## Archimath Blog

1. Mission Impossible

Every year, for the past 20 years, while working* with 500 students in grades 5-12 at the Berkeley Math Circle (BMC), I have been asked over and over again by parents, students, and teachers alike:

”Can you give us good textbooks for our middle and high school math curriculum? What we have is not working and it hasn’t worked for years!“

And for these 20 years I could think of only two sources that I would also give to my own children: my old Bulgarian middle school textbooks, and the Art of Problem Solving books (AoPS). But neither of these choices would have worked, despite the fact that they are, in my opinion, among the best middle school textbooks in the world. Indeed, they are written for advanced audiences as measured by U.S. standards:

The very first sentence in my old Bulgarian 6 th Grade Algebra textbook would stump many of my freshman college students: ”Since we have worked [in 5 th grade] with the set of rational numbers Q and are familiar with its properties, we move on.“ And without further ado, Lesson 9 went on to studying the subtle differences between functions and relations!

My old Bulgarian 6 th Grade Geometry textbook did not lag behind. On page 42 it asked:

”Is the relation ‘is tangent to’ on the set of circles in the plane reflexive, symmetric, or transitive?“

Just to explain what the question means to an unsuspect-ing, typical 6 th grader would take a whole lot of time! Although the AoPS books are a wonderful resource, they are not designed as “textbooks” in a regular math class environment. I tend to recommend these books for my more advanced Math Circle students, who are guaranteed to have extra support outside of a typical school classroom.

1. The Find

….That is, until I asked my mom to send me some recent Bulgarian math textbooks, to teach my own kids in the U.S. She found the most popular textbooks in Bulgaria: the present series from “Archimedes.” When I opened them, I was immediately taken by a main feature: Each lesson is on exactly 2 pages!

And each lesson consists of:

• Several examples of the new concept(s) that illustrate how to understand the new material and how to solve related problems.
• A clear summary of new theory, formulae, algorithms, and consequences to be learned.
• Up to 10 exercises that differ from each other and that further strengthen the understanding and develop the skills of the reader.

* I am a Teaching Professor of Mathematics at UC Berkeley. Originally from Bulgaria, I went through its rigorous mathematics curriculum. I founded BMC and has been running it since 1998 to help pass the math enthusiasm on to the younger generation in the San Francisco Bay Area.

The textbooks are written in a straightforward and simple manner, capturing the elegance and robustness of mathematics itself. There is no “hide-and-seek”: everything is crystal clear; one look at a lesson suffices to see what it will teach you; there is no need to look around on the Internet for other sources: the textbook, workbook, and problem collection are all one needs for a successful course in middle school mathematics.

1. Origins

This middle and high school math program is based on one of the leading programs of today in Eastern Europe, developed and implemented by the well-known Bulgarian mathematician and educator Professor Georgi Paskalev. His relatively new mathematical program specifically targeting middle and high schools in Bulgaria has conquered 60-70% of the Bulgarian education market during the single decade since its conception.

The new program:

has partially departed from the traditional “old school” Bulgarian program, and has crossed boundaries to meet the standards and demands of a global educational market, such as in the U.S.; yet, has preserved the intrinsic pedagogical and mathematical values of the old curriculum.

Quite appropriately, the company founded by Professor Paskalev is called “Archimedes”: its publications live up to the name of the great Greek mathematician, scientist, and inventor from antiquity.*