Examples of Alternative Capstone Courses
Below is an illustrative, though not exhaustive, list of some alternative capstone courses that are being used in various states, districts, and schools around the United States. All of the courses included here scored within an acceptable range (1.5 to 2.0) when reviewed using our 15 criteria. Those offering the widest variety of topics and most extensive descriptions fared best in this review. However, those few with narrower scopes or less detail in their descriptions still show promise as high-interest courses for fourth-year high school students who plan to study or work in fields that are not necessarily mathematically oriented. The courses that did not make this list tended to be repeated overviews of a typical Algebra I and II curriculum. While such review may be seen as necessary, it is advised that the review come in a instructional form different from the one already experienced by the student. These students crave relevance in their mathematical studies and are more likely to thrive in a course that applies to their lives and interests.
International Baccalaureate’s Mathematical Studies Standard Level (available online to IB school personnel)
This international curriculum includes a strong review of algebraic processes and provides an opportunity for students to statistically investigate a topic of their choosing in depth. It is targeted at students with varied backgrounds and abilities. More specifically, it is designed to build confidence and encourage an appreciation of mathematics in students who do not anticipate a need for mathematics in their future studies. Students taking this course need to be already equipped with fundamental skills and a rudimentary knowledge of basic processes. The course concentrates on mathematics that can be applied to contexts related to other subjects being studied and common real-world occurrences. Students must produce a project—a piece of written work based on personal research, guided and supervised by the teacher. The project provides an opportunity for students to carry out a mathematical investigation in the context of another course being studied or another interest, using skills learned before and during the course.
The students most likely to select this course are those whose main interests lie outside the field of mathematics, and for many students this course will be their final experience of being taught formal mathematics. All parts of the syllabus have therefore been carefully selected to ensure that an approach starting with first principles can be used. As a consequence, students can use their inherent logical thinking skills and do not need to rely on standard algorithms and remembered formulas. Because of the nature of this course, teachers may find that less formal, shared learning techniques can be more stimulating and rewarding for students. Lessons that use an inquiry-based approach, starting with practical investigations where possible, followed by analysis of results and leading to the understanding of a mathematical principle and its formulation into mathematical language are often most successful in engaging the interest of students. Furthermore, this type of approach is likely to assist students in their understanding of mathematics by providing a meaningful context and by leading them to understand more fully how to structure their work for the project.
Advanced Placement Computer Science A
AP Computer Science A builds upon a foundation of mathematical reasoning that should be acquired before attempting the course. Students should be able to design and implement computer-based solutions to problems in a variety of application areas, implement commonly used algorithms and data structures, and develop and select appropriate algorithms and data structures to solve problems. The necessary prerequisites for entering AP Computer Science A include knowledge of basic algebra and experience in problem solving. A student in the course should be comfortable with functions and the concepts found in the uses of functional notation.
A large part of this course is built around the development of computer programs that correctly solve a given problem. These programs should be understandable, adaptable, and, when appropriate, reusable. At the same time, the design and implementation of computer programs is used as a context for introducing other important aspects of computer science, including the development and analysis of algorithms, the development and use of fundamental data structures, the study of standard algorithms and typical applications, and the use of logic and formal methods.
The goals of an AP course in computer science are comparable to those in the introductory sequence of courses for computer science majors offered in college and university computer science departments. It is not expected or intended, however, that all students in the AP Computer Science A course will major in computer science at the university level. AP Computer Science A is intended to serve both as an introductory course for computer science majors and as a course for people who will major in other disciplines that require significant involvement with technology. It is not a substitute for the usual college-preparatory mathematics courses.
Advanced Placement Statistics
The Advanced Placement Program offers a course and exam in statistics to secondary school students who wish to complete studies equivalent to a one-semester, introductory, non-calculus-based college course in statistics concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: Exploring Data (describing patterns and departures from patterns), Sampling and Experimentation (planning and conducting a study), Anticipating Patterns (exploring random phenomena using probability and simulation), and Statistical Inference (estimating population parameters and testing hypotheses).
