Dan Cunningham, Buffalo State College
Why Does Trigonometric Substitution Work?
Modern calculus textbooks carefully illustrate how to perform integration by trigonometric substitution. Unfortunately, most of these books do not adequately justify this powerful technique. Integration by trigonometric substitution is an application of a general technique of integration called inverse substitution. We shall present an accessible theorem and proof that establishes the validity of integration by inverse substitution. The proof offers calculus instructors a simple argument that can be used to show their students that trigonometric substitution is a valid technique of integration.
Heng Li, Hartwick College
A Free Boundary Problem Arising from Ductal Carcinoma in Situ Model
Ductal carcinoma in situ — a special cancer that is confined within the breast ductal only. We derive the mathematical ductal carcinoma in situ model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Well-posedness theorem of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique.
Jonathan Lopez, Canisius College
Graph Labelings, Eigenvectors, and Their Use in the Classification of Small Operators
A finite graph can be represented by a square matrix known as the adjacency matrix of the graph. Often, the eigenvalues and eigenvectors of an adjacency matrix are of interest. We discuss how certain labelings of the graph’s vertices can be used to identify eigenvalues and eigenvectors for its adjacency matrix. For some labelings, invoking the Perron-Frobenius Theorem leads to the spectral radius of the graph. These and other related ideas come up in an ADE-classification result in which “small” operators are classified by disjoint unions of graphs from the ADE-series. This is joint work with Terrence Bisson.
James Marengo, RIT
A Practical Modification of a Poisson Process
Suppose that we have a countably infinite supply of identical components, each of which has a lifetime which is exponentially distributed with the same failure rate. Suppose that the first component is placed into operation and is replaced by the second when it either lasts for its expected lifetime or when it actually fails. The same is true for the second component and so on. In this talk we use the inclusion-exclusion principle to find the joint probability distribution of the numbers of components which have been replaced while still functioning, and those which have failed by any specified time t. We also compute expected values for these two numbers. This talk should be accessible to anyone who has had a first course in probability.
Gabriel Prajitura, SUNY Brockport
Math and Literature: A Tom and Jerry Story
We will discuss what math can do for literature and what literature can do for math with special references to the East European approach.
Richard Spindler, SUNY Plattsburgh
Assessing Different Cognitive Tasks Using Bloom’s Taxonomy
Do we want students to learn critical thinking? Yes! Do we want students to think deeply about the content? Of course! But what exactly do we mean by “critical thinking” and “thinking deeply”? How do we measure and assess those tasks? This talk will describe Bloom’s taxonomy, present the author’s efforts to use Bloom’s taxonomy to assess and guide his courses, and planned further efforts.
Matt Thomas and Osman Yurekli, Ithaca College
Divination Process to Explore Ethnomathematics
This presentation will provide an outline of an introductory liberal arts mathematics course where students develop different methods of understanding mathematics as a human creation through looking at the development of mathematical ideas in different cultures at different times. Students experience various social activities such as divination and marking time. They learn how to reflect on broader ideas about how we discover and create mathematical knowledge and understand the world around us. As a demonstration of our ideas, we will present a divination process that originated in Madagascar centuries ago and is still practiced there to this day. Furthermore, we discuss a generalization of the divination process developed in a capstone course.
Jayleen Wangle, SUNY Oneonta
An APOS Analysis of Calculus I Students’ Conception of Continuity and Related Topics
Continuity is a central yet subtle concept in Calculus I, which students often struggle to understand. Dubinsky’s (1991) Action Process Object Schema (APOS) Theory provides a valuable framework for investigating student depth of understanding. I will discuss participants’ displayed depth of understanding of function, limit, and continuity in terms of the constructs described in APOS Theory. One primary finding was that participants’ who demonstrated a strong conception of continuity also showed a strong understanding of the concept of function.
Elizabeth Wilcox, SUNY Oswego
A Quick Intro to Elliptic Curve Cryptography
Ever wonder what elliptic curve (EC) cryptography is all about? This talk will give a super-fast intro to the basic mathematical underpinnings of a super-fast cryptosystem that’s in use in your smartphone and internet browsers right now.
Aside from inspiring you to feel modern and “in the know”, an intro-level familiarity with EC cryptography comes in handy. EC groups, the foundation of EC cryptosystems, are an interesting tidbit, fit for inclusion in an undergraduate abstract algebra class — who doesn’t love an abstract group operation? — and with ties to number theory. We’ll see these groups and learn about some neat features that Sage has for doing calculations in EC groups.
