transitioning to the common core: changing the definition of mathematical proficiency

Transitioning to the Common Core: Changing the Definition of Mathematical

Proficiency

Patrick CallahanStatewide Co-Director, California Mathematics Project

UCLA

SCOE 30 April 2014

What do we mean by implementing the Common Core?

Many districts and even states are claiming or planning to fully implement the Common Core by 2014 or 2015.

“fully implemented?”

From a student’s perspective the first time the Common Core could be fully implemented is a student graduating in 2024.

Before that time every student will experience a hybrid of Common Core and previous mathematics.

“fully implemented?”

From a student’s perspective the first time the Common Core could be fully implemented is a student graduating in 2024.

Before that time every student will experience a hybrid of Common Core and previous mathematics.

CONGRATS CLASS OF 2014 !

You have experienced about 7.692% Common Core!

“Fully implementing” and text books

With a little $$$ we took our old textbook…

And bought new Common Core

textbooks!

Implementation and Textbooks

Implementation vs Transition

The word “implementation” tends to refer to the policy aspects of adopting the Common Core.

In a policy sense you can be “fully implemented” right away.

Another, more student-centric, approach is to think in terms of “transition” rather than “implementation”.

This is a pragmatic approach that acknowledges that student, parents, teachers, and systems are where they are now and that it will take time to move the system to the Common Core.

Transition to What?

We use the phrase “implement the Common Core”

or “transition to the Common Core”

but what does that mean?

What exactly are the Common Core Standards?

Common Core Standards, what they are NOT and what they ARE:

The Common Core standards are not a list of topics to be covered or taught.

The Common Core State Standards are a description of the mathematics students are expected to understand and use, not a curriculum. The standards are not the building blocks of curriculum, they are the achievements we want students to attain as the result of curriculum.

How are the CCSS different?The CCSS are reverse engineered from an analysis of what students need to be college and career ready.

The design principals were focus and coherence. (No more mile-wide inch deep laundry lists of standards)

The CCSS in Mathematics have two sections:CONTENT and PRACTICES

The Mathematical Content is what students should know.The Mathematical Practices are what students should do.

Real life applications and mathematical modeling are essential.

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.

Mathematical Practice

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REASONING AND EXPLAINING2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of others

MODELING AND USING TOOLS4. Model with mathematics5. Use appropriate tools strategically

SEEING STRUCTURE AND GENERALIZING7. Look for and make use of structure8. Look for and express regularity in repeated reasoning

Constructing viable arguments

3 Construct viable arguments and critique the reasoning of others.Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Constructing viable arguments

• use stated assumptions, definitions, and previously established results in constructing arguments.

• make conjectures

• build a logical progression of statements

• analyze situations by breaking them into cases

• recognize and use counterexamples

• justify their conclusions, communicate them to others, and respond to the arguments of others

• distinguish correct logic or reasoning from that which is flawed

Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.

Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Viable arguments are important beyond mathematics

21st Century Skills

Common Core Standards for English Language Arts

Career and College Readiness Anchor Standards for Writing

Text types and Purposes*1. Write arguments to support claims in an analysis of substantive topics or texts, using valid reasoning and relevant and sufficient evidence.2. Write informative/explanatory texts to examine and convey complex ideas and information clearly and accurately through the effective selection, organization, and analysis of content.3. Write narratives to develop real or imagined experiences or events using effective technique, well-chosen details, and well-structured event sequences.

Production and distribution of Writing4. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.5. Develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach. 6. Use technology, including the Internet, to produce and publish writing and to interact and collaborate with others.

Research to Build and Present Knowledge7. Conduct short as well as more sustained research projects based on focused questions, demonstrating understanding of the subject under investigation.8. Gather relevant information from multiple print and digital sources, assess the credibility and accuracy of each source, and integrate the information while avoiding plagiarism.9. Draw evidence from literary or informational texts to support analysis, reflection, and research.

Range of Writing10. Write routinely over extended time frames (time for research, reflection, and revision) and shorter time frames (a single sitting or a day or two) for a range of tasks, purposes, and audiences.

Arguments to support claims

1. Write arguments to support claims in an analysis of substantive topics or texts, using valid reasoning and relevant and sufficient evidence.

2. Write informative/explanatory texts to examine and convey complex ideas and information clearly and accurately through the effective selection, organization, and analysis of content.

Practices for Next Generation Science Standards

1. Asking questions (for science) and defining problems (for engineering)

2. Developing and using models3. Planning and carrying our investigations4. Analyzing and interpreting data5. Using mathematics and computational thinking6. Constructing explanations (for science) and designing

solutions (for engineering)7. Engaging in argument from evidence8. Obtaining, evaluating and communicating information

Shifts in Content

Because the Common Core were reverse engineered from a definition of Career and College Ready, there were shifts in content.

How is Algebra different?

More applications, modeling, equivalence

Less algorithms, answer-getting, simplifying

Sample Algebra Worksheet

This should look familiar.What do you notice?

What is the mathematical goal?

