Facing Choices about
the Fourth Year in Arizona:
Achieve’s Fourth-Year
Capstone Course Criteria
October 18, 2008
Institute for Mathematics & Education
Tucson, Arizona
Who is Achieve, Inc.?

Created by the nation’s governors and
business leaders, Achieve, Inc., is a
bipartisan non-profit organization that
helps states raise academic standards,
improve assessments and strengthen
accountability to prepare all young people
for postsecondary education and training,
careers, and citizenship.
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Presentation Overview
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
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
History of the American Diploma Project (ADP)
Mathematics Graduation Requirements
Criteria for High-Quality Capstone Courses
Contact Information
Questions
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History of the
American Diploma Project
American Diploma Project

The American Diploma Project (ADP) was
created to ensure all graduates leave high
school ready for college and careers.

Early research by ADP sought to identify
“must-have” knowledge and skills graduates
will need to be successful in college and
careers.
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American Diploma Project

Found a convergence between the skills that
high school graduates need to be successful in
college and those they need to be successful
in a career that supports a family and offers
career advancement.

Developed ADP benchmarks that include the
core content and skills in mathematics and
English all students should have when they
graduate high school.
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Key findings

In mathematics, graduates need strong
computation skills, ability to solve challenging
problems, reasoning skills, geometry, data
analysis, statistics, and advanced algebra.

Essentially, they need the knowledge and
skills typically taught in courses such as
Algebra I, Algebra II and Geometry, as well
as data analysis and statistics.

In English, graduates need strong reading,
writing and oral communication skills equal to
four years of grade-level coursework, as well
as research and logical reasoning skills.
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The ADP Benchmarks:
Challenging content for all
students
To cover the content in the ADP benchmarks,
high school graduates need:

In Mathematics:


A rigorous four-year
course sequence
Content* equivalent to
a sequence that
includes Algebra I and
II, Geometry, and Data
Analysis & Statistics
*can be taught via different
pathways

In English:


Four courses
Content equivalent to
four years of gradelevel English or higher
with a strong focus on
oral and written
communication skills
and considerable
research and analysis
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Closing the Expectations Gap:
ADP Policy Agenda

In 2005, Achieve launched the ADP Network, a
group of states committed to taking four
college and career readiness action steps:


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
Align high school standards with college and career
expectations.
Require all students to take a college- and careerready curriculum, aligned with standards, to earn a
diploma.
Build “college-ready” measures, aligned with state
standards, into high school assessment systems.
Hold high schools accountable for graduating
students college- and career-ready, and hold
postsecondary institutions accountable for
student success.
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ADP Network launched at 2005
Summit: 13 states committed to
improving student preparation
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ADP Network today: thirty-four
states now committed to
improving student preparation
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Mathematics Graduation
Requirements
An Expectations Gap

Historically, we haven’t expected all
students to graduate from high school
college- and career-ready

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
State standards reflect consensus about
what is desirable, not what is essential
Few states required Algebra II or its
equivalent for graduation
State tests measure 8th and 9th grade
knowledge and skills
High school accountability rarely focuses
on graduation rates or on college- and
work-readiness
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Arizona Mathematics
Graduation Requirement


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The State Board of Education raised the
state’s graduation requirements in 2007.
A default diploma was established.
The new requirements will affect the
incoming freshman class of 2009-2010
(graduating class of 2013).
“Math courses shall consist of Algebra I,
Geometry, Algebra II (or its equivalent)
and an additional course with substantial
math content as determined by districts or
charter schools.”
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Minimum Diploma by State
2005
15
Minimum Diploma by State
2008
As of October 2008
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Minimum Diploma by State
Type of Diploma
Number of States*
College and Career-Ready
Diploma
8
College and Career-Ready
Diploma with Opt-Out
13
General Diploma
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No State Requirements
4
* “States” includes the District of Columbia
As of October 2008
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State Mathematics
Requirements 2005
2
9
30
Algebra II
Algebra I
10
Geometry
Not Specified
18
State Mathematics
Requirements 2008
18
21
6
Algebra II
Algebra I
6
Geometry
Not Specified
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Mathematics Credits Required
for Graduation
Number of Mathematics
Units Required for
Graduation
Number of
States*
4 Units
19
3 Units
22
2 Units
6
0 Units/No Statewide
Requirements
4
* “States” includes the District of Columbia
As of October 2008
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Highest Mathematics Courses
Required
Highest Mathematics Course
(or its equivalent) Required
for Graduation
Number of
States*
Beyond Algebra II
2
Algebra II
19
Geometry
6
Algebra I
6
Not Specified
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* “States” includes the District of Columbia
As of October 2008
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The Majority of Graduates Would
Have Taken Harder Courses,
Particularly in Mathematics
Knowing what you know today about the expectations of college/work …
College students
Students who did not go to college
Would have taken more
challenging courses in at
least one area
Would have taken
more challenging
courses in:
Math
Science
62%
72%
34%
48%
32%
41%
English
29%
38%
Source: Peter D. Hart Research Associates/Public Opinion Strategies. (2005) Rising to the
Challenge: Are High School Graduates Prepared for College and Work? Washington, DC: Achieve.
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The Senior Year Is One Step Along
the Education Pipeline, Not an End
Point

Researchers who study learning and cognition
describe mathematical learning as a
progression in which conceptual understanding
builds logically, and expertise is developed
gradually.

Students not intending to pursue mathintensive majors should be able to select from
a number of fourth year “capstone” courses to
maintain and extend their prior mathematical
knowledge and connect mathematics
instruction with other interests.
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Criteria for High-Quality
Capstone Courses
Examples of Alternative Capstone
Courses (State Curricula)

Computer Mathematics (AR)

Data Analysis (CA)

Advanced Functions and Modeling (NC)

Discrete Mathematics (IN)

Probability and Statistics (IN)
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Modeling and Quantitative Reasoning (OH)

Advanced Mathematical Decision Making (TX)

Computer Mathematics (VA)

Discrete mathematics (VA)

Probability and Statistics (VA)
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Examples of Alternative Capstone
Courses (National/International
Curricula)

International Baccalaureate’s Mathematical Studies
Standard Level

Advanced Placement Computer Science A

Advanced Placement Statistics
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Criteria for High-Quality
Capstone Courses
I. Students should solidify and increase mathematical
knowledge and skills at and above the level of
Algebra II or its equivalent.
a. The course description indicates that opportunities will be
provided for students to reinforce and increase their fluency
with arithmetic and algebraic processes.
b. The course description indicates that opportunities will be
provided for continued experience with functions that include
linear, quadratic, and exponential and possibly extend to some
advanced functions, such as logarithmic, trigonometric,
higher-degree polynomial, or piecewise-defined functions.
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Criteria for High-Quality
Capstone Courses
I. Students should solidify and increase mathematical
knowledge and skills at and above the level of
Algebra II or its equivalent.
c.
The course description offers students new insight into
mathematics by including topics from non-traditional areas—
such areas might include finite or discrete mathematics,
statistical reasoning and inference, computer applications,
analytic geometry, non-Euclidean geometry, or geometric
probability.
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Criteria for High-Quality
Capstone Courses
II. Students should deepen and enrich the ways they
think about mathematics to elevate its study well
beyond rote memorization to a process of analysis
and interpretation that enables the learner to
grapple with a range of complex questions, topics,
and issues.
a. The course description provides opportunities for students to
think conceptually, in addition to procedurally, about
mathematics.
b. The course description requires students to justify approaches
to and results of problems with compelling mathematical
arguments and encourages the application of solid reasoning
in multiple contexts and across disciplines.
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Criteria for High-Quality
Capstone Courses
II. Students should deepen and enrich the ways they
think about mathematics to elevate its study well
beyond rote memorization to a process of analysis
and interpretation that enables the learner to
grapple with a range of complex questions, topics,
and issues.
c.
The course description provides students with opportunities to
deepen their understanding of mathematical reasoning and
supports the development of expertise in using the
appropriate type of reasoning for a given situation.
d. The course description encourages experimental thinking,
inquisitiveness, and evaluation of problem solving processes.
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Criteria for High-Quality
Capstone Courses
II. Students should deepen and enrich the ways they
think about mathematics to elevate its study well
beyond rote memorization to a process of analysis
and interpretation that enables the learner to
grapple with a range of complex questions, topics,
and issues.
e. The course description includes situations that engage
students in the use of abstraction and generalization.
f.
The course description includes opportunities to see
connections among the branches of mathematics.
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Criteria for High-Quality
Capstone Courses
II. Students should deepen and enrich the ways they
think about mathematics to elevate its study well
beyond rote memorization to a process of analysis
and interpretation that enables the learner to
grapple with a range of complex questions, topics,
and issues.
g. The course description provides opportunities for the effective
use of modern technologies, such as graphing and algebraic
calculators or software, data gathering probes, or computerassisted sampling.
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Criteria for High-Quality
Capstone Courses
III. Students should develop an appreciation for and
experience with a variety of applications of
mathematics across disciplines and in practical
situations.
a. The course description includes a focus on solving nonroutine problems that are of interest to students who do not
currently plan to follow a mathematics-intensive
postsecondary pathway in college or work.
b. The course description provides opportunities for students to
determine the key elements of a problem, translate them into
related mathematics, apply relevant mathematical strategies
to solve problems, and communicate results using
terminology that is understandable, correct, and appropriate
for the situation.
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Criteria for High-Quality
Capstone Courses
III. Students should develop an appreciation for and
experience with a variety of applications of
mathematics across disciplines and in practical
situations.
c.
The course description provides students with problems that
can be solved in more than one way and with the opportunity
to evaluate the effectiveness and efficiency of particular
solution methods.
d. The course description encourages student persistence when
solving problems, including those problems that require
extended time and/or the gathering of information for their
solutions.
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Criteria for High-Quality
Capstone Courses
III. Students should develop an appreciation for and
experience with a variety of applications of
mathematics across disciplines and in practical
situations.
e. The course description requires students to visually represent
a situation using mathematical diagrams or representations
and to apply common sense to evaluate the reasonableness
of solutions for practical problems in terms of the context.
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Criteria for High-Quality
Capstone Courses
The following rating system will be used to judge the
quality of a course curriculum:

