Life of Fred: Five Days of Upper Division Math
For the dedicated math student who wants to push the envelope, here you go. Join Fred at Kittens University for the summer session at least for the first five days in which he lectures (from 4am-8am) on, respectively, Set Theory, Modern Algebra, Abstract Arithmetic, and Topology. Prerequisites for this course include more than a passing knowledge of algebra as well as a course in Geometry that taught and required proofs. Prior algebra instruction should have included:
- Union of sets
- Math induction
- Associative property
- One-to-one functions
- Inverse functions
- Multiplying matrices
As you can see, this includes most high school seniors. However, final prerequisites I would add are an appreciation for math, an ability to think outside the book, ability for creative thought, and a willingness to take a peek at "real" math. I got this opportunity when I took some upper-level university math courses. There is truly a "magic" and beauty to math (which Prof. Schmidt alludes to in the introduction to this book) that most people never get to experience. If your student would like to experience the same, I heartily recommend this summer session with Fred.
Unlike other LOF books, there is no real storyline outside of Fred's lectures. There are fun puzzles (which look suspiciously like proofs, for the most part) to solve - 139 of them that require thought, creativity, and, for some, dedication. Some are easily solved, others are going to take some time. Solutions to all exercises are in the back of the book. If your student loves math or just wants to get a glimpse of higher level math, this course is a great introduction to math beyond calculus.
Rather than a course of study, this book is a sampler of advanced math for students who love math and Fred and want to preview courses they may take later.
Upper division (college junior/senior) pure math is much different than calculus. No "word problems, no formulas to memorize, no concrete applicationsjust puzzles to solve. Instead of learning procedures, students create definitions, theorems, and proofs.
These are the first five days of Fred's teaching set theory, modern algebra, abstract arithmetic, and topology. Each of the 139 assignments/puzzles/questions that he gives his students calls for creativity rather than doing drill work. Some of these can be done in a minute. Some will take several hours to complete. They are all meant to be enjoyed.
The first day of set theory: cardinality of a set, set builder notation, naive set theory, modus ponens, seven possible reasons to give in a math proof, the high school geometry postulates are inconsistent, the proof that every triangle is isosceles, normal sets.
The first day of modern algebra: definition of a math theory, six properties of equality, formal definition of a binary operation, formal definition of a function, definition of a group, right cancellation law, left inverses, commutative law.
The first day of abstract arithmetic: circular definitions, unary operations, the successor function, natural numbers, the five Peano postulates, mathematical induction.
The first day of topology: topology is all about friendship, listed and counting subsets, open sets, the discrete topology, the three axioms of a topology, models for a topology, open intervals.
By the fifth day Fred will have covered the Schrder-Bernstein theorem (set theory), proved Lagrange's theorem for subgroups of any group (modern algebra), defined the real numbers based only on the concept of "adding one (abstract arithmetic), and explored continuous images of compact sets (topology).
Life of Fred (LOF) is an unconventional series of math texts that seem accessible and friendly, especially for students who dislike traditional programs. According to Dr. Schmidt, the author, this series is designed to teach you the math you need to know without repetition, redundancy, and a multitude of problems to work. LOF follows the storyline of Fred's life while incorporating solid math concepts and skills. Motivated or independent students will appreciate this series, as well as gifted math students who might need a challenge. Written to the student and intended to be self-teaching, the author prefers that students use these with very little help from you, so they can learn to study and understand on their own. Even the solutions (found in the text) are addressed to the student. In fact, there's a lot of actual instruction in the solutions, which students should read after trying to solve problems on their own. Math in a story context can sometimes make more sense to students than stand-alone math concepts - especially if it's an entertaining story! Students do two books per year up through pre-algebra; beginning with algebra, one book per year.