What is the expectation of the student?

Look at the circled answers.What do you notice?

“Answer Getting”As Phil Daro has mentioned:There is a difference between

using problems to “get answers” and to learn mathematics.

This algebra exam sends a clear message to students:

Math is about getting answers.

Note also that there is no context, just numbers and expressions

What are these assessing?

SBAC Claims

Another shift:

Just in Time

vs

Just in Case

New expectations require new Pathways

Changing expectationsThe trouble with course names

In the particular case of mathematics, there is a “vocabulary” around the names of mathematics courses that is likely to cause confusion not only for educators, but also for parents. “Algebra 1” is a course that, prior to CA CCSSM, has been taught in 8th grade to an increasing number of students. That same course name will be the default for ninth grade for most students who moving forward will complete the CA CCSSM for grade eight – a course that is more rigorous and more demanding than the earlier versions of “Algebra 1.” Even so, we expect the changes to cause confusion. The single most practical solution is to describe detailed course contents, in addition to course names, as a way of clearing up confusion until “Algebra I” as commonly used, refers to a ninth grade and not an eighth grade course

An important equation:

Algebra 1 ≠ Algebra 1

Previous 8th grade CA standards

Crosswalks are not the answer

Changing expectations:Middle School is key

When the expectations for middles school mathematics were about speed and accuracy of computations it made sense to accelerate in middle school, and even skip grades.

This no longer makes sense.

Middle school mathematics is the key to success for all students. Rushing or skipping is a bad idea for almost all students.

NCEE Report (May, 2013)http://www.ncee.org/college-and-work-ready/

NCEE Summary Findings: Career and College Ready

1. Many community college career programs demand little or no use of mathematics. To the extent that they do use mathematics, the mathematics needed by first year students in these courses is almost exclusively middle school mathematics. But the failure rates in our community colleges suggest that many of them do not know that math very well. A very high priority should be given to the improvement of the teaching of proportional relationships including percent, graphical representations, functions, and expressions and equations in our schools, including their application to concrete practical problems.

NCEE Summary Findings: Career and College Ready

3. It makes no sense to rush through the middle school mathematics curriculum in order to get to advanced algebra as rapidly as possible. Given the strong evidence that mastery of middle school mathematics plays a very important role in college and career success, strong consideration should be given to spending more time, not less, on the mastery of middle school mathematics, and requiring students to master Algebra I no later than the end of their sophomore year in high school, rather than by the end of middle school. This recommendation should be read in combination with the preceding one. Spending more time on middle school mathematics is in fact a recommendation to spend more time making sure that students understand the concepts on which all subsequent mathematics is based. It does little good to push for teaching more advanced topics at lower grade levels if the students’ grasp of the underlying concepts is so weak that they cannot do the mathematics. Once students understand the basic concepts thoroughly, they should be able to learn whatever mathematics they need for the path they subsequently want to pursue more quickly and easily than they can now

Common Core Grade 8 Curriculum Plan

Common Core is much more rigorous than previous middle school expectations.

CA Framework on Acceleration

1. Decisions to accelerate students into the Common Core State Standards for higher mathematics before ninth grade should not be rushed.

Placing students into an accelerated pathway too early should be avoided at all costs. It is not recommended to compact the standards before grade seven to ensure that students are developmentally ready for accelerated content. In this document, compaction begins in seventh grade for both the traditional and integrated sequences.

CA Framework on Acceleration

2. Decisions to accelerate students into higher mathematics before ninth grade must require solid evidence of mastery of prerequisite CA CCSSM.

3. Compacted courses should include the same Common Core State Standards as the non-compacted courses.

4. A menu of challenging options should be available for students after their third year of mathematics—and all students should be strongly encouraged to take mathematics in all years of high school.

Framework Suggested Pathways

Better than accelerating Middle School.

But doubling up is not necessary!

Framework Suggested Pathways

Better than accelerating Middle School.

But doubling up is not necessary! “Pre-calculus” is not necessary!

A better pathway:

Enhanced means: Include the (+) standards, go deeper, more rigorous, not skim faster!

Pathways and curricular CONTENT changes will require time to roll out

The Common Core Standards for Mathematical Practice (and the SBAC Claims) can start happening immediately.

But, remember, students will need to transition from previous expectations to the new expectations.

3 Year Content Roll Out9th 10th 11th

Year 1 Common Core Math Practices+Common CoreAlg 1 or Math 1

Common Core Math Practices+Whatever Content was in place



Common Core Math Practices+Common CoreGeom or Math 2



Common Core Math Practices+Common CoreGeom or Math 2

Common Core Math Practices+Common CoreAlg 2 or Math 3

This applies to other grade bands too: K-2, 3-5, and 6-8

Transitioning to Common Core:Focus and Purpose

Transitions are like home remodels: they take time and they can be messy along the way, but it is worth the effort if you have a vision for improving mathematics for all students.

Real-life?

A template for planning

transitioning to the common core: changing the definition of mathematical proficiency

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