Marlo Brown, Niagara University
Analytical and Numerical Study of Detachment Effects for the Upscaled Porous Medium Biofilm Reactor Model
Starting from the traditional mesoscopic one-dimensional biofilm model we derive a macroscopic model of a simple porous medium biofilm reactor in the convection dominated, laminar regime. The mesoscopic processes included in this model are biofilm growth due to substrate consumption and biomass loss due to cell death and biofilm detachment. The upscaling to the macroscale leads to a stiff quasilinear hyperbolic system of balance laws. In numerical simulations we investigate the role of the mesoscopic detachment description for the macroscopic model. To this end we compare four mesoscopic detachment models that are based on different model assumptions and lead to different mathematical expressions. We find that the particular choice plays only a minor role for macroscopic behavior, both from a quantitative and qualitative aspect. Similarly, we find that the overall reactor performance is rather insensitive with respect to the parameters of the detachment rate expressions.
Special Session “IBL for All” in Action
In the Randolph Lecture, Yousuf George talks about “IBL for All.” To reach all, IBL must be flexible enough to fit various situations, personalities, and constraints. Everyone will develop their own style and make mistakes along the way. In this session, we’ll hear the voices of several instructors in our section who are working toward the “IBL for All” vision. We’ll hear about their stories, about their differences, and about their successes and failures in hope that we can learn something from the community that has formed around us.
Jonathan Cox, SUNY Fredonia
The Joys and Challenges of Implementing IBL in Calculus Courses with a Required Textbook
Research indicates that we learn more from making (or seeing) and then fixing mistakes, rather than seeing tasks done perfectly. Implementing IBL in calculus has been particularly challenging for me, partly due to structure imposed by the mandated textbook that conflicts with my goals. Every day I am reminded how short I fall of being the teacher I aspire to be, and realize ways in which I could lead the class more effectively. I’ll give some examples. Learn from my mistakes and subsequent insights!
Ralph Craig, Utica College
I Don’t Want to Teach You Math
In my sections of Excursions in Mathematics, a course for liberal arts students at Utica College, the content is almost completely irrelevant. On the first day I tell the students, “I don’t want to teach you math. I want to teach you how to think like a mathematician.” We will look at some of the activities used to pursue that goal.
Ryan Gantner, St. John Fisher
Course Notes Transformed into Class Journal
In an IBL class, students often work on a sequence of problems from a set of IBL course notes. They produce solutions to the problems and present them, either orally or in writing (often both). This semester, my Real Analysis II students are tasked with taking the problem sequence and turning it into a “journal” (in the academic sense). That is, they are combining their solutions into a single document, adding transitions from one to the next when necessary, developing a common notation and vocabulary, and developing a sense of “big picture” to accompany the problems. The process forces them to focus on their writing in a powerful way. It also helps them to work as a team in a way I’ve never experienced before, as students are given tasks (author, referee, content editor, copy editor) which rotate as the semester goes on. As with other talks in this session, this talk will give an update on how this process is working.
Ketih Jones and Toke Knudsen, SUNY Oneonta
Engaging Students with the Pell Equation Through Inquiry Using Primary Historical Sources from Medieval India
TRIUMPHS (Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources) is an NSF-funded project to develop inquiry-based modules for student engagement in mathematical topics by working with and interpreting primary historical sources. As a part of this project, we have created a module, “The Pell Equation in India,” which guides students through source translations of two of medieval India’s most important mathematicians: Brahmagupta and Bhaskara II. This module begins with a simple exploration of the Pell equation, Nx2 + 1 = y2, and moves through crucial tools developed by Brahmagupta for understanding this equation, into Bhaskara’s “Cyclic Method” for solving the Pell equation. The student discovery process is guided by tasks which ask students to explore calculations, engage with the primary source, and make, test, and prove conjectures.
Joel Louwsma, Niagara University
Logistics of Student Presentations
In presentation-driven courses, I have students indicate online which problems they are prepared to present in class. I will discuss how this works and how I select problems to be presented and students to present them. I will also indicate how I factor volunteering and presenting into course grades.
Olympia Nicodemia, SUNY Geneseo
To IBL or Not to IBL
I would like to start a conversation among those of us who, like me, are not-fully IBL. Why and when do we use IBL techniques? What is an effective mix for us? Why not all the time?
Matt Thomas, Megan Martinez and Aaron Weinberg, Ithaca College
Using Standard/Specifications Grading to Complement IBL
Standards-based and specifications-based grading methods are gaining popularity as a method for assessing students’ understanding of topics. After a set of meetings outlining a set of both content and process learning goals, three calculus 1 instructors decided to reinvent our course, now in its third iteration. One aspect was to incorporate standard/specifications based grading into the course featuring IBL components. In this talk, we’ll discuss some of the ways we were able to combine these paradigms, as well as some challenges that we faced.