Elementary Mathematics from an Advanced Standpoint Appendix C

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

Both computers and calculators are used to develop mathematical ideas, solve problems, and to study the structure of mathematics.

Facilitating the building of student conceptual and procedural understanding.

Homework, tests, and labs using computers, models, and manipulatives focus on conceptual understanding and how/why procedures/algorithms work. Students are assessed on their understanding and ability to model the use of manipulatives both in lab/homework and on tests.

Manipulatives such as algebra tiles, geoboards, 3-d polyhedra, and 2-d tiles are used to model mathematical concepts and principles. Both real and virtual manipulatives are utilized in labs and homework.

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.

The focus of the course is building understanding of mathematics and is demonstrated through labs in class, in the computer lab, homework, tests, and projects. Applications, especially through projects and labs build confidence and awareness of the usefulness of mathematics.

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

Through labs, homework, and tests, students demonstrate their knowledge of heuristics, and labs provide an environment for exploring, conjecturing, and solving problems. Problem solving is utilized throughout the semester.

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.

As pre-service teachers, students create mathematical tasks/problems for future students to solve both in lab and on written assessments (homework, tests) Students also share their problems and solutions in class and compare strategies.

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

Proofs without words, using models and diagrams in labs, homework, and tests. Inductive and deductive arguments are developed when focusing on patterns (specific to general) shared in small group discussion, homework and tests.

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

Student questions/responses in class discussion and lab setting. Student responses demonstrating correct use of appropriate mathematical language, symbolism, and notation on homework, labs, projects, and tests.

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

Students ability to solve practical problems using technology and paper and pencil methods on labs, homework, and tests. Completion (and demonstration) of a technology integrated project connecting mathematics and other disciplines and daily life.

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

Students share their representation methods used in solving problems and discuss these in a lab setting. Discussing advantages and disadvantages of these approaches.

Mathematical processes including:

. Problem solving.

. Communication.

. Reasoning and formal and informal argument.

. Mathematical connections.

. Representations.

. Technology.

As a standards based math curriculum, students exhibit their abilities related to problem solving, communication, reasoning, representations, connections, and use of technology through discussion, labs, projects, homework, and tests.

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.

Number and operations, finite number systems, and computational procedures are a focus of this course. Students demonstrate their knowledge in class discussion, through homework, labs, projects, and tests.

Composition and decomposition of numbers, as applied to number theory, and sequences are assessed through homework, labs, and tests.

Computational procedures related to iteration and recursion are assessed through homework, labs, and tests.

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).

Approximation techniques as a basis for numerical integration, fractals, and numerical-based proofs are assessed through homework, labs, and tests.

Estimation and approximation techniques are used to check the reasonableness of results, especially in problem solving situations in the homework and in the lab.

Calculators and computers are used to carry out complicated computations.

Situations in which numerical arguments presented in a variety of situations and representations are discussed and compared in class, homework, and lab.

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.

Informal argument, proofs without words, through models and diagrams are used by the student in lab, homework, and tests.

Properties and relationships of two- and three -dimensional objects, transformations of these objects, and representation using coordinate geometry systems to discuss the relations between coordinate and synthetic geometry are assessed through labs using models such as geoboards, tiles, as well as the use of computer software, homework, and tests.

Transformations of two- and three- dimensional objects and the ways in which rotation, reflection, and translation of shapes can illustrate concepts, properties, and relationships are assessed on homework, labs, and tests.

Concepts of measurement and estimation, including measuring attributes, standard and non-standard units to length, perimeter, area, and circumference are assessed through discussion in class lab, use of computer software, homework, and tests.

Understanding and application of the Pythagorean Theorem is assessed using homework and exams.

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.

Euclidean and transformational geometry concepts, procedures, and connections are assessed with two and three - dimensional objects in class discussion, lab, through the use of computer software such as Logo, Kaleidomania, and Geometer's Sketchpad, homework, projects, and tests.

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 are given in class, lab, through the use of computer software, homework, and in tests.

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.

Use of data to explore real-world issues is assessed through in- class labs and homework.

The process of investigation including formulation of a problem, collecting, recording, and organizing data is assessed through labs, homework, and tests.

Data representation through graphs and tables are assessed through labs, homework, and exams.

Randomness, sampling, probability as a way to describe chances or risk, and outcome analysis and prediction based on experimentation or theoretical probabilities are assessed through labs, through the use of software, and homework, projects, and tests.

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.

Descriptive statistics and situations where students analyze, evaluate, and critique methods and conclusions based on probability experiments are used in labs, homework, and projects.

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 and algebra from both a concrete and abstract

perspective are used in the study of patterns, relations, representatives of real world situations.

Multiple representations and models of relations and the

advantages/disadvantages are discussed in class and lab.

Attributes of polynomial, algebraic, geometric/exponential functions are a focus of the sequences and series unit of study and are assessed by labs, homework, and tests.

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.

Concrete and virtual models are used to solve problems, demonstrate mathematical concepts and procedures.

These models are demonstrated and discussed by students in lab using manipulatives and computer software, as well as being assessed in homework, and tests.

Sequences and series, and numerical iteration are assessed through labs, homework, and tests.

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.

Basic counting techniques, study of sequences and analysis of patterns are used in solving problems and related to function analysis. Labs, homework, and tests are used to assess these ideas.

Iteration and recursion are important units of this course and are assessed by homework, labs, and on tests.

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 develop and analyze algorithms for probability

applications and to study geometric principles. These are

assessed using labs, homework, and exams.

Students develop and analyze algorithms for computer solution using software such as logo and are assessed by labs, homework, and on tests