8710.4600 TEACHERS OF MATHEMATICS                                                                                                   FORM I-C MATRIX

Professional Education Program Evaluation Report     (PEPER II)

MATRIX   Form I-C

8710.4600 Teachers of Mathematics

Identify coding used to indicate placement or assignment of standards here: K=Knowledge, A= Assessed

 

Math 120

Math 126

Math 220

Math 232

Math 244

Math 252

Math 262

Math 356

Stat 212/272

Ed 350

Subp. 3.  Subject matter standard.  A candidate for licensure as a teacher of mathematics must complete a preparation program under subpart 2, item C, that must include the candidate's demonstration of the knowledge and skills in items A to I.

A.  A teacher of mathematics understands patterns, relations, functions, algebra, and basic concepts underlying calculus from both concrete and abstract perspectives and is able to apply this understanding to represent and solve real world problems.  The teacher of mathematics must demonstrate knowledge of the following mathematical concepts and procedures and the connections among them: 

(1) recognize, describe, and generalize patterns and build mathematical models to describe situations, solve problems, and make predictions;

 

 

 

KA

 

 

 

 

 

 

(2) analyze the interaction between quantities and variables to model patterns of change and use appropriate representations including tables, graphs, matrices, words, ordered pairs, algebraic expressions, algebraic equations, and verbal descriptions;

KA

 

KA

 

 

 

 

 

 

 

(3) represent and solve problem situations that involve variable quantities and use appropriate technology;

KA

 

 

 

 

 

 

 

 

 

(4) understand patterns present in number systems and apply these patterns to further investigations;

 

 

 

KA

 

 

 

 

 

 

(5) apply properties of boundedness and limits to investigate problems with sequences and series;

 

 

 

 

KA

 

 

 

 

 

(6) apply concepts of derivatives to investigate problems involving rates of change;

KA

 

 

 

 

 

 

 

 

 

(7) apply concepts and standard mathematical representations from differential, integral, and multivariate calculus; linear algebra, including vectors and vector spaces; and transformational operations to solve problems; and

 

KA

KA

 

 

 

 

 

 

 

 

 

Math 120

Math 126

Math 220

Math 232

Math 244

Math 252

Math 262

Math 356

Stat 212/272

Ed 350

(8) apply properties of group and field structures to mathematical investigations.

 

 

 

 

 

KA

 

 

 

 

B.  A teacher of mathematics understands the discrete processes from both concrete and abstract perspectives and is able to identify real world applications; the differences between the mathematics of continuous and discrete phenomena; and the relationships involved when discrete models or processes are used to investigate continuous phenomena.  The teacher of mathematics must demonstrate knowledge of the following mathematical concepts and procedures and the connections among them:

(1) the application of discrete models to problem situations using appropriate representations such as sequences, vertex-edge graphs and trees, matrices, and arrays;

 

 

 

KA

 

 

 

 

 

 

(2) application of systematic counting techniques to problem situations including determination of the existence of a solution, the determination of the number of possible solutions, or the optimal solution;

 

 

 

KA

 

 

 

 

 

 

(3) application of discrete mathematics strategies, for example, pattern searching, organization of information, sorting, case-by-case analysis, iteration and recursion, and mathematical induction, to investigate, solve, and extend problems;

 

 

 

KA

 

 

 

 

 

 

(4) exploration, development, analysis, and comparison of algorithms designed to accomplish a task or solve a problem;

 

 

 

KA

 

 

 

 

 

 

(5) application of additional discrete strategies including symbolic logic and linear programming;

 

 

 

KA

 

 

 

 

 

 

(6) matrices as a mathematical system and matrices and matrix operations as tools to record information and find solutions of systems of equations; and

 

 

KA

 

 

 

 

 

 

 

(7) analysis of iterative and recursive algorithms to estimate the time needed in order to execute the algorithms for data likely to be encountered in problem situations.

 

 

 

KA

 

 

 

 

 

 

 

 

Math 120

Math 126

Math 220

Math 232

Math 244

Math 252

Math 262

Math 356

Stat 212/272

Ed 350

C.  A teacher of mathematics understands that number sense is the underlying structure that ties mathematics into a coherent field of study, rather than an isolated set of rules, facts, and formulae.  The teacher of mathematics must demonstrate knowledge of the following mathematical concepts and procedures and the connections among them: 

(1) an intuitive sense of numbers including a sense of magnitude, mental mathematics, place value, and a sense of reasonableness of results;

 

 

 

 

 

 

 

 

 

KA

(2) an understanding of number systems, their properties and relations including whole numbers, integers, rational numbers, real numbers, and complex numbers;

 

 

 

 

 

 

 

 

 

KA

(3) translation among equivalent forms of numbers to facilitate problem solving;

 

 

 

 

 

 

 

 

 

KA

(4) application of appropriate methods of estimation of quantities and evaluation of the reasonableness of estimates;

 

 

 

 

 

 

 

 

 

KA

(5) a knowledge of elementary operations, application of properties of operations, and the estimation of results;

 

 

 

 

 

 

 

 

 

KA

(6) geometric and polar representation of complex numbers and the interpretation of complex solutions to equations;

 

 

 

 

 

 

 

 

 

KA

(7) algebraic and transcendental numbers;

 

 

 

 

 

 

 

 

 

KA

(8) numerical approximation techniques as a basis for numerical integration, numerical-based proofs, and investigation of fractals; and

 

KA

 

 

 

 

 

KA

 

 

(9) number theory divisibility, properties of prime and composite numbers, and the Euclidean algorithm.

 

 

 

 

 

KA

 

 

 

 

 

 

Math 120

Math 126

Math 220

Math 232

Math 244

Math 252

Math 262

Math 356

Stat 212/272

Ed 350

D.  A teacher of mathematics understands geometry and measurement from both abstract and concrete perspectives and is able to identify real world applications and to use geometric learning tools and models, including geoboards, compass and straight edge, rules and protractor, patty paper, reflection tools, spheres, and platonic solids.  The teacher of mathematics must demonstrate knowledge of the following mathematical concepts and procedures and the connections among them: 

(1) shapes and how shapes can be derived/described in terms of dimension, direction, orientation, perspective, relationships among these properties;

 

 

 

 

 

 

 

KA

 

 

(2) spatial sense and the ways shapes can be visualized, combined, subdivided, and changed to illustrate concepts, properties, and relationships;

 

 

 

 

 

 

 

KA

 

 

(3) spatial reasoning and the use of geometric models to represent, visualize, and solve problems;

KA

 

 

 

 

 

 

 

 

 

(4) motion and the ways in which rotation, reflection, and translation of shapes can illustrate concepts, properties, and relationships;

 

 

 

 

 

 

 

KA

 

 

(5) formal and informal argument, including the processes of making assumptions; formulating, testing, and reformulating conjectures; justifying arguments based on geometric figures; and evaluating the arguments of others;

 

 

 

 

 

 

 

KA

 

 

(6) plane, solid, and coordinate geometry systems including relations between coordinate and synthetic geometry, and generalizing geometric principles from a two-dimensional system to a three-dimensional system;

 

 

 

 

 

 

 

KA

 

 

(7) attributes of shapes and objects that can be measured, including length, area, volume, capacity, size of angles, weight, and mass;

 

 

 

 

 

 

 

 

 

KA

(8) the structure of systems of measurement, including the development and use of measurement systems and the relationships among different systems;

 

 

 

 

 

 

 

 

 

KA

 

 

Math 120

Math 126

Math 220

Math 232

Math 244

Math 252

Math 262

Math 356

Stat 212/272

Ed 350

(9) measuring, estimating, and using measurements to describe and compare geometric phenomena;

 

 

 

 

 

 

 

 

 

KA

(10) systems of geometry, including Euclidean, non-Euclidean, coordinate, transformational, and projective geometry;

