IBL Math course materials are available for download on Dropbox to instructors at other institutions. Please contact Ralf Spatzier for permissions.
Content: This course gives a historical introduction to Cryptology, from ancient times up to modern public key encryption, particularly RSA, and introduces a number of mathematical ideas involved in the development and analysis of codes. Mathematical topics include some enumeration, probability, and statistics, but the bulk of the course is devoted to elementary number theory. Students also work throughout the course on effectively communicating mathematics, both written and orally. Moreover, students will develop rigorous mathematical proof writing skills, and a primary goal of the course is to not only understand how various cryptosystems work, but why.
Structure: The course has two components, classroom and computer lab. The classroom component meets three days each week, and is driven by in-class worksheets students complete in small groups. Each worksheet consists of definitions, examples, problems, and mathematical results that students attempt to understand through discussion with their peers and the instructor. As students solve problems from the worksheet, they present their solutions to the rest of the class. In the computer lab, various discovery-based projects allow the students to explore the ideas developed in the classroom and cryptosystems not covered in the worksheets. No previous experience with computer programming is necessary.
In the computer lab, various discovery-based projects are designed to allow students to explore the ideas developed in the classroom.
Materials: The detailed description “Inquiry Based Learning in Cryptology” of the course by Kyle Petersen provides more information about the course and inquiry based teaching in general. For other course materials, please contact Ralf Spatzier.
Math 175/275 – Introduction to Cryptology (link to the “Courses” page)
This course is an Inquiry-Based version of Honors Calculus I and II (such as Math 185/186) and provides the necessary preparation for Calculus III (Math 215 or the honors version, Math 285). A student who has had some exposure to calculus (e.g. AB or BC in high school, or Math 115) will be well-prepared for this course.This is a calculus course from a theoretical perspective, and we will study concepts like continuity, derivatives, and integrals, as well as investigate some of the properties of the real numbers that make these things work.As an IBL course, we will spend the majority of class time working in groups and presenting ideas and solutions to problems. I will give occasional short lectures to set the context for the class activities.
Materials: A. Uribe and D. Visscher’s book, Explorations in Analysis, Topology, and Dynamics: An Introduction to Abstract Mathematics
Math 176/276 – Explorations in Calculus (link to the “Courses” page)
This is a course about the nuts and bolts of mathematical writing. The course introduces the fundamentals of mathematical communication (e.g., sets, functions, quantifiers) and explores various proof techniques (e.g., contrapositive, contradiction, induction). Most importantly, it provides guided practice in rudimentary proof writing. Topics include elementary set theory, functions, existential and universal quantifiers, and proof techniques.
The course has no lecture; students work through the worksheets in groups.
Content: This course provides a rigorous introduction to linear algebra, as well as an introduction to reading and writing mathematical proofs. It is intended for potential math majors and those interested in theoretical mathematics, and is taught using IBL methods in an interactive classroom. Topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and their subspaces; geometry of Rn linear dependence; bases, dimension, and coordinates; linear transformations; eigenvalues and eigenvectors; diagonalization; inner products and orthogonality. We study not only these concepts and their applications, but also methods of proof and the written communication of mathematics. Students should leave this course prepared to use linear algebra as well as to succeed in further theoretical courses in mathematics.
Prerequisites: Math 215, 255, or 285. Incoming students are not expected to have experience writing proofs, but should have some interest in theory and abstract concepts; those who are more interested in computation, concrete problems, and applications should consider alternative courses such as Math 214 or 417.