0 = There is no basis in the course description for evaluation of
this criterion.

1 = Criterion is not suggested overtly but is implied in the
course description. OR = Criterion is partially suggested in the
course description.

2 = Criterion is clearly or strongly suggested by the course
description.
In a few cases there may be a need for a higher rating:

3 = Course description requirements satisfy and go beyond the
scope of this criterion reference.
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Capstone Course Rubric
Criteria for High-Quality Capstone Courses
Sample
Capstone #1
Course Title
Course Title
Course Title
Course Title
Rating system:
0 = There is no basis in the course description for evaluation of this criterion.
1 = Criterion is not suggested overtly but is implied in the course description OR Criterion is partially suggested by the course description
2 = Criterion is clearly or strongly suggested by the course description
In a few cases there may be a need for a higher rating:
3 = Course description requirements satisfy and go beyond the scope of this criterion reference
Directions to an evaluator: Enter the title of the course to be evaluated at the top of the column and the rating for each criterion in the
white cells of the column below the title. An average score will be computed automatically in the last cell of each column. An adequate
capstone course would rate somewhere around an average score of 1.5 and would ideally approach an average of 2. A course description
may rate lower than average for many reasons. Some course descriptions do not provide enough detail to make an evaluation in some
areas, and some intentionally have a narrower focus than others. A lower score should not be construed as an indicator of a sub-standard
course. Analysis of the scores of individual criterion is needed to make a judgment regarding a course's strengths and weaknesses. It is
not the intent of this tool to make a judgment regarding the overall worth of any course but rather to simply provide a system for
evaluating and comparing capstone courses relative to these criteria.
I. Students should solidify and increase mathematical knowledge and skills at and above the level of Algebra II or its equivalent.
A. The course description indicates that opportunities will be
provided for students to reinforce and increase their fluency with
arithmetic and algebraic processes.
2
B. The course description indicates that opportunities will be
provided for continued experience with functions that include
linear, quadratic, and exponential and possibly extend to some
advanced functions, such as logarithmic, trigonometric, higherdegree polynomial, or piecewise-defined functions.
2
C. . The course description offers students new insight into
mathematics by including topics from non-traditional areas–such
areas might include finite or discrete mathematics, statistical
reasoning and inference, computer applications, analytic
geometry, non-Euclidean geometry, or geometric probability.
1
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For More Information…
www.UTDanaCenter.org
Under “More
P-16 Projects”
select
“Mathematics
Benchmarks,
Grades K–12”.
Then select
“Supporting
Resources”.
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www.UTDanaCenter.org
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More Information
For more information on Achieve,
please visit Achieve, Inc., on the Web at
http://www.achieve.org
For more information on Fourth Year Capstone
Courses,
please visit The Charles A. Dana Center on the
Web at
www.UTDanaCenter.org
Contact Achieve: Tracy Halka
thalka@achieve.org
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Questions???