 

 

 

 

 

 

 

KA

 

 

(11) transformations, coordinates, and vectors, including polar and parametric equations, and the use of these in problem solving;

 

 

KA

 

 

 

 

 

 

 

(12) three-dimensional geometry and its generalization to other dimensions;

 

 

 

 

 

 

 

KA

 

 

(13) topology, including topological properties and transformations;

 

 

 

 

 

 

 

KA

 

 

(14) extend informal argument to include more rigorous proofs; and

 

 

 

 

 

 

 

KA

 

 

(15) extend work with two-dimensional right triangles including unit circle trigonometry.

KA

 

 

 

 

 

 

 

 

 

E.  A teacher of mathematics uses a variety of conceptual and procedural tools for collecting, organizing, and reasoning about data; applies numerical and graphical techniques for representing and summarizing data; and interprets and draws inferences from these data and makes decisions in a wide range of applied problem situations.  The teacher of mathematics must demonstrate knowledge of the following mathematical concepts and procedures and the connections among them: 

(1) data and its power as a way to explore questions and issues in our world;

 

 

 

 

 

 

 

 

KA

 

(2) investigation through data including formulating a problem; devising a plan to collect data; and systematically collecting, recording, and organizing data;

 

 

 

 

 

 

 

 

KA

 

(3) data representation to describe data distributions, central tendency, and variance through appropriate use of graphs, tables, and summary statistics;

 

 

 

 

 

 

 

 

KA

 

 

 

Math 120

Math 126

Math 220

Math 232

Math 244

Math 252

Math 262

Math 356

Stat 212/272

Ed 350

(4) analysis and interpretation of data, including summarizing data, and making or evaluating arguments, predictions, recommendations, or decisions based on an analysis

of the data; and

 

 

 

 

 

 

 

 

KA

 

(5) descriptive and inferential statistics, including validity and reliability.

 

 

 

 

 

 

 

 

KA

 

F.  A teacher of mathematics understands how to reduce the uncertainties through predictions based on empirical or theoretical probabilities.  The teacher of mathematics must demonstrate knowledge of the following mathematical concepts and procedures and the connections among them:

(1) inference, and the role of randomness and sampling in statistical claims about populations;

 

 

 

 

 

 

 

 

KA

 

(2) probability as a way to describe chance or risk in simple and compound events;

 

 

 

 

 

 

KA

 

 

 

(3) predicting outcomes based on exploration of probability through data collection, experiments, and simulations;

 

 

 

 

 

 

 

 

KA

 

(4) predicting outcomes based on theoretical probabilities, and comparing mathematical expectations with experimental results;

 

 

 

 

 

 

KA

 

 

 

(5) random variable and the application of random variable to generate and interpret probability distributions;

 

 

 

 

 

 

KA

 

 

 

(6) probability theory and the link of probability theory to inferential statistics; and

 

 

 

 

 

 

 

 

KA

 

(7) discrete and continuous probability distributions as a basis for making inferences about population.

 

 

 

 

 

 

 

 

KA

 

G.  A teacher of mathematics is able to reason mathematically, solve problems mathematically, and communicate in mathematics effectively at different levels of formality and knows the connections among mathematical concepts and procedures as well as their application to the real world.  The teacher of mathematics must be able to: 

(1) solve problems in mathematics by: 

(a) formulating and posing problems;

 

 

 

 

 

 

 

 

 

KA

 

 

Math 120

Math 126

Math 220

Math 232

Math 244

Math 252

Math 262

Math 356

Stat 212/272

Ed 350

(b) solving problems using different strategies, verifying and interpreting results, and generalizing the solution;

 

 

 

 

 

 

 

 

 

KA

(c) using problem solving approaches to investigate and understand mathematics; and

 

 

 

 

 

 

 

 

 

KA

(d) applying mathematical modeling to real world situations;

 

 

 

 

 

 

 

 

 

KA

(2) reason in mathematics by: 

(a) examining patterns, abstracting and generalizing based on the examination, and making convincing mathematical arguments;

 

 

 

 

 

 

 

KA

 

 

(b) framing mathematical questions and conjectures, formulating counter-examples, and constructing and evaluating arguments; and

 

 

 

 

KA

 

 

 

 

 

(c) using intuitive, informal exploration, and formal proof.