Structure: Students learn both content and proof-writing through collaborative in-class problem sessions, instructor coaching, and opportunities for individualized peer-tutoring outside of class. Students are encouraged to write notes, scratch work, and solutions to problems on the dry-erase boards in our classrooms for their peers and instructor to view, question, and discuss. A typical class may consist of a short introductory lecture, a challenging and sometimes open-ended problem set to be worked collaboratively in groups for the majority of the session, and a brief summary of material or explanation of solutions at the end. Grades are determined by in-class quizzes, weekly written homework sets, frequent short web assignments, two midterm exams, and a final exam.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Math 217 – Linear Algebra & Introduction to Proofs (link to the “Courses” page)
Math 285 – Honors Calculus III (IBL modules in some sections)
…coming soon.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Math 297 – Introduction to Analysis
Math 297 is an honors course in analysis, the study of how/why calculus works. Students develop an appreciation for the importance of the completeness and ordering of the real numbers. Students also learn how to read, write, and understand “epsilon-delta” style arguments as they cover topics including basic topology, uniform continuity, and the properties of derivatives, definite integrals, and infinite sequences and series. To the extent possible, they carry out their investigations in the setting of (finite dimensional) inner product spaces.
The class meets forty-one times over the course of the semester. There is very little lecture during each of the eighty minute classes; instead, students work in groups of two or three at the board on carefully crafted worksheets that guide them as they learn and internalize the material.
It is assumed that students have acquired a solid foundation in the theoretical aspects of linear algebra (as in the IBL course Math 217). Throughout the semester there will be many homework problems that review and extend the ideas students learned in Math 217.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Content: Every day the media showers us with news, analysis, and op-eds, which use and misuse numbers to arrive at various far-reaching conclusions. The objective of the course is to help students to acquire some basic mathematical skills to navigate in the sea of numbers. Often, this boils down to understanding a few fundamental, ancient, and deep concepts: randomness, fairness, coincidence, and bias.
We will study what “probability”, “events”, and “independence” mean, how to compute some basic probabilities and why it can be costly to assume that events are independent when in fact they are not, as illustrated by recent and not so recent events in the insurance industry and the stock market. We will also discuss why randomized strategies in games can be quite helpful.
The course is somewhat in spirit of the book “A Mathematician Reads the Newspaper” by J.A. Paulos appended by examples taken from the “Chance” database at Dartmouth http://www.dartmouth.edu/~chance/
Time permitting, we will discuss some remarkably exact predictions of the 2008 elections results made with the help of some statistical analysis.
Materials: There is no recommended text. Materials will be prepared and distributed in class. Please contact Ralf Spatzier for more information.
Math 310 – Choice and Chance (link to the “Courses” page)
Content: Math 351 is a junior-level course in real analysis that emphasizes proof writing and mathematical communication. IBL aspects include team homework, student presentations at the blackboard, and in-class discussion, with limited lecture. The course satisfies the real analysis requirement for several concentrations of the mathematics major.
Structure: The course covers logic and techniques of proof, sequences, continuous functions, uniform continuity, differentiation, integration and the Fundamental Theorem of Calculus.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Math 351 – Principles of Analysis (link to the “Courses” page)
Content: This course discusses mathematics for elementary school teachers, taught with IBL methods. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Topics covered typically include numeration systems, arithmetic (including estimation, mental arithmetic, and alternative algorithms), divisibility and fractions, sets, and problem solving.
Structure: Primary goals of the course include developing students’ problem solving skills and giving students practice in explaining/presenting problems to others, both one-on-one and to larger groups. To this end the classes typically include only a brief review lecture; most of the class time is spent working on challenging problems in groups, followed by student presentations of the problems to the class. Class participation is a significant portion of the course grade.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Math 385 – Mathematics for Elementary School Teachers (link to the “Courses” page)
Course Description: This course is not like any other in the mathematics department. The course is designed to show you how new mathematics is actually created: how to take a problem, make models and guesses, experiment with them, and search for underlying structure. It is suitable for students at many levels. This course serves also to develop useful skills, including how to write and typeset a math paper, making an oral presentation, and computing with a mathematics software system such as Mathematica or Maple.
The class will be split into groups, typically of 3 students, who will choose a project, work on it, and submit a written report describing their findings. We will also have oral reports of some of the projects. Often, though not necessarily always, your research will involve computer experiments. Findings should be stated precisely, either as facts or as conjectures; proofs will be viewed favorably, but are not required.
Topics range from ones that are treated in mathematics books to ones that lead to open problems. Few are well defined. In contrast to homework assignments for your other classes, you will not be told precisely what to compute or to prove. The topics have been chosen because they display interesting phenomena, but we do not necessarily have a particular result in mind. And if we did, you might discover something else.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Math 389 – Explorations in Mathematics (link to Course webpage) course webpage
…coming soon.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
…coming soon.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Math 412 is a junior-level course in abstract algebra that emphasizes proof writing and mathematical communication. IBL aspects include collaborative problem solving, student discovery and exploration of the material during class in small groups. There is no lecturing by the professor in Math 412.
The course satisfies the algebra requirement for several concentrations of the mathematics major.The course covers the definitions and basic properties of groups, rings and fields. Topics include: the Euclidean algorithm, the fundamental theorem of arithmetic, modular arithmetic, finite fields, polynomial rings, the Chinese Remainder theorem, symmetric groups, linear groups, homomorphism and isomorphism of groups and rings, quotient groups and rings.
Materials: We have developed materials for the course. Please contact Ralf Spatzier (spatzier@umich.edu) for more information.
Math 412 – Introduction to Modern Algebra (link to the “Courses” page)
Course Description: The mathematical discipline of Geometry is immense. Even with the focused goal of preparing future teachers of high school geometry, the relevant material could last several semesters. This course is guided by the principle of “depth over breadth”, with the goal of giving students the tools to tackle an unpredictable array of geometric topics they might encounter in their future teaching practice. The aims of the course include that (1) students become comfortable operating within an axiomatic development of Euclidean Geometry; (2) students become critical, proficient provers and problem-solvers; and (3) students improve their mathematical communication. The class is taught using inquiry-based learning. Students start to work through non-routine problem sets before class and class time is spent discussing problems in groups and as a class.
Content: Math 486 can be described as the rudiments of analysis and algebra underlying theorems used in secondary mathematics. Math486 examines the principles of analysis and algebra underlying theorems concerning fields — especially the rationals, reals, and complex numbers; and concerning functions — especially polynomials, exponential functions, and logarithmic functions. Mathematical underpinnings of these ideas can serve as intellectual resources for secondary teachers. Major topics covered by Math 486-W11 vary from year to year, but have included:
- Properties of fields, including the parallels between Q[√p] and C=R[i], for positive integers p such that √p is irrational.
- Properties of the rational numbers, including density.
- A rigorous description of the long division algorithm for integers and for polynomials; invariance of the sequence of remainder terms for a particular integer dividend, assuming an integer divisor.
- The proof that a degree n polynomial must have exactly n complex roots (counting with multiplicity); the proof establishing an equivalence between factors of a polynomial and roots of a polynomial.
- Limits, convergence, and divergence of sequences of real and complex numbers; accumulation points of sets of real and complex numbers.
- Limits, convergence, and divergence of sequences of real functions; uniform convergence, pointwise convergence.
- Properties of functions including injectivity, surjectivity, invertibility, continuity, and periodicity.
- Definition of the exponential function and logarithmic function on a complex domain.
Students are engaged via constructing collective explanations of key mathematical topics. Mathematical practices emphasized by Math486 include:
- Assessing the completeness and soundness of explanations and proofs of mathematical ideas.
- Reading, explaining, and writing conjectures and proofs.
- Alertness to mathematical language and precision; how small differences in phrasing may have significant mathematical implications (e.g., the phrase “the solution” versus “a solution”).
Giving mathematical motivations for different given formulations of equivalent mathematical ideas; using different ways of representing the same mathematical idea, and giving explanations of why they are equivalent.Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Course Description: This course has two mutually supportive aims: (1) To cultivate what can be called “connected mathematical thinking,” largely through ambitious problem solving activities. (2) To provide a rigorous and coherent treatment of some of the foundational domains of the school mathematics curriculum, especially those coming from abstract algebra. The course should be helpful for those planning to teach mathematics (at any level, including college) as well as for mathematics and science majors who want to deepen their knowledge of some fundamental mathematical concepts and to develop productive mathematical practices (habits of mind). This course is taught using inquiry-based learning; students start to work through non-routine problem sets before class and class time is spent discussing problems in groups and as a class.
Math 489 – Math for Elementary & Middle School Teachers
Content: This is the sequel to MATH 385, continues with mathematics for elementary and middle school teachers, and is taught with IBL methods. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Math 385 is a prerequisite for this class. Topics covered typically include decimals, ratios and percent, statistics and probability, and geometry.
Structure: Primary goals of the course include developing students’ problem solving skills and giving students practice in explaining/presenting problems to others, both one-on-one and to larger groups. To this end the classes typically include only a brief review lecture; most of the class time is spent working on challenging problems in groups, followed by student presentations of the problems to the class. Class participation is a significant portion of the course grade.
Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Math 489 – Mathematics for Elementary & Middle School Teachers (link to the “Courses” page)
Math 490 – Introduction to Topology
Content: Topology is a fundamental area of mathematics that provides a foundation for analysis and geometry. Once a set has a topology (so it becomes a “topological space”), we can start to build on it. For example, the notion of a continuous function makes sense on a topological space, and in fact, this is the most general setting where the idea of a continuous function makes sense. One theme in topology is trying to distinguish topological spaces from each other. A topologist will tell you that all pentagons look the same, and in fact, that all pentagons look like all triangles! One thing we will do in this course is rigorously explore what it means for two topological spaces to be “the same”. We will also develop tools that will help to distinguish topological spaces from each other. Topics include: open/closed sets, metric spaces, continuity, homeomorphisms, connectedness, compactness, Euler characteristic. This course is taught in the IBL style. This means that the instructor will speak only for a few minutes at the beginning of class, and then the students will work in groups (guided by worksheets) to explore and develop the material. There is no textbook; the main reference for course material is the compilation of the worksheets from class. By group discussions and problem-solving, students will discover the world of topology.
The prerequisites for Math 490 are: Math 351, Math 451 or previous exposure to real analysis. Even if you have not taken these courses, it is still possible to take 490 with permission of the instructor. Lastly, group work is a large part of this course, so it is important that you can work effectively with your peers.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Content: This is an elective course for elementary teaching certificate candidates that extends and deepens the coverage of mathematics begun in the required two-course sequence Math 385-489. Topics are primarily chosen from the middle-school geometry curriculum, but topics from algebra and number theory are also included. The logical structure of mathematics is emphasized: The notions of axiom, definition, theorem and proof are used throughout. This course is taught using IBL methods. Students work in groups developing concepts, formulating and criticizing definitions, advancing the mathematical content and solving problems.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
Content: MATH 593 is a beginning graduate class in algebra, which concentrates on rings, modules and their properties and constructions with universal properties as a guiding principle.
Topics: We will discuss localization of rings and modules, tensor products, alternating products and their universal properties. Structure theory for modules, especially over a PID, classification of finitely generated abelian groups, Jordan and rational canonical form. Bilinear algebra. Some homological algebra.
Prerequisites: It is assumed that you have acquired a solid foundation in the theoretical aspects of linear algebra and have taken an undergraduate abstract algebra course on group theory.
Pedagogy: The course will be taught in an Inquiry Based Learning style, a teaching method that emphasizes discovery, analysis and investigation to deepen students’ understanding of the material and its applications. There will be very little lecture; instead, you will work through worksheets that guide you as you learn and internalize the material.
Materials: For course materials, please contact Ralf Spatzier (spatzier@umich.edu) for permissions to Math IBL Resources.
The Department offers other courses with significant use of IBL techniques. In particular, our Introductory Mathematics Program offers precalculus MATH 105, mainstream calculus MATH 115 and MATH 116 that use group work and presentations. In addition, the Department together with the College of LSA, sponsors the the Douglass Houghton Scholars Program in which students working hard calculus problems in groups in workshop classes taken along with MATH 115 or MATH 116. For more detailed information on this, please consult the links.