math Standards

1. Mathematical Performance: what kids should be able to do

2. Mathematical Understanding: standards for what kids need to understand

3. Mathematical Practices: varieties of expertise that math educators should seek to develop in their students.

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Take the number apart?

Tina, Emma, and Jen discuss this expression:

6×5 1/3

Tina: I know a way to multiply with a mixed number that is different from the one we learned in class. I call my way “take the number apart.” I’ll show you. First, I multiply the 5 by the 6 and get 30. Then I multiply the 1/3 by the 6 and get 2. Finally, I add the 30 and the 2 to get my answer, which is 32.

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Take the number apart?

Tina: It works whenever I have to multiply a mixed number by a whole number.

Emma: Sorry Tina, but that answer is wrong!

Jen: No, Tina’s answer is right for this one problem, but “take the number apart” doesn’t work for other fraction problems.

Which of the three girls do you think is right?Justify your answer mathematically?

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Table Talk

• What are your reactions to the sample assessment item?

• How does the sample assessment item compare to tasks being assigned in your school?

• How does the sample assessment item assesses the three types of standards (performance, understanding and practices)?

• What are the implications for mathematics teaching and learning?

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Overview of K-8 Mathematics Standards

•The K-5 standards provide students with a solid foundation in whole numbers, addition, subtraction, multiplication, division, fractions and decimals

•The 6-8 standards describe robust learning in geometry, algebra, and probability and statistics

•Modeled after the focus of standards from high-performing nations, the standards for grades 7 and 8 include significant algebra and geometry content

•Students who have completed 7th grade and mastered the content and skills will be prepared for algebra, in 8th grade or after

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How to Read the Standards: K-8

• introduction (see page 13)

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• Overview (see page 14)

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• Domains are larger groups of related standards. Standards from different domains may sometimes be closely related.

DOMAIN

STANDARD

CLUSTER

• Standards define what students should understand and be able to do.

• Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject.

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Overview of High School Mathematics Standards

The high school mathematics standards:

–Call on students to practice applying mathematical ways of thinking to real world issues and challenges

–Require students to develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly are called to do

–Emphasize mathematical modeling, the use of mathematics and statistics to analyze empirical situations, understand them better, and improve decisions

–Identify the mathematics that all students should study in order to be college and career ready.

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How to Read the Standards: High School• Additional mathematics that students should learn in order to

take advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by (+), as in this example: – (+)Represent complex numbers on the complex plane in

rectangular and polar form (including real and imaginary numbers).

• All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students.

• The high school standards are listed in conceptual categories:• Number and Quantity • Algebra • Functions • Modeling • Geometry • Statistics and Probability

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How to Read the Standards: High School

• Introduction (p. 62)

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How to Read the Standards: High School

• Overview (p. 63)

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Four Model Course Pathways

• A more traditional approach, with two algebra courses and a geometry course and data included in each;

• An integrated approach, with three courses that each includes number, algebra, geometry, and data;

• A “compacted” version of each pathway that begins in Grade 7 and allows students to study Calculus or other college level courses in high school.

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In considering the pathways, there are several things important to note:

• The pathways and courses are models, not mandates. They illustrate possible approaches to organizing the content of the CCSS into coherent and rigorous courses that lead to college and career readiness. States and districts are not expected to adopt these courses as is; rather, they are encouraged to use these pathways and courses as a starting point for developing their own.

• All college-and career-ready standards (those without a +) are found in each pathway. A few (+) standards are included to increase the coherence of a course.

• The course descriptions delineate the mathematics standards to be covered in a course but they are not prescriptions for curriculum or pedagogy. Additional work will be needed to create coherent instructional programs that help students achieve these standards.

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Activity: Becoming Familiar with the Standards

• Review the Mathematics Standards by marking them up with questions/comments that focus on changes that these standards will require of teachers across all disciplines.

• Think through the instructional changes that will arise as a result of the CCSS by talking through the issues that these standards will engender and the problems with resources, including time and the need for professional development.Encouraging Challenges Questions Instructional

Changes

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Analyzing Math Tasks from the Lens of the CCSS

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Analyzing Student Work from the Lens of the CCSS

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CCSS Math Assessments

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A TEST THAT IS WORTH TEACHING TO

SHOULD…

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• The through-course components in mathematics will be administered online to students in grade 6 through high school, using an equation editor-type program that allows students to enter responses to mathematical problems via the computer.

• Through-course components for grades 3–5 will be administered in paper-and-pencil format because of concerns about young students‘ lack of familiarity with pull-down menus and online mathematics tools.

• The Partnership will study the efficacy of online administration of the through-course components to students in grades 3–5 over time. Additionally, as in ELA/literacy, the end-of-year mathematics component will be delivered via computer to students in all grades.

Math Assessment: Mode of Administration

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• The first two through-course components emphasize standards or clusters of standards (i.e., one to two essential topics) from the CCSS that play a central role during the first stages of mathematics instruction over the school year.

• These include standards that are prerequisites for others at the same grade level, as well as standards or clusters of standards for fields of study that first appear during the grade in question. Thus, instead of surveying an overly broad mathematical landscape as typical ―interim assessments‖ currently do, these components will promote the coherent curricular structure embedded in the CCSS.

• This approach also will enable the through-course components to provide more useful results to teachers across the range of performance from a blend of one to two brief constructed-response items per topic and one extended constructed-response per topic.

• Over time, the Partnership will refine the selection of standards measured by the focused components based on which mathematical topics prove most predictive of success later in the school year.

Math-1 and Math-2. Focused Assessments of

Essential Topics.

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Math-1 and Math-2. Focused Assessments of Essential Topics.

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Math-1 and Math-2. Focused Assessments of Essential

Topics.

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Math-3. Extended Mathematics Assessment.

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– This component will leverage technology to administer innovative, computer-enhanced items that measure the extent to which students have mastered important knowledge and skills identified in the CCSS and critical for college and career readiness, including reasoning and problem solving.

– The assessment will sample a range of important topics for the year, as well as sampling material from the previous year, and use items with an appropriate range of cognitive demand to provide a better measure of achievement for students at the very low and very high end of the performance spectrum

– . The items for this component will include next-generation selected-response items that signal not only whether students provided a correct answer but also help analyze why some students might have provided an incorrect answer (i.e., by identifying common mathematical errors that suggest common mathematics misunderstandings). The component will also include innovative item types like using drag and drop or graphing tools.

•

•

Math-4. End-of-Year Mathematics Assessment.

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•

Math-4. End-of-Year Mathematics Assessment.

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Reflection

• Write one action step you will take to have your math instruction be more aligned with the CCSS.

• Share your action step with the person sitting next to you.

1. Curriculum: What are the academic tasks (content, knowledge, skills) that we ask students to do?

2. Pedagogy: How do teachers support student learning?

3. Assessment: How do we know students are learning?

4. Collaboration: How do adults learn and improve their practice?

5. Structure: How do we use time, space, technology, and other resources to enable student learning?

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Challenges Ahead

60

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There are a number of technical/technological challenges that PARCC is currently facing including:•Developing an interoperable technology platform that meets the needs of all PARCC states •Transitioning states to an computer-based assessment system

– Will provide state and district needs assessment– Will support state and district transition planning

• Developing and implementing Artificial Intelligence (AI) scoring systems and processes

• Identifying innovative item types that are effective measures

Key Technical Challenges for PARCC

61

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Developing and implementing next generation, K-12 assessment system in just four years will be a major challenge for state leaders, district and school leaders, and educators alike. Challenges include:•Estimating administrative costs over time, including long-term budgetary planning

– How can states use existing sources of funding to support implementation of the new assessment system?

•Transitioning to the new assessments, including “through-course” components, and what the impact will be at the classroom level

– Providing tools, resources and supports to districts and schools to ease this transition

•Ensuring long-term sustainability

Key Implementation Challengesfor PARCC

62

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The implementation of CCSS and PARCC will not happen in a vacuum and require states to address a number of related policies, such as:•High school course requirements

– What courses need to be required to ensure there is alignment with the Common Core and high school PARCC assessments?

– In what courses should the assessments be given in high school?•Accountability

– How will states’ accountability systems need to evolve to take into account PARCC assessments?

