Geometry for Educators Appendix C

The structures within the discipline, the historical roots and evolving nature of mathematics, and the interaction between technology and the discipline.

Teachers' knowledge of axiomatic structures assessed through exam items; students useGeometer's Sketchpad to explore geometric ideas, and are assessed via exams, assignments, projects and presentations.

Facilitating the building of student conceptual and procedural understanding.

Students explore relationships between Euclidean geometry and Finite, Spherical, Hyperbolic, and Taxicab geometries, and conceptual understanding is assessed on exams, assignments, projects, and presentations

Helping all students build understanding of the discipline including:

. Confidence in their abilities to utilize mathematical knowledge.

. Awareness of the usefulness of mathematics.

. The economic implications of fine mathematical preparation.

Confidence levels in dealing with a variety of axiomatic systems are assessed primarily through presentations and comments made in class and to individual instructors. Awareness of usefulness of geometry is assessed in presentations and class discussions.

Exploring, conjecturing, examining and testing all aspects of problem solving.

Exploring and testing conjectures is an integral part of the instruction of this course; student ability to explore geometric ideas in a problem solving manner is assessed during class discussions, on homework, and in written projects and oral presentations.

Formulating and posing worthwhile mathematical tasks, solving problems using several strategies, evaluating results, generalizing solutions, using problem solving approaches effectively, and applying mathematical modeling to real-world situations.

Teachers' problem solving abilities are assessed during class laboratory activities, on exams, in homework, and in projects and presentations.

Making convincing mathematical arguments, framing mathematical questions and conjectures, formulating counter-examples, constructing and evaluating arguments, and using intuitive, informal exploration and formal proof.

Teachers' ability to provide convincing informal and formal arguments based on appropriate geometric axiomatic systems are an integral part of the course, and are assessed during class discussions, homework, exams, projects, and presentations.

Expressing ideas orally, in writing, and visually-, using mathematical language, notation, and symbolism; translating mathematical ideas between and among contexts.

Mathematical communication skills are assessed during class discussions, on homework assignments and exams, on written projects and oral presentations.

Connecting the concepts and procedures of mathematics, drawing connections between mathematical strands, between mathematics and other disciplines, and with daily life.

Projects provide the main opportunity for teachers to show their understanding of connections between geometry and real life. The concepts and procedures of non-Euclidean and Euclidean geometries are connected within and between topics through an emphasis on which axioms hold in all systems, and which are unique to a system; teachers' understanding is assessed through exams, homework, and projects and presentations.

Selecting appropriate representations to facilitate mathematical problem solving and translating between and among representations to explicate problem-solving situations.

Different geometries use different representations depending on their axiomatic systems and the real life problems that are being solved; teachers' abilities to choose appropriate representations and to move back and forth between representations are assessed on homework, exams, in class discussions, and in projects and presentations.

Mathematical processes including:

. Problem solving.

. Communication.

. Reasoning and formal and informal argument.

. Mathematical connections.

. Representations.

. Technology.

In their study of every type of geometry in this course (Euclidean, finite, Spherical, Hyperbolic, and Taxicab), teachers' apply each and every one of these processes, as noted in previous sections. All are assessed through class discussions, homework, exams, projects, and presentations.

Number operations and relationships from both abstract and concrete perspectives identifying real world applications, and representing and connecting mathematical concepts and procedures including:

. Number sense.

. Set theory.

. Number and operation.

. Composition and decomposition of numbers, including place value, primes, factors, multiples, inverses, and the extension of these concepts throughout mathematics.

. Number systems through the real numbers, their properties and relations.

. Computational procedures.

. Proportional reasoning.

. Number theory.

Students use numbers, number systems, and operations as needed in this geometry course, but the course is not focused on the teaching of number systems, and so, they are not assessed per se.

Mathematical concepts and procedures, and the connections among them for teaching upper level number operations and relationships including:

. Advanced counting procedures, including union and intersection of sets, and parenthetical operations.

. Algebraic and transcendental numbers.

. The complex number system, including polar coordinates.

. Approximation techniques as a basis for numerical integration, fractals, and numerical-based proofs.

. Situations in which numerical arguments presented in a variety of classroom and real-world situations (e.g., political, economic, scientific, social) can be created and critically evaluated.

. Opportunities in which acceptable limits of error can be assessed (e.g., evaluating strategies, testing the reasonableness of results, and using technology to carry out computations).

Number systems are used as needed in the study of geometry, but are not assessed directly in this course.

Geometry and measurement from both abstract and concrete perspectives and to identify real world applications, and mathematical concepts, procedures and connections among them including:

. Formal and informal argument.

. Names, properties, and relationships of two- and three-dimensional shapes.

. Spatial sense.

. Spatial reasoning and the use of geometric models to represent, visualize, and solve problems.

. Transformations and the ways in which rotation, reflection, and translation of shapes can illustrate concepts, properties, and relationships.

. Coordinate geometry systems including relations between coordinate and synthetic geometry, and generalizing geometric principles from a two-dimensional system to a three-dimensional system.

. Concepts of measurement, including measurable attributes, standard and non-standard units, precision and accuracy, and use of appropriate tools.

. The structure of systems of measurement, including the development and use of measurement systems and the relationships among different systems. Measurement including length, area, volume, size of angles, weight and mass, time, temperature, and money.

. Measuring, estimating, and using measurement to describe and compare geometric phenomena.

. Indirect measurement and its uses, including developing formulas and procedures for determining measure to solve problems.

