Did ancient cave-people have insights into infinity? How did pre-historic people balance their checkbooks before we had the notion of "number"? Was the Pythagorean Brotherhood really a "brotherhood"? Have you ever prayed to "10"? Does the square-root of 2 really exist? Can you count to 5? Will a randomly selected real number ever equal 1/3? Was Cantor as kooky as his colleagues conjectured? If you've answered "yes" or "no" to any of these questions, then this breathless journey through the history of numbers is for you!

Have you ever gone out with someone for a while and asked yourself: "How close are we?" This presentation will address that question by answering: What does it mean for two things to be close to one another? We'll take a strange look infinite series, dare to mention a calculus student's fantasy, and momentarily consider transcendental meditation. In fact, we'll even attempt to build some very exotic series that can be used if you ever have to flee the country in a hurry: we'll either succeed or fail... you'll have to come to the lecture to find out. Will you be at the edge of your seats? Perhaps; but if not, then you'll probably fall asleep and either way, after the talk, you'll feel refreshed. No matter what, you'll learn a sneaky way to always win at Limbo.

Simple physical systems that obey Newton's laws can be written as systems of differential equations whose solution, for a given initial condition, is unique. In other words, if one can specify the initial position and velocity of every component, then the state of the system can be predicted, in principle, at every instant in the future. Let us call such a system deterministic. Suppose we want to predict the future state of a deterministic system where the initial state must be measured somehow. No measurement is perfect, but suppose that, given a positive integer $d$ and enough time and money, it is possible to build an apparatus to measure position and velocity with $d$ decimal digits of accuracy. Furthermore, assume that the ultimate future state of the deterministic system is one of two simple periodic motions. Would it be possible, given these circumstances and unlimited resources, to correctly predict---say 90 percent of the time---which of the two eventual behaviors one will observe given a suitable measurement of the initial condition? I will describe a deterministic system for which the answer is no.

The purpose of this report is to describe how several seemingly different phenomenon in mathematical reasoning can be seen through one unifying lens with the use of Fauconnier and Turner's theory of conceptual blending. This theory describes how humans reason and learn by combining familiar mental spaces or frames into a new blended space or an integrated network of blended spaces. I use examples of theoretical frameworks from the mathematics education research literature, including data analyzed within these frameworks, and show how they may be reframed using conceptual blending. For this presentation these examples will draw mainly from undergraduate students working to construct a proof that two parallel postulate of Euclidean geometry are equivalent. We will see blending occurring within verbal, symbolic and pictoral reasoning. By explicating these examples I hope to illustrate how conceptual blending may play a role in our understanding of how our students learn and how we may use this knowledge to aid in curriculum design.

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