SCOE 30 April 2014
Transitioning to the Common Core:
Changing the Definition of
Mathematical Proficiency
Patrick Callahan
Statewide Co-Director, California Mathematics Project
UCLA
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.
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 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 milewide 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.
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.
REASONING AND EXPLAINING
1. Make sense of problems and persevere in
solving them
6. Attend to precision
OVERARCHING HABITS OF MIND
CCSS Mathematical Practices
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
MODELING AND USING TOOLS
4. Model with mathematics
5. Use appropriate tools strategically
SEEING STRUCTURE AND GENERALIZING
7. Look for and make use of structure
8. 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 wellstructured event sequences.
Production and distribution of Writing
4. 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 Knowledge
7. 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 Writing
10. 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 models
3. Planning and carrying our investigations
4. Analyzing and interpreting data
5. Using mathematics and computational thinking
6. Constructing explanations (for science) and designing
solutions (for engineering)
7. Engaging in argument from evidence
8. 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 expectations
The 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
Changing expectations
The 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 Out
9th
10th
11th
Year 1
Common Core Math
Practices
+
Common Core
Alg 1 or Math 1
Common Core Math
Practices
+
Whatever Content was in
place
Common Core Math
Practices
+
Whatever Content was
in place
Year 2
Common Core Math
Practices
+
Common Core
Alg 1 or Math 1
Common Core Math
Practices
+
Common Core
Geom or Math 2
Common Core Math
Practices
+
Whatever Content was
in place
Year 3
Common Core Math
Practices
+
Common Core
Alg 1 or Math 1
Common Core Math
Practices
+
Common Core
Geom or Math 2
Common Core Math
Practices
+
Common Core
Alg 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