Facing Choices about the Fourth Year in Arizona: Achieve’s Fourth-Year Capstone Course Criteria October 18, 2008 Institute for Mathematics & Education Tucson, Arizona

2 Who is Achieve, Inc.? n 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.

3 Presentation Overview n History of the American Diploma Project (ADP) n Mathematics Graduation Requirements n Criteria for High-Quality Capstone Courses n Contact Information n Questions

History of the American Diploma Project

5 American Diploma Project n The American Diploma Project (ADP) was created to ensure all graduates leave high school ready for college and careers. n Early research by ADP sought to identify “must-have” knowledge and skills graduates will need to be successful in college and careers.

6 American Diploma Project n 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. n Developed ADP benchmarks that include the core content and skills in mathematics and English all students should have when they graduate high school.

7 Key findings n In mathematics, graduates need strong computation skills, ability to solve challenging problems, reasoning skills, geometry, data analysis, statistics, and advanced algebra. n 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. n 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.

8 The ADP Benchmarks: Challenging content for all students In Mathematics: l A rigorous four-year course sequence l Content* equivalent to a sequence that includes Algebra I and II, Geometry, and Data Analysis & Statistics *can be taught via different pathways In English: l Four courses l Content equivalent to four years of grade- level English or higher with a strong focus on oral and written communication skills and considerable research and analysis To cover the content in the ADP benchmarks, high school graduates need:

9 Closing the Expectations Gap: ADP Policy Agenda n In 2005, Achieve launched the ADP Network, a group of states committed to taking four college and career readiness action steps: l Align high school standards with college and career expectations. l Require all students to take a college- and career- ready curriculum, aligned with standards, to earn a diploma. l Build “college-ready” measures, aligned with state standards, into high school assessment systems. l Hold high schools accountable for graduating students college- and career-ready, and hold postsecondary institutions accountable for student success.

10 ADP Network launched at 2005 Summit: 13 states committed to improving student preparation

11 ADP Network today: thirty-four states now committed to improving student preparation

Mathematics Graduation Requirements

13 An Expectations Gap n Historically, we haven’t expected all students to graduate from high school college- and career-ready l State standards reflect consensus about what is desirable, not what is essential l Few states required Algebra II or its equivalent for graduation l State tests measure 8 th and 9 th grade knowledge and skills l High school accountability rarely focuses on graduation rates or on college- and work-readiness

14 Arizona Mathematics Graduation Requirement n The State Board of Education raised the state’s graduation requirements in n A default diploma was established. n The new requirements will affect the incoming freshman class of (graduating class of 2013). n “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.”

15 Minimum Diploma by State 2005

16 Minimum Diploma by State 2008 As of October 2008

17 Minimum Diploma by State Type of DiplomaNumber of States* College and Career-Ready Diploma 8 College and Career-Ready Diploma with Opt-Out 13 General Diploma26 No State Requirements4 * “States” includes the District of Columbia As of October 2008

18 State Mathematics Requirements 2005

19 State Mathematics Requirements 2008

20 Mathematics Credits Required for Graduation Number of Mathematics Units Required for Graduation Number of States* 4 Units19 3 Units22 2 Units6 0 Units/No Statewide Requirements 4 * “States” includes the District of Columbia As of October 2008

21 Highest Mathematics Courses Required Highest Mathematics Course (or its equivalent) Required for Graduation Number of States* Beyond Algebra II2 Algebra II 19 Geometry 6 Algebra I 6 Not Specified18 * “States” includes the District of Columbia As of October 2008

22 The Majority of Graduates Would Have Taken Harder Courses, Particularly in Mathematics Knowing what you know today about the expectations of college/work … Would have taken more challenging courses in: Would have taken more challenging courses in at least one area Math Science English 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.

23 The Senior Year Is One Step Along the Education Pipeline, Not an End Point n Researchers who study learning and cognition describe mathematical learning as a progression in which conceptual understanding builds logically, and expertise is developed gradually. n Students not intending to pursue math- intensive 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.

Criteria for High-Quality Capstone Courses

25 Examples of Alternative Capstone Courses (State Curricula) n Computer Mathematics (AR) n Data Analysis (CA) n Advanced Functions and Modeling (NC) n Discrete Mathematics (IN) n Probability and Statistics (IN) n Modeling and Quantitative Reasoning (OH) n Advanced Mathematical Decision Making (TX) n Computer Mathematics (VA) n Discrete mathematics (VA) n Probability and Statistics (VA)

26 Examples of Alternative Capstone Courses (National/International Curricula) n International Baccalaureate’s Mathematical Studies Standard Level n Advanced Placement Computer Science A n Advanced Placement Statistics

27 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.

28 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.

29 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.

30 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.

31 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.

32 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 computer- assisted sampling.

33 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 non- routine 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.

34 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.

35 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.

36 Criteria for High-Quality Capstone Courses The following rating system will be used to judge the quality of a course curriculum: n 0 = There is no basis in the course description for evaluation of this criterion. n 1 = Criterion is not suggested overtly but is implied in the course description. OR = Criterion is partially suggested in the course description. n 2 = Criterion is clearly or strongly suggested by the course description. In a few cases there may be a need for a higher rating: n 3 = Course description requirements satisfy and go beyond the scope of this criterion reference.

37 Capstone Course Rubric Criteria for High-Quality Capstone Courses Sample Capstone #1 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, higher- degree 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

For More Information…

39 Under “More P-16 Projects” select “Mathematics Benchmarks, Grades K–12”. Then select “Supporting Resources”.

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41 For more information on Achieve, please visit Achieve, Inc., on the Web at For more information on Fourth Year Capstone Courses, please visit The Charles A. Dana Center on the Web at Contact Achieve: Tracy Halka More Information

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