
PATTERNS, RELATIONS, AND FUNCTIONS 
Required Course 
Elsewhere: Portfolio 
Strongly Recc. 


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: 

A1 
(1) recognize, describe, and generalize patterns and build mathematical models to describe situations, solve problems, and make predictions; 
Calculus Discrete 
Precollege 


A2 
(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; 
Calculus Linear 

Multi 

A3 
(3) represent and solve problem situations that involve variable quantities and use appropriate technology; 
Calculus Linear 
Precollege 
Multi 

A4 
(4) understand patterns present in number systems and apply these patterns to further investigations; 
Discrete 
Precollege 
Number Theory 

A5 
(5) apply properties of boundedness and limits to investigate problems involving sequences and series; 
Calculus ERA 



A6 
(6) apply concepts of derivatives to investigate problems involving rates of change; 
Calculus 



A7 
(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 
Calculus Linear 

Multivariable 

A8 
(8) apply properties of group and field structures to mathematical investigations. 
Abstract Algebra 



DISCRETE MATHEMATICS 
Required Course 
Elsewhere: Portfolio 
Strong Recc. 

B 
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: 

B1 
(1) the application of discrete models to problem situations using appropriate representations such as sequences, vertexedge graphs and trees, matrices, and arrays; 
Discrete 



B2 
(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; 
Discrete Probability 



B3 
(3) application of discrete mathematics strategies, for example, pattern searching, organization of information, sorting, casebycase analysis, iteration and recursion, and mathematical induction, to investigate, solve, and extend problems; 
Discrete 

CS172 

B4 
(4) exploration, development, analysis, and comparison of algorithms designed to accomplish a task or solve a problem; 
Discrete 

CS172 

B5 
(5) application of additional discrete strategies including symbolic logic and linear programming; 
Discrete 

CS172 OR 

B6 
(6) matrices as a mathematical system and matrices and matrix operations as tools to record information and find solutions of systems of equations; and 
Linear 



B7 
(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. 
Discrete 

CS172 

NUMBER SENSE 
Required Course 
Elsewhere: Portfolio 
Strong Recc. 

C 
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: 

C1 
(1) an intuitive sense of numbers including a sense of magnitude, mental mathematics, place value, and a sense of reasonableness of results; 

Precollege 


C2 
(2) an understanding of number systems, their properties and relations including whole numbers, integers, rational numbers, real numbers, and complex numbers; 
ERA 
Precollege 


C3 
(3) translation among equivalent forms of numbers to facilitate problem solving; 

Precollege 


C4 
(4) application of appropriate methods of estimation of quantities and evaluation of the reasonableness of estimates; 

Precollege 


C5 
(5) a knowledge of elementary operations, application of properties of operations, and the estimation of results; 

Precollege 


C6 
(6) geometric and polar representation of complex numbers and the interpretation of complex solutions to equations; 

Precollege 


C7 
(7) algebraic and transcendental numbers; 
Abstract 

Number Theory 

C8 
(8) numerical approximation techniques as a basis for numerical integration, numericalbased proofs, and investigation of fractals; and 
Calculus Discrete Geometry 



C9 
(9) number theory divisibility, properties of prime and composite numbers, and the Euclidean algorithm. 
Discrete Abstract 

Number Theory 

SPACE AND SHAPE 
Required Course 
Elsewhere: Portfolio 
Strong Recc. 


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: 

D1 
(1) shapes and the ways shapes can be derived and described in terms of dimension, direction, orientation, perspective, and relationships among these properties; 
Linear, Geometry 
Precollege 
Multi 

D2 
(2) spatial sense and the ways shapes can be visualized, combined, subdivided, and changed to illustrate concepts, properties, and relationships; 
Calculus Geometry 
Precollege 


D3 
(3) spatial reasoning and the use of geometric models to represent, visualize, and solve problems; 
Calculus Geometry 
Precollege 


D4 
(4) motion and the ways in which rotation, reflection, and translation of shapes can illustrate concepts, properties, and relationships; 
Geometry 
Precollege 


D5 
(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; 
Geometry ERA 

Structures & Proof 

D6 
(6) plane, solid, and coordinate geometry systems including relations between coordinate and synthetic geometry, and generalizing geometric principles from a twodimensional system to a threedimensional system; 
Geometry 

Multi 

D7 
(7) attributes of shapes and objects that can be measured, including length, area, volume, capacity, size of angles, weight, and mass; 

Precollege 


D8 
(8) the structure of systems of measurement, including the development and use of measurement systems and the relationships among different systems; 

Precollege 


SPACE AND SHAPE  II 
Required Course 
Elsewhere: Portfolio 
Strongly Recc. 
D9 
(9) measuring, estimating, and using measurements to describe and compare geometric phenomena; 

Precollege 

D10 
(10) systems of geometry, including Euclidean, nonEuclidean, coordinate, transformational, and projective geometry; 
Geometry 


D11 
(11) transformations, coordinates, and vectors, including polar and parametric equations, and the use of these in problem solving; 
Linear Geometry 
Precollege 
Multi 
D12 
(12) threedimensional geometry and its generalization to other dimensions; 
Calculus, Linear 

Multi 
D13 
(13) topology, including topological properties and transformations; 
Geometry ERA 


D14 
(14) extend informal argument to include more rigorous proofs; and 
ERA Abstract, Geometry 


D15 
(15) extend work with twodimensional right triangles including unit circle trigonometry. (Old language: ... to include introduction to trigonometry.) 

Precollege 


DATA INVESTIGATIONS 
Required Course 
Elsewhere: Portfolio 
Strong Recc. 

E 
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: 

E1 
(1) data and its power as a way to explore questions and issues in our world; 
Statistics 212 
Science lab 
Practicum 

E2 
(2) investigation through data including formulating a problem; devising a plan to collect data; and systematically collecting, recording, and organizing data; 
Statistics 212 

Practicum 

E3 
(3) data representation to describe data distributions, central tendency, and variance through appropriate use of graphs, tables, and summary statistics; 
Statistics 212 or 312* 



E4 
(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 
Statistics 212 or 312 



E5 
(5) descriptive and inferential statistics, including validity and reliability. 
Statistics 212 or 312 
Ed Psych 


RANDOMNESS AND UNCERTAINTY 
Required Course 
Elsewhere: Portfolio 
Strong Recc. 

F 
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: 

F1 
(1) inference, and the role of randomness and sampling in statistical claims about populations; 
Statistics 212 or 312 



F2 
(2) probability as a way to describe chance or risk in simple and compound events; 
Statistics 212 or 312 



F3 
(3) predicting outcomes based on exploration of probability through data collection, experiments, and simulations; 
Statistics 212 or 312 



F4 
(4) predicting outcomes based on theoretical probabilities, and comparing mathematical expectations with experimental results; 
Statistics 212 Probability 



F5 
(5) random variable and the application of random variable to generate and interpret probability distributions; 
Statistics 212 Probability 



F6 
(6) probability theory and the link of probability theory to inferential statistics; and 
Statistics 212 Probability 



F7 
(7) discrete and continuous probability distributions as a basis for making inferences about population. 
Statistics 212 Probability 



MATHEMATICAL PROCESSES 
Required Course 
Elsewhere: Portfolio 
Strong Recc. 

G 
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: 

G1 
(1) solve problems in mathematics by: (a) formulating and posing problems; (b) solving problems using different strategies, verifying and interpreting results, and generalizing the solution; (c) using problem solving approaches to investigate and understand mathematics; and (d) applying mathematical modeling to real world situations; 
Math Major Senior Seminar 
Portfolio 
Practicum 

G2 
2) reason in mathematics by: (a) examining patterns, abstracting and generalizing based on the examination, and making convincing mathematical arguments; (b) framing mathematical questions and conjectures, formulating counterexamples, and constructing and evaluating arguments; and (c) using intuitive, informal exploration, and formal proof. 
ERA or Abstract and Writing Projects Senior Seminar 
Portfolio 


G3 
3) communicate in mathematics by: (a) expressing mathematical ideas orally, visually, and in writing; (b) using the power of mathematical language, notation, and symbolism; and (c) translating mathematical ideas into mathematical language, notations, and symbols; and 
Group Learning in Geometry and Other Courses Writing Throughout the Major Senior Seminar Presentations in Geometry and Other Courses 



G4 
(4) make mathematical connections by: (a) demonstrating the interconnectedness of the concepts and procedures of mathematics; (b) making connections between mathematics and other disciplines; (c) making connections between mathematics and daily living; and (d) making connections between equivalent representations of the same concept. 
Math Ed (Ed 350) Senior Seminar 
Portfolio 


MATHEMATICAL PERSPECTIVES 
Required Course 
Elsewhere: Portfolio 
Strong Recc. 
H1 
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; 
Geometry Senior Seminar 
Portfolio: Independent Reading and Writing 
Masterpieces Centrality of Math in Human Intellectual Thought 
H2 
(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; 
Senior Seminar Human Issues: (Ed 385) 

Centrality of Math in Human Intellectual Thought 
H3 
(3) understand the overall framework of mathematics including the: (a) processes and consequences of expanding mathematical systems; (b) examination of the effects of broad ideas, including operations or properties, as these ideas are applied to various systems; (c) examination of the same object from different perspectives; and (d) investigation of the logical reasoning that takes place within a system; and 
Senior Seminar Math Ed 
Portfolio 

H4 
(4) understand the role of technology, manipulatives, and models in mathematics. 
Math Major Calculus Geometry Math Ed 


I 
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: 

I1 
(1) understand and apply educational principles relevant to the physical, social, emotional, moral, and cognitive development of preadolescents and adolescents; 
Ed Courses 



I2 
(2) understand and apply the research base for and the best practices of middle level and high school education; 
Ed Courses 



I3 
(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; 
Math Ed 



I4 
(4) understand the role and alignment of district, school, and department mission and goals in program planning; 
Ed Courses 



I5 
(5) understand the need for and how to connect students' schooling experiences with everyday life, the workplace, and further educational opportunities; 
Ed Courses 



I6 
(6) know how to involve representatives of business, industry, and community organizations as active partners in creating educational opportunities; 
Ed Courses 



I7 
(7) understand the role and purpose of cocurricular and extracurricular activities in the teaching and learning process; 
Ed Courses 



I8 
(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 
Ed Courses Math Ed 



I9 
(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. 
Ed Courses 


Minnesota BOT rules, Current as of 07/20/99