In this course, students are expected to construct their own knowledge by working individually or in small groups to plan and perform data collection and analyses while the teacher serves in the role of a consultant, rather than as a director. This approach gives students ample opportunities to think through problems, make decisions, and share questions and conclusions with other students as well as with the teacher. Important components of the course include, as part of the concept-oriented instruction and assessment, the use of technology, projects and laboratories, cooperative group problem solving, and writing. This approach to teaching AP Statistics will allow students to build interdisciplinary connections with other subjects and with their world outside school. Students who successfully complete the course and exam may receive credit for a one-semester introductory college statistics course.
The AP Statistics course depends heavily on the availability of technology suitable for the interactive, investigative aspects of data analysis. Therefore, schools should make every effort to provide students and teachers easy access to graphing calculators and/or computers to facilitate the teaching and learning of statistics.
Advanced Functions and Modeling (NC)
Content for this course consists of situations that generate data, which are then modeled using elementary (linear, quadratic, exponential) or more advanced (polynomial, logarithmic, trigonometric) functions and their transformations. Students are expected to evaluate the goodness of fit of such models and use them to make predictions, understanding the power and limitations of such predictions. This curriculum offers a review of key functions of algebra and encourages the use of technology. Non-traditional or extended topics are included. Prerequisites for the course include content typically found in courses in Geometry and Algebra II or their equivalents.
Advanced Quantitative Decision-Making (TX) (proposed)
Emphasizing practical and engaging problem solving through the use of technology in the context of functions and their applications, this course was designed to provide an appropriate fourth-year mathematics option for students planning to enter the workforce or a training program after high school or for students not intending to pursue a mathematics-intensive field in college. It may also provide a rich mathematics experience as an elective course for students following a pre-calculus/calculus path. Course topics are drawn from the areas of descriptive statistics, financial/economic literacy, and basic trigonometry. Students are expected to make informed decisions that make use of the underlying mathematics or statistics. A particular goal of the course is that students will formulate and solve problems and verify solutions in multiple ways, connecting mathematics to other disciplines and connecting mathematical topics with each other. Prerequisites include fluency in subject matter drawn from the Texas Essential Knowledge and Skills (TEKS) for Algebra I and Geometry, completion of Algebra II, and experience with mathematics drawn from authentic contexts.
Discrete mathematics (VA)
This course involves applications using discrete variables rather than continuous variables. It relies on modeling and understanding finite systems that are central to the development of the economy, the natural and physical sciences, and mathematics itself, to introduce the topics of social choice as a mathematical application, matrices and their uses, graph theory and its applications, and counting and finite probability. In addition, the processes of optimization, existence, and algorithm construction mathematics content develop sequentially in concert with a set of processes that are common to different bodies of mathematics knowledge. Students become mathematical problem solvers, communicate mathematically, reason mathematically, make mathematical connections, and use mathematical representations to model and interpret practical situations. Teachers help students make connections and build relationships among algebra, arithmetic, geometry, discrete mathematics, and probability and statistics. Connections can be made to other subject areas and fields of endeavor through applications. Using manipulatives, graphing calculators, and computer applications to develop concepts allows students to develop and attach meaning to abstract ideas. Topics include modeling and solving problems using vertex-edge graphs, problems in election theory and fair division, computer mathematics (including various sorting methods, coding systems, and Boolean logic), and recursion and optimization.
Probability and Statistics (VA)
This course presents basic concepts and techniques for collecting and analyzing data, drawing conclusions, and making predictions. Applications may be drawn from a wide variety of disciplines ranging from the social sciences of psychology and sociology to education, health fields, business, economics, engineering, the humanities, the physical sciences, journalism, communications, and liberal arts. Students should be able to design an experiment, collect appropriate data, select and use statistical techniques to analyze the data, and develop and evaluate inferences based on the data. The mathematics content develops sequentially in concert with a set of processes that are common to different bodies of mathematics knowledge. Connections can be made to other subject areas and fields of endeavor through applications. Use of technology is imperative and should help students develop and attach meaning to abstract ideas. Students are encouraged to talk about mathematics, use the language and symbols of mathematics, communicate, discuss problems and problem solving, and develop competence and confidence in their mathematics skills. Content is drawn from the areas of descriptive statistics, data collection, probability, and inferential statistics.