 

 

 

 

 

KA

 

 

 

 

(3) communicate in mathematics by: 

(a) expressing mathematical ideas orally, visually, and in writing;

 

 

 

 

 

 

 

KA

 

 

(b) using the power of mathematical language, notation, and symbolism; and

 

 

 

 

 

KA

 

 

 

 

(c) translating mathematical ideas into mathematical language, notations, and symbols; and

 

 

 

 

 

KA

 

 

 

 

(4) make mathematical connections by: 

(a) demonstrating the interconnectedness of the concepts and procedures of mathematics;

 

 

 

 

 

 

 

 

 

KA

(b) making connections between mathematics and other disciplines;

 

 

 

 

 

 

 

 

 

KA

(c) making connections between mathematics and daily living; and

 

 

 

 

 

 

 

 

 

KA

(d) making connections between equivalent representations of the same concept.

 

 

 

 

 

 

 

 

 

KA

 

 

Math 120

Math 126

Math 220

Math 232

Math 244

Math 252

Math 262

Math 356

Stat 212/272

Ed 350

H.  A teacher of mathematics must: 

(1) understand the historical bases of mathematics, including the contributions made by individuals and cultures, and the problems societies faced that gave rise to mathematical systems;

 

 

 

 

 

 

 

KA

 

 

(2) recognize that there are multiple mathematical world views and how the teacher's own view is similar to or different from that of the students;

 

 

 

 

 

 

 

 

 

KA

(3) understand the overall framework of mathematics including the: 

 

 

 

 

 

 

 

 

 

 

(a) processes and consequences of expanding mathematical systems;

 

 

 

 

 

 

 

 

 

KA

(b) examination of the effects of broad ideas, including operations or properties, as these ideas are applied to various systems;

 

 

 

 

 

 

 

 

 

KA

(c) examination of the same object from different perspectives; and

 

 

 

 

 

 

 

 

 

KA

(d) investigation of the logical reasoning that takes place within a system; and

 

 

 

 

 

 

 

 

 

KA

 

I.  A teacher of mathematics must demonstrate an understanding of the teaching of mathematics that integrates understanding of mathematics with the understanding of pedagogy, students, learning, classroom management, and professional development.  The teacher of mathematics to preadolescent and adolescent students shall:

 

Ed 290

Ed 330

Ed 375

Ed 350

(1)  understand and apply educational principles relevant to physical, social, emotional, moral, cognitive development of preadolescents and adolescents;

KA

 

 

KA

(2) understand and apply the research base for and the best practices of middle level and high school education;

 

 

 

KA

(3) develop curriculum goals and purposes based on the central concepts of mathematics and know how to apply instructional strategies and materials for achieving student understanding of this discipline;

 

 

 

KA

 

 

Ed 290

Ed 330

Ed 375

Ed 350

(4) understand the role and alignment of district, school, and department mission and goals in program planning;

 

 

 

KA

(5) understand the need for and how to connect students' schooling experiences with everyday life, the workplace, and further educational opportunities;

 

KA

 

KA

(6) know how to involve representatives of business, industry, and community organizations as active partners in creating educational opportunities;

 

 

 

KA

(7) understand the role and purpose of cocurricular and extracurricular activities in the teaching and learning process;

 

 

 

KA

(8) understand the impact of reading ability on student achievement in mathematics, recognize the varying reading comprehension and fluency levels represented by students, and possess the strategies to assist students to read mathematical content materials more effectively; and

 

 

KA

KA

(9) apply the standards of effective practice in teaching students through a variety of early and ongoing clinical experiences with middle level and high school students within a range of educational programming models.

KA

KA

KA

KA