Math 499 The Foundations of Mathematics and the Acquisition of Mathematical Knowledge Spring 2014

Why is addition commutative but its ‘inverse’ subtraction is not? Why is a equal to a÷b? b

Whyis a ÷c equalto a ×d? Whyisthisstilltrueevenifa,b,canddaren’tintegers? bd bc

What does e + π mean and how can we evaluate it? What is the difference in the meaning of the equals sign between x2 −1 = 0, x2 −1 = (x−1)(x+1), (x2 −1)/(x−1) = x+1 and √x2 = x? What does it mean for a line to be straight? Are there lines that are not straight? In Math 499 we will be addressing these questions and more!

In this class we will explore the foundations of mathematics and how we acquire and process mathematical knowledge. We will revisit K-12 mathematics from the point of view of a mathematician. We will explore the roles of metaphors, models, and definitions. We will discuss the use of symbols and see that even in mathematics their meanings are often contextual. We will compare and contrast proofs and convincing arguments and think about the roles they play in developing and understanding mathematics. We will discuss the relationship between mathematics and our physical world and how we use mathematics to understand the physical world. We will consider various algorithms common in K- 12 mathematics and discuss why and how they work. We also will read and discuss the literature on how K-12 mathematics is taught and how we learn and process that knowledge. Throughout the semester, you will also the opportunity to observe and participate in classes at AUGUSTUS HAWKINS High School. This is a new school with a modern curriculum implementing an initiative called the Algebra Project.

This class has no prerequisites. In particular, it is not necessary to have taken any college level math classes; you are only expected to know how to count (albeit fairly well!). However, students must be willing to engage with the material at a mathematically sophisticated level. There will be very little lecturing. There will be a lot of discussion, group work, and both oral and written presentations. This class will be valuable for math majors, anyone with an interest in teaching mathematics, and sociology and psychology majors interested in the science of learning.

David Crombecque

Lecturer

Mathematics Department

crombecq@usc.edu

Course offering: has a service learning component

relevant majors: mathematics, sciences, psychology, sociology,