Computer Mathematics (VA)
This course provides students with experiences in using the computer to solve problems that can be set up as mathematical models. Students will develop and refine skills in logic, organization, and precise expression, thereby enhancing learning in other disciplines. Programming is introduced in the context of mathematical concepts and problem solving. Students will define a problem; develop, refine, and implement a plan; and test and revise the solution. Content includes applications in number sense, algebraic functions, graphics, and coordinate geometry. Teachers should help students make connections and build relationships among algebra, arithmetic, geometry, discrete mathematics, and probability and statistics. Connections can be made to other subject areas and fields of endeavor through applications. Throughout the study of mathematics, students should be encouraged to talk about mathematics, use the language and symbols of mathematics, communicate, discuss problems and problem solving, and develop their competence and confidence in their mathematics skills.
Discrete Mathematics (IN) (pdf 108kb)
This course consists of units in counting techniques, matrices, recursion, graph theory, social choice, linear programming, and game theory. Mathematical reasoning and problem solving are emphasized along with connections to other branches of mathematics and other disciplines. The course develops students’ ability to read, write, listen, ask questions, think, and communicate about math to deepen their understanding of mathematical concepts. Proper communication using the language of mathematics—its terminology, symbols, formulas, graphics, displays, etc.—is a dynamic tool for solving problems and communicating and expressing mathematical ideas
Probability and Statistics (IN) (pdf 100kb)
This course consists of units in descriptive statistics, probability, and statistical inference. Woven throughout Indiana’s mathematics standards are the following processes: mathematical reasoning and problem solving, communication, representation, and connections. The ability to read, write, listen, ask questions, think, and communicate about math develops and deepens students’ understanding of mathematical concepts. Students are expected to read text, data, tables, and graphs with comprehension and understanding. Their writing should be detailed and coherent, and they should use correct mathematical vocabulary. Students are expected to explain answers, justify mathematical reasoning, and describe problem solving strategies.
Modeling and Quantitative Reasoning (OH)
This course prepares students to investigate contemporary issues mathematically and apply the mathematics learned in earlier courses to answer questions that are relevant to their civic and personal lives. It reinforces student understanding of percentages, functions and their graphs, probability and statistics, and multiple representations of data and data analysis. It also introduces functions of two variables and graphs in three dimensions. The applications provide an opportunity for students to gain a deeper understanding and expand upon material from previous math instruction. This course also shows the connections between the branches of mathematics and the various applications of mathematics. Today, numbers and data are critical parts of public and private decision making. Decisions about health care, finances, science policy, and the environment are decisions that require citizens to understand information presented in numerical form and in tables, diagrams, and graphs. Students must develop skills to analyze complex issues using quantitative tools. In addition to a textbook, teachers are advised to use online resources, newspapers, and magazines to identify problems that are appropriate for the course. Students should be encouraged to find issues that can be represented in a quantitative way and shape them for investigation. Appropriate use of available technology is essential as students explore quantitative ways of representing and presenting the results of their investigations.
Computer Mathematics (AR) (102 kb)
This course provides students with experiences in using the computer to solve problems that can be set up as mathematical models. Students entering this course must be currently enrolled in or have already taken Algebra II. Students will develop and refine skills in logic, organization, and precise expression, thereby enhancing learning in other disciplines. Programming is introduced in the context of mathematical concepts and problem solving. Students will define a problem; develop, refine, and implement a plan; and test and revise the solution. It is recommended that class size be no larger than 20 students because of the hands-on projects and activity nature of the course. Teachers help students make connections to other subject areas and fields of endeavor through applications. Use of manipulatives, graphing calculators, and computer spread sheet applications helps students develop and attach meaning to abstract ideas.
Data Analysis (CA)
This course covers the basic principles of descriptive statistics, exploratory data analysis, design of experiments, sampling distributions and estimation, and fitting models to data. Statistical concepts are studied in order to understand related methods and their applications. Other topics include probability distributions, sampling techniques, binomial distributions, and experimental design. The course also looks extensively at the principles of hypothesis testing and statistical inference. Measuring the probability of an event, interpreting probability, and using probability in decision making are central themes of this course. Examples from games of chance, business, medicine, policymaking, the natural and social sciences, and sports will be explored. Use of computers and graphing calculators exposes students to the power and simplicity of statistical software for data analysis. The Texas Instruments TI-83+ is used extensively as a learning tool and is required for the course. Prerequisite: full year of Algebra II with passing grade.