UNIVERSITY OF WISCONSIN River Falls

# Mathematical Statistics Appendix C

All professional education content courses leading to certification shall include teaching and assessment of the Wisconsin Content Standards in the content area.

In this column, list the Wisconsin Content Standards that are included in this course. The Standards for each content area are found in the Wisconsin Content Standards document.

In this column, indicate the nature of the performance assessments used in this course to evaluate student proficiency in each standard.

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

Examples appear in class, on tests, and in assignments that show the use of technology in simulating random processes; a description is given of the increasing sophistication of the tests used to make statistical inference.

Facilitating the building of student conceptual and procedural understanding.

In class-responses gauge the development of student intuition for the major issues that arise in a given situation involving statistical inference; the execution of the statistical tests that are developed in class demonstrate a procedural understanding on the part of students.

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 problems on homework assignments and exams require students to integrate a rich background of material from Calculus and Discrete Mathematics. Many "real-world" problems can be rephrased as problems that require the application of a statistical test. These tests are used throughout industry.

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

A strong emphasis is placed on problem solving in this course. The use of computers to simulate random processes yield important opportunities to carry out the multiple stages of active problem solving.

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.

In Mathematical Statistics, as in Probability more generally, problems oftentimes admit a "rigorous" solution through the use of Calculus as well as a heuristic solution obtained through the use of probabilistic "intuition." Most of the statistic theory developed in this course arose from the need to make statistical inferences in situations that arise quite naturally from modeling certain real-world phenomena.

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

Certain examples allow students the opportunity to develop their own (mathematical) statistics that estimate parameters of some kind, with the goal of achieving sufficiency, consistency, etc.

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

Students regular submit homework assignments are take exams; in-class participation is strongly encouraged.

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

An important example of this concerns the identification of certain underlying assumptions that must be satisfied before a given statistical test can be performed or deemed to be valid.

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

Where possible, issues in Mathematical Statistics are translated into models that are encountered in Calculus and in Discrete Mathematics.

Mathematical processes including:

. Problem solving.

. Communication.

. Reasoning and formal and informal argument.

. Mathematical connections.

. Representations.

. Technology.

In class work and out-of-class assignments lead to the development of problem solving and communication on the part of the student. Problems such as the computation of bias require students to identify the need to use integration to perform an "averaging."

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.

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).

In Mathematical Statistics, the question of whether or not a test has been applied in a legitimate manner is of paramount importance, and is addressed frequently. The inappropriate use of a test leads to an invalid numerical argument.

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.

The expected values of various order statistics for uniform distributions can be discussed in terms of a uniform partition of the underlying interval for the distribution; these examples have a strong geometric flavor.

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.

An ambitious discussion of the rigorous proof of the independence of the sample mean and the sample variance for a random sample requires the use of linear algebra reasoning and concepts.

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.

The uses of randomness, sampling, and inference are basic to all that we do in this course. By pursuing a rigorous treatment of these concepts, their connections with purely mathematical concepts become very well established.

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.

The uses of random variables, descriptive and inferential statistics, and discrete probability theory are basic to all that we do in this course. By pursuing a rigorous treatment of these concepts, their connections with purely mathematical concepts become very well established.

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.

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.

Integrals are used almost daily to compute expected values; series are used to evaluate probabilities of events that are described in terms of discrete distributions

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.

Counting techniques arise in a natural way in the rigorous computation of the order statistics.

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.

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