Modern Algebra 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.

Students use the axiomatic structure to prove results about algebraic structures. Students investigate topics of historical significance (e.g. the trisection of the angle and solvability of polynomials) using modern algebra tools. Students use computer tools to investigate geometric properties of algebraic systems including symmetry groups, linear transformations, and complex solutions to polynomials. *

Facilitating the building of student conceptual and procedural understanding.

Students show understanding of modern algebra topics from algebraic and geometric perspectives, by investigating concrete examples and presenting abstract arguments. *

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.

Students integrate their knowledge from previous experiences in Algebra into the abstract structures they are learning, gaining confidence in their ability to utilize mathematical knowledge. Students investigate applications of modern algebra to other disciplines and contexts. *

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

Students explore mathematics, and make and test conjectures in investigating properties of algebraic structures (e.g. determining which classes of permutations ar of which rank), and explicitly make, test and present a conjecture in modern algebra in their final project. *

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.

Students pose worthwhile mathematical tasks both for themselves and for a high school level student as part of their final project. Students solve problems in general as well as specific contexts (e.g. determine which groups or have the given properties). *

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

Students write mathematical arguments and construct counterexamples on modern algebra topics including verifying properties of polynomials and algebraic systems. Students frame mathematical questions or conjectures as part of their final project. Students write formal proofs as well as engaging in investigative activities. *

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

Students present mathematical arguments about algebraic structures, and applications of modern algebra orally, visually and in writing, using mathematical language and symbolism. Students translate between geometric, concrete algebraic, and abstract algebraic contexts throughout the course. Specific examples of geometric and algebraic representations arise in the study of symmetry groups, low dimensional linear algebra, and the Fundamental Theorem of Algebra. Specific examples of concrete and abstract representations arise in the study of group, field and vector space structures. *

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

Students demonstrate knowledge of connections between geometry and algebra in their work with low dimensional vector spaces and symmetry groups. Students demonstrate knowledge of connections to other disciplines and daily life in their work with applications of modern algebra. In their work with algebraic structures students investigate the conceptual properties that define these structures and apply these in making arguments about properties and procedures in concrete examples and generalizations. *

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

Students solve problems that involve choosing and using appropriate notation and representations for specific groups, fields and vector spaces. Students show that they can translate between representations in their work with symmetry and permutation groups, linear transformations, and the fundamental theorem of algebra. *

Mathematical processes including:

. Problem solving.

. Communication.

. Reasoning and formal and informal argument.

. Mathematical connections.

. Representations.

. Technology.

Students solve problems that require them to relate concrete examples to a generalized context and axiom structure. Students make formal arguments about modern algebra questions. Students communicate their results mathematically; making use of algebraic, geometric and generalized representations. Students use computer tools to investigate some elements of modern algebra (see above). *

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 properties of numbers such as primes and factorization in their work on modular numbers, extension fields and polynomials. Students investigate, discuss and define number systems in terms of their operation properties including closure, inverses, commutativity and assosciativity. Students investigate and solve problems about number systems including Integer, Rational and Real numbers in terms of their properties and relations. *

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).

Students use and identify algebraic numbers. Students investigate polynomial solutions in the complex number system. *

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.

Students use linear algebra techniques to analyze lines, planes and spaces from a vector standpoint, including angle measure relationships and perpendicularity. Students study rotations, reflections and dilations in the context of linear transformations, and translate between the geometric and matrix form of these transformations. Students use vector-defined coordinate systems in two and three dimensions, and more generally in higher dimensions. Students use the geometry of the complex number system to understand complex solutions to polynomial equations. Students demonstrate knowledge of the algebraic structures relating transformations including rotations and reflections. Students make formal and informal mathematical arguments about the algebraic structures relating to symmetries of geometric objects. *

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.

Students use coordinate geometry from a vector perspective. Students apply, analyze and describe linear transformations, coordinates and coordinate systems and vectors. Students solve problems and write mathematical arguments about transformations, coordinates and coordinate systems and vectors. Students create and share mathematical arguments, proof s and counterexamples in the context of vector spaces and linear transformations. *

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 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 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.

Students use functions and algebraic structures to model situations in the study of applications of linear algebra. Students demonstrate knowledge of linear transformations and their attributes as functions on vector spaces. Students translate between standard and vector forms of equations for lines and planes. Students solve and manipulate systems of linear equations using matrix methods. Students demonstrate understanding of algebraic attributes of polynomial and rational functions, including solvability and divisibility. *

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.

Not assessed in this course.

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.

Students do introductory work on iterations in working on higher powers of matrices. Students make limited use of graphs or networks in their work on linear algebra applications. *

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.

Students demonstrate knowledge of matrices computationally and in vector space contexts. *