•Student supports and interventions– How/when will supports and interventions be triggered for students

not meeting proficiency/readiness scores on the PARCC assessments?

Key Policy Challenges for PARCC

63

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Aligning Tasks to theCommon Core State Standards:

Mathematics

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Comparing Two Mathematical Tasks

MAKING CONJECTURES Complete the conjecture based on the pattern you

observe in the specific cases.

29. Conjecture: The sum of any two odd numbers is ______?

1 + 1 = 2 7 + 11 = 181 + 3 = 4 13 + 19 = 323 + 5 = 8 201 + 305 = 506

30. Conjecture: The product of any two odd numbers is ____?1 x 1 = 1 7 x 11 = 771 x 3 = 3 13 x 19 = 2473 x 5 = 15 201 x 305 = 61,305

TASK A

Think privately about how you would go about solving task A (solve them if you have time)

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TASK B

Think privately about how you would go about solving task B (solve them if you have time)

TASK BFor problems 29 and 30, complete the

conjecture based on the pattern you observe in the

examples. Then explain why the conjecture is always true

or show a case in which it is not true.

MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases.

29. Conjecture: The sum of any two odd numbers is ______?

1 + 1 = 2 7 + 11 = 181 + 3 = 4 13 + 19 = 323 + 5 = 8 201 + 305 = 506

30. Conjecture: The product of any two odd numbers is ____?1 x 1 = 1 7 x 11 = 771 x 3 = 3 13 x 19 = 2473 x 5 = 15 201 x 305 = 61,305

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Draw a Picture

Every odd number (like 11 and 13) has one loner number. Add the two loner numbers and you will get an even number (24). Now add all together the loner numbers and the other two (now even) numbers.

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Build a Model

If I take the numbers 5 and 11 and organize the counters as shown, you can see the pattern.

You can see that when you put the sets together (add the numbers), the two extra blocks will form a pair and the answer is always even. This is because any odd number will have an extra block and the two extra blocks for any set of two odd numbers will always form a pair.

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Logical Argument

An odd number = [an] even number + 1. e.g. 9 = 8 + 1

So when you add two odd numbers you are adding an even no. + an even no. + 1 + 1. So you get an even number. This is because it has already been proved that an even number + an even number = an even number.

Therefore as an odd number = an even number + 1, if you add two of them together, you get an even number + 2, which is still an even number.

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Use Algebra

If a and b are odd integers, then a and b can be written a = 2m + 1 and b = 2n + 1, where m and n are other integers.

If a = 2m + 1 and b = 2n + 1, then a + b = 2m + 2n + 2.

If a + b = 2m + 2n + 2, then a + b = 2(m + n + 1).

If a + b = 2(m + n + 1), then a + b is an even integer.

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How are the two versions of the task the same and how are they different?

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Tasks A and B

Same Both ask students to

complete a conjecture about odd numbers based on a set of finite examples that are provided

Different Task B asks students to develop

an argument that explains why the conjecture is always true (or not)

Task A can be completed with limited effort; Task B requires considerable effort – students need to figure out WHY this conjecture holds up

The amount of thinking and reasoning required

The number of ways the problem can be solved

The range of ways to enter the problem

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Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively3. Construct viable arguments and critique the

reasoning of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning

Common Core State Standards for Mathematics, 2010, pp.6-7

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Tasks A and B

Same Both ask students to

complete a conjecture about odd numbers based on a set of finite examples that are provided

Different Task B asks students to develop

an argument that explains why the conjecture is always true (or not)

Task A can be completed with limited effort; Task B requires considerable effort – students need to figure out WHY this conjecture holds up

The amount of thinking and reasoning required

The number of ways the problem can be solved

The range of ways to enter the problem

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Mathematical Tasks:A Critical Starting Point for Instruction

Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.

Stein, Smith, Henningsen, & Silver, 2000

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The level and kind of thinking in which students engage determines what they will learn.

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

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There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.

Lappan & Briars, 1995

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If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks.

Stein & Lane, 1996

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If we want students to develop the capacity to think, reason, and problem solve then we need

to start with high-level, cognitively complex tasks.

Stein & Lane, 1996

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K-12 Mathematics Common Core State Standards

what difference will they make?

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