Vermont Mathematics Initiative

Building Capacity across Vermont for High-Quality Mathematics Instruction

Course 2: Functions and Algebra for Teachers-This course builds upon the prior course Mathematics as a Second Language and extends and reinforces the learning from that course.  Participants will obtain deep understanding of the concept of a function, utilize functions in problem solving, appreciate the pervasiveness of the function idea in the K-8 mathematics curriculum as well as everyday life, and engage in a variety of problem-solving activities that relate directly to the K-8 mathematics classroom. Topics include functions, graphs, inverse functions, linear functions, the algebra and geometry of straight lines, solving linear equations and inequalities, and an introduction to nonlinear functions.  This course, together with Mathematics as a Second Language, serve as the mathematical foundation for a K-8 lesson study project that participants will undertake during the school year.

Course 3: Trigonometry for Teachers, and Algebra and Geometry II-This course builds on the arithmetic, algebra, and geometry developed in prior courses. The first part of the course develops the subject of trigonometry from the perspective of the K-8 mathematics classroom. Topics include similar triangles, the trigonometric functions and their graphs, the number pi, and applications to measurement, wave motion, and problem solving. The second part of the course continues the study of algebra from the perspective of K-8 mathematics. Topics include quadratic functions, parabolas, and related problem solving. Teachers will relate their learning to the K-8 classroom by examining the Vermont Grade Expectations to identify the foundational concepts of Trigonometry.  Also, teachers will develop effective content-based questioning techniques and explore the components of building successful mathematics lessons.  Participants will begin to receive the first of a series of classroom observations by their VMI mentor.

Course 4: Measurement, Geometry, and Probability for Teachers-This course builds on prior courses in algebra and geometry.  Topics include measurement (length, area and volume), experimental and theoretical probability, and the ways in which these concepts develop across the elementary, middle and high school curricula. Topics are presented in the context of problem solving, and there is an emphasis on reinforcing one’s understanding of functions, function notation, and topics from algebra.

Course 5: Number Theory for Teachers-This course introduces teachers to the branch of mathematics known as number theory, in which one studies properties of positive integers with respect to the operations of multiplication and division.  Emphasis in this course is placed on the mathematical content of number theory and on how number theory is taught in grades K-8, with particular attention to student learning of number theory in these grades.  Topics include the division algorithm, properties of prime and composite numbers, the sieve of Eratosthenes as a way of understanding distributions of primes and composites, the infinitude of primes, the fundamental theorem of arithmetic, properties of the greatest common factor and methods of computing the greatest common factor including the Euclidean algorithm, properties of least common multiples, use of base ten and expanded notation, writing numbers and computing in different bases, and arithmetic progressions.

 

Course 6: Statistics, Action Research, and Inquiry into Effective Practice, I-This course provides an introduction to statistics and begins to incorporate research in mathematics education. Topics include graphical and numerical organization and presentation of data, summary statistics for quantitative data, measures of relationship between variables, and inference from sample data to populations.  This course forms the foundation for later work in statistics and school-based research, and is followed by the completion of a small-scale classroom inquiry.  The inquiry allows participants to bring together the research they read with the statistics they learn to formulate the study, develop an intervention, and analyze the resulting data. 

 

Course 7: Statistics, Action Research, and Inquiry into Effective Practice, II-This course is designed to build upon previously completed introductory work in statistics.  Teachers will apply their understanding of statistics to interpret and critique educational research studies, to develop and analyze the effectiveness of classroom interventions and to analyze and interpret local assessment results.  This course will prepare teachers to lead their schools in understanding the meaning and appropriate uses of assessment data and in working with colleagues around making data-driven classroom decisions.  Statistics topics include measures of central tendency and variability; representations of data; probability distributions; normal curve, stanine; estimation-standard error, margin of error, confidence intervals; and hypothesis tests.

 

Course 8: Algebra and Geometry for Teachers, III-This course continues the study of functions, algebra, and geometry from prior VMI courses. In this course the focus is on exponential processes and inverse processes, with an emphasis on problem solving.  Topics include the laws of exponents; the transition from simple to compound interest; calculations with compound interest; exponential functions, including domain, range, graph, and different bases; logarithm functions; the natural base e; applications to growth and decay; applications of logarithms in everyday life; and the history of exponential functions and logarithms.  Participants also study current research on mathematics education and analyze the mathematics content and teaching skills necessary to help students develop additive, multiplicative, and proportional reasoning. 

 

Course 9: Statistics, Action Research, and Inquiry into Effective Practice, III-This course builds on prior courses in statistics and action research.  The course reviews earlier concepts in descriptive and inferential statistics, and includes additional topics in the analysis of cross-tabulated data and in the analysis of correlational relationships between dependent and independent variables.  Teachers will do critical reading of research on instructional practices in elementary mathematics, and will complete the design of their own action research investigations.

 

Course 10: Calculus for Teachers, I-This course builds upon prior courses in arithmetic, algebra, and geometry. It is designed to introduce teachers to the branch of mathematics known as calculus in a way that relates calculus to the mathematics taught in the K-8 classroom. Topics include the idea of a limit, the role limits play in K-8 mathematics, the concept of instantaneous change, the derivative of a function, and applications to optimization. Course goals include reinforcing and extending arithmetic, algebra, and geometry knowledge and skills through problem solving involving calculus, and empowering teachers with a deep understanding of how capability in K-8 arithmetic and algebra is foundational for success in higher-level mathematics. This course also includes an analysis of the Vermont Grade Expectations and of the various curricula used in Vermont’s schools to identify the underlying role or appearance of ideas from calculus.  Participants will discuss ways to build such foundational skills and concepts into K-8 lessons.

 

Course 11: Calculus for Teachers II-This course continues the study of calculus and its relationship to the K-8 classroom.  Topics include infinite series, calculation of area, the definite integral, and the Fundamental Theorem of Calculus – all viewed from the perspective of the K-8 classroom teacher.  This second course in Calculus prepares the participant to develop a ‘Calculus in the Classroom’ lesson ­­to be taught during the academic school year.

 

Course 12: Capstone VMI experienceThe Capstone is the final course of the Vermont Mathematics Initiative and provides opportunities for participants to synthesize the coursework, field experiences, leadership, and research components of the VMI.  Teachers will revisit key mathematical concepts from basic arithmetic through calculus, study advanced topics in mathematics education and leadership, re-examine curriculum and instruction based on their VMI learning, and complete their school-based research projects and share findings with colleagues. Additionally, mini-workshops presented by the VMI instructional staff will provide further mathematics content enrichment that draws upon teachers’ prior VMI course experience. Teachers will complete their action research projects and share findings with colleagues..

The VMI Master’s Degree Curriculum

These 12 courses comprise the 36-hour VMI Masters Degree curriculum. 

Course 1: Mathematics as a Second Language-This course lays the groundwork for all the Vermont Mathematics Initiative courses that follow. A major theme is understanding algebra and arithmetic through language. The objective is to provide a solid conceptual understanding of the operations of arithmetic, as well as the interrelationships among arithmetic, algebra, and geometry. Topics include arithmetic vs. algebra; solving equations; place value and the history of counting; inverse processes; the geometry of multiplication; the many faces of division; rational vs. irrational numbers; and the one-dimensional geometry of real numbers.  In K-12 application of content, teachers will examine the Vermont Grade Expectations for the strand of Arithmetic, Number, and Operations and demonstrate an understanding of how the concepts associated with this strand of mathematics develop across the grades.