## Bulgarian Math

## Archimath Publishing LLC

The Bulgarian Math series focuses on building a truly solid foundation of mathematical knowledge and skill, with deep explorations into algebra, geometry, and general logic and problem solving. In addition to the core content, these textbooks place a strong emphasis on communicating mathematics clearly, doing computations efficiently, and using common sense to check if answers and results make sense. The series has been written, translated, and edited by a group of talented mathematics educators with a focus on developing strong mathematical intuition and preparing students for high level mathematics.

This affordable textbook series is structured to provide a smooth experience for students and teachers alike. Each set consists of a textbook and a workbook, providing a full math curriculum for one semester. In the textbooks, each lesson fits onto a two-page spread, including the practice problems, making it easier to focus on the topic at hand. The workbooks provide space for students to try the solved examples from the reading on their own, plus additional worksheets to further student understanding. In both books, the exercises contain problems that are extremely similar to the solved examples, as well as variations that require critical thinking skills to transform difficult problems into more manageable ones.

These books cultivate a strong relationship between the students and the material; by using helpful color coding, symbols, and by providing appropriate scaffolding for students throughout.

They have been used with great success in :

- Tehiyah Day School (El Cerrito, CA), a private non-denominational Jewish K-8 school (grades kindergarten through 8th grade) http://tehiyah.org

Head of School: Deborah Massey

- “Math Taught the Right Way”, a new weekly program in conjunction with the Berkeley Math Circle-Upper at UC Berkeley (grades 5-12) http://mathcircle.berkeley.edu

Director: Prof. Zvezdelina Stankova

– Geffen Academy, a new middle/high school at UCLA, to commence in Fall 2017 (grades 6-12) http://geffenacademy.ucla.edu

Head of Math Department: Dr. Oleg Gleizer

Many students who expand their understanding using Archimath see growth in areas of algebra, geometry, and creative problem solving. It is the best preparation for middle and high school students to tackle college mathematics and beyond. Materials that should be in school, but are often not!

**Our Program**

is systematic, excellently organized, and easy to use;

is comprised of 50-minute lessons, each taking only 2 pages in the textbook;

has separate worksheets for class work and for homework;

has an introduction chapter that smoothly leads into the material each year, review lessons and tests of various lengths and difficulty for each chapter, and an extensive review chapter that summarizes and further tests the covered material at the end of each year;

is ambitious, and gratifying to both teachers and students and has been successfully implemented in several schools in the California Bay Area.

**The Value of a Teacher**

As with any textbooks, the present textbooks are just that: a basis for learning. They will come alive in the classroom only through the teacher. Let us not deceive ourselves: there is no universal panacea for the middle school math problems in the U.S. And although the present textbooks have been hailed by pilot parents as:

- the best textbooks they have ever seen;
- the most suitable for their children;
- a resource to easily follow and learn from at home if a child is absent from school or did not understand something in class …

…still, a mediocre teacher will do a mediocre job even with the best textbook in hand. The fact that many middle school math positions are filled by non-specialists in math is alarming and is one of the major reasons for the current general downfall of U.S. middle school math. A non-specialist does not have an “eagle view” of the full middle and high school math curriculum. Although locally, he/she might be able to teach some of the material well, a lot of damage is done by a teacher’s limited knowledge and lack of understanding of what is important in the long run, even if hard to teach. The situation is similar to dropping by parachute the whole math class into the middle of a jungle, giving each student a flashlight, and waiting out- side the jungle to see who will come out alive. A better chance of “survival” will be ensured if the students are given a map of the area, taught how to read and use it, and equipped with the necessary tools, knowledge, and skills to efficiently cope with the difficulties that will inevitably occur on the way out of the jungle. And the teacher has to be there, with the students, every step of the way,… not just waiting for them outside of the “danger zone.” I personally never thought of school math in this “jungle vision” until my own children experienced several different U.S. math programs and the image of the “jungle” resembled more and more powerfully the reality of pre-college math education.

More often than not, non-specialist math teachers in middle and elementary school, for a variety of reasons, decide to change the order, depth, and emphasis of the math program they are teaching from. Not knowing the “big picture” of how mathematics will develop over high school and then college years (and perhaps, not caring about this big picture, since it is not part of their day-to-day job duties), their actions unintentionally destroy the logical structure and goals of the program, whether it is a good program or not. (See earlier discussion of this situation in Section 5: The Temple of Mathematics.) Fortunately, the new math program:

Is straightforward to follow: every lesson is on 2 pages only! (Most importantly!)

Ensures accountability from the teachers.

The program is so crystal clear that anyone can track where the students are and what they have covered at any time during the school year. By “anyone” we mean parents, administrators, and random visitors alike. It is essential, though, that a copy of the textbook resides at every student’s home: there is nothing to hide in the program, while there is so much to learn from reading the textbook at home!

When each lesson in the textbook is on 2 pages only, with 6-to-10 exercises, and when the concepts are practiced in the next lessons and incorporated in new situations for better understanding, it is very hard to justify staying on one particular lesson for a week or so (and killing everyone’s enthusiasm in the process. Due to the natural transparency of the material, it is impossible to hide behind phrases like “too much,” “too hard,” or “not well organized” “to be covered all in one year”. And the program does depend on the steady and full completion of the material in every middle school year, in order for the next year to be successful.

A Good Choice for a Teacher for this curriculum would take into account:

The Mathematical Level A teacher with a solid math background who can see several steps beyond middle school, e.g., the high school curriculum, and can recognize:

- the math concepts,
- types of solutions, and
- features of analytical thinking that are indispensable and must be taught and reinforced in middle school, despite how hard it might be to teach them to a young audience unaware of them.

The Pedagogical Level A teacher who can quickly adapt to new situations and turn them into an educational advantage for the students. A teacher who is constantly attuned to the weaknesses and strengths in students’ background and skills, and who emphasizes accordingly features of the curriculum in a balanced way to match and improve the classroom situation. A teacher who can communicate mathematics to young students.

The Level of Dedication A teacher who is dedicated to the well-being of the students and who will sacrifice his/her own desire to “shine on the career ladder” to providing a robust math education to his/ her young charges. A teacher who believes in and is ready to undertake and follow closely the new curