Mathematical knowledge for teaching examples in pre-calculus: a collective case study
The purpose of this collective case study is to examine mathematical knowledge for teaching examples in pre-calculus. The instructors involved in the study were experienced graduate teaching assistants who were teaching their course for the third time who had been identified by their department as good teachers. Utilizing a cognitive theory approach, I analyzed video recordings of enacted examples. The central question that guided this analysis was: What is the mathematical knowledge for teaching examples in pre-calculus? The goal of this study is to examine postsecondary mathematical knowledge for teaching from the perspective of practice, instead of relying on existing frameworks. As a result of this study, the author developed a model of mathematical knowledge for teaching examples in pre-calculus that includes knowledge of connections, representations, students, instruction, and content when enacting high cognitive demand examples.
Turning hard problems into (infinitely) many easy problems
In mathematics, we tend to study only a few classes of functions. Examples include polynomials, rational functions, trigonometric functions, exponentials, and step functions. Why is this? Certainly other, more complex functions appear in nature. Could it be that any other type of function is too hard to study in generality? A pragmatic answer to this question is the following: the “complicated functions” are lots of simple functions in disguise.
This talk is primarily about the approximation of complicated functions using simpler functions. We will begin by reviewing the types of approximations learned in calculus – tangent line and Taylor series approximations. We will discuss what we mean by approximation, and the flaws that these techniques have. We will then move on to talk a bit about a different type of approximation, and how linear algebra can solve many of these problems. For better or for worse, this will take us to a world of infinite dimensional spaces. Applications of this approach are vast, including machine learning, signal processing, and a formulation of quantum mechanics to name a few.
The card game SET and tic-tac-toe on a torus
SET is a card game played with a special deck of 81 cards. Each card contains four features: color (red, purple or green), shape (oval, squiggle or diamond), number (one, two or three) and shading (solid, striped or outlined). The game begins by setting out 12 of these cards and the goal of the game is to quickly find a set of 3 cards such that each individual feature is completely uniform or completely different. If no such set exists with the initial 12 cards then 3 additional cards are added until a set can be made. This leads to a natural question. What is the minimum number of cards that must be played in order for us to be certain a SET exists? We will answer this question by relating a set of cards to a configuration on a tic-tac-toe whose edges are glued together and counting something in this new setting. The talk will be accessible to a general audience.
Removing excess surface area with a little soap, water, and the Divergence Theorem
Suppose you want to find a surface that will enclose a certain amount of volume while having as little surface area as possible. You might be quick to suggest a sphere. But how do you know a sphere is the best shape? And what if you have to meet other conditions for the surface as well? Maybe you need that surface to contain two separate volumes or maybe you want it to have a specific boundary. As it turns out soap bubbles and soap films solve these problems naturally. In this talk we will discuss how we mere mortals can learn from the bubbles and use a little Calculus 3 magic to actually prove which surfaces are best.
Fractional derivatives on a discrete domain
In your calculus course, you learned a lot about derivatives. But how do we make sense of taking a half derivative of a function? L'Hopital asked this very question in his 1695 letter to Leibniz, to which Leibniz responded "It will lead to an apparent paradox, from which one day useful consequences will be drawn." Since then, fractional calculus has developed to extend derivatives to be of any fractional order in many different ways. (In fact, we call them fractional derivatives only for historical reasons; we can make sense of a square-root-of-2-order derivative or a pi-th order derivative.)
We will focus on derivatives on a discrete domain, i.e. differences. We will start by investigating whole-order differences and observing various results which have analog counterparts in calculus. Then we will develop the Caputo fractional difference and consider its properties. Finally, we will describe problems involving the Caputo fractional difference that are analogous to problems in ordinary differential equations.
Triphos: an alternative coordinate system
In this talk, we investigate characteristics and properties of the Triphos coordinate system, an alternative, two-dimensional coordinate plane consisting of three axes evenly spaced 120 degrees apart. We will examine similarities and differences between the Triphos system and the Cartesian coordinate system, explore its algebraic and geometric properties, and discuss some advantages and applications of this nonconventional coordinate system.
Introduction to cluster algebras
An n x m matrix is said to be totally positive, if all of the n x n minors of the matrix are positive. Then in the case of 2 x m matrix, there are m choose 2, or m(m-1)/2 minors to check before determining whether the matrix is totally positive. As m gets larger this number grows fast. However, we only really need to check 2m-3 minors to determine the total positivity of 2 x m matrix. We'll extend the idea of the proof to find the generators of cluster algebras given by a quiver. The cluster algebra was introduced by Fomin and Zelevinsky in 2002 and it has been studied widely since then. The type A_n cluster algebras have been studied well; we can find the cluster variables of these cluster algebras in multiple ways including T-path and snake graph.
This title is false: Hilbert, Godel, Turing and the beautiful futility of mathematics
In a well known XKCD comic, scientific fields are arranged by purity with mathematics all the way at the end labeled "more pure." We love math because it is rigorous and clear cute; there's a right answer and we know how to find it. We deal with truth and certainty, right?
In this talk, we examine the foundation of these claims. Does math rest on a solid bedrock of truth, or is it turtles all the way down? What does it mean to be "true" anyway? We will investigate the very heart of mathematics and find it is not the well-oiled machine we pretend it is. After pulling back the curtain and seeing math for what it really is, will you still find it beautiful?
An exploration of the noncommutative world
Quantum mechanics is the best theory we have to describe micro-scale interactions in our universe. Many consider the mathematical language of quantum mechanics to be "non-commutative algebra." You might ask, "Why non-commutative?" Well, you may have heard of the Heisenberg Uncertainty Principle, which asserts that we cannot simultaneously know where a quantum particle is and how fast it is going. This mathematical principle is manifested in the strange phenomenon of measuring the position of a particle and then its momentum, and getting a totally different result than if the particle's momentum had first been measured and then its position. In this talk, we will prove the Heisenberg Uncertainty Principle in a certain case using methods from calculus, and we will explore several more abstract geometric settings where commutativity fails, and non-commutativity prevails!
Telling physical objects apart using mathematics
Most people think of math as a way to use numbers or letters to represent or model real world situations. However, in some branches of mathematics, the focus shifts to actually dealing with the physical objects themselves, such as a donut, a coffee mug, or a knotted up pair of headphone wires, and using math to attach a number, variable, or certain property to the object to be able to differentiate the objects from one another. In this talk we will give a very light introduction to what this field of math (usually called topology) is all about without getting too bogged down by the numbers and details. This talk will be geared towards undergraduates, and no prior math background will be assumed.
Prime vertex labelings of some families of graphs
A simple n-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers 1, 2, 3, ... , n such that adjacent vertices have relatively prime labels. There are many known families of graphs that have a prime vertex labeling (and some that don't). I will present previously unknown prime vertex labelings for new families of graphs, some of which are special cases of Seuod and Youssef's 1999 conjecture that all unicycle graphs have a prime labeling. No prior graph theory knowledge will be assumed.
Knot theory and beyond
Knot theory has been of interest to mathematicians since the late 19th century, yet knot diagrams are simple enough to be understood by anyone -- mathematician or otherwise -- who sees them. While this doesn't mean knot theory is simple or that knots are easy to work with, you might ask: what other areas of topology deal with complicated objects that can be represented by relatively simple diagrams? This talk will touch on a few such areas, and naturally will involve a number of pictures.