All of these content areas are prerequisite knowledges for the current course, but they are not a focus of the course, since they appear to reflect lower level geometric knowledge and this is a graduate level course; consequently, they are not formally assessed in this course per se, although teachers cannot demonstrate their complete understanding of this courses' topics without having a certain amount of proficiency with these lower level topics.

There are two exceptions to the above statement; the first occurs in the area of spatial reasoning and the use of geometric models to represent, visualize and solve problems. In the study of spherical geometry, teachers use Lenart spheres and spherical compasses to visualize spherical geometry concepts and solve problems. They are assessed on their use of these models and their problem solving ability in spherical geometry on homework, exams, projects, and presentations.

The other exception is in the area of using measurement to compare geometric phenomena. Angle sizes and sums of angles in various spherical and hyperbolic polygons are compared to results from Euclidean geometry. Teachers knowledge is assessed on homework, exams, and projects.

Mathematical concepts, procedures, and the connections among them for teaching upper level geometry and measurement including:

. Systems of geometry, including Euclidean, non-Euclidean, coordinate, transformational, and projective geometry.

. Transformations, coordinates, and vectors and their use in problem solving. Three-dimensional geometry and its generalization to other dimensions. Topology, including topological properties and transformations.

. Opportunities to present convincing arguments by means of demonstration, informal proof, counter-examples, or other logical means to show the truth of statements and/or generalizations.

Systems of geometry (finite, Euclidean, Spherical, Hyperbolic and Taxicab), transformational geometry, and the development of the ability to use convincing arguments and proof, are the main components of this course.

Teachers' ability to analyze systems, solve problems, use technology, and make and prove hypotheses are assessed throughout the course activities: class discussions, homework, exams, projects, and presentations.

Statistics and probability from both abstract and concrete perspectives and to identify real world applications, and the mathematical concepts, procedures and the connections between them including:

. Use of data to explore real-world issues.

. The process of investigation including formulation of a problem, designing a data collection plan, and collecting, recording, and organizing data.

. Data representation through graphs, tables, and summary statistics to describe data distributions, central tendency, and variance.

. Analysis and interpretation of data.

. Randomness, sampling, and inference.

. Probability as a way to describe chances or risk in simple and compound events.

. Outcome prediction based on experimentation or theoretical probabilities.

Not covered or assessed in this course.

Mathematical concepts, procedures, and the connections among them for teaching upper level statistics and probability including:

. Use of the random variable in the generation and interpretation of probability distributions.

. Descriptive and inferential statistics, measures of disbursement, including validity and reliability, and correlation.

. Probability theory and its link to inferential statistics.

. Discrete and continuous probability distributions as bases for inference.

. Situations in which students can analyze, evaluate, and critique the methods and conclusions of statistical experiments reported in journals, magazines, news media, advertising, etc.

Not covered or assessed in this course.

Functions, algebra, and basic concepts underlying calculus from both abstract and concrete perspectives and to identify real world applications, and the mathematical concepts, procedures and the connections among them including:

. Patterns.

. Functions as used to describe relations and to model real world situations.

. Representations of situations that involve variable quantities with expressions, equations and inequalities and that include algebraic and geometric relationships.

. Multiple representations of relations, the strengths and limitations of each representation, and conversion from one representation to another.

. Attributes of polynomial, rational, trigonometric, algebraic, and exponential functions.

. Operations on expressions and solution of equations, systems of equations and inequalities using concrete, informal, and formal methods.

. Underlying concepts of calculus, including rate of change, limits, and approximations for irregular areas.

Functions, algebra, and calculus are not a focus of this course; some aspects of these areas are incorporated as needed into teachers' projects, but they are not assessed per se.

One notable exception is in the area of trigonometric functions and their definition and applications to spherical geometry. Teachers compare and contrast Euclidean trigonometry with spherical trigonometry, and are assessed on their knowledge and understanding via homework, exams, projects, and presentations.

Mathematical concepts, procedures, and the connections among them for teaching upper level functions, algebra, and concepts of calculus including:

. Concepts of calculus, including limits (epsilon-delta) and tangents, derivatives, integrals, and sequences and series.

. Modeling to solve problems.

. Calculus techniques including finding limits, derivatives, integrals, and using special rules.

. Calculus applications including modeling, optimization, velocity and acceleration, area, volume, and center of mass.

. Numerical and approximation techniques including Simpson's rule, trapezoidal rule, Newton's Approximation, and linearization.

. Multivariate calculus.

. Differential equations.

Functions, algebra, and calculus are not a focus of this course; some aspects of these areas are incorporated as needed into teachers' projects, but they are not assessed per se.

Discrete processes from both abstract and concrete perspectives and to identify real world applications, and the mathematical concepts, procedures and the connections among them including:

. Counting techniques.

. Representation and analysis of discrete mathematics problems using sequences, graph theory, arrays, and networks.

. Iteration and recursion.

The processes of discrete mathematics are not a focus of this course; some aspects of these areas are incorporated as needed into teachers' projects, but they are not assessed per se.

Mathematical concepts, procedures, and the connections among them for teaching upper level discrete mathematics including:

. Topics, including symbolic logic, induction, linear programming, and finite graphs.

. Matrices as a mathematical system, and matrices and matrix operations as tools for recording information and for solving problems.

. Developing and analyzing algorithms.

The processes of discrete mathematics are not a focus of this course; some aspects of these areas are incorporated as needed into teachers' projects, but they are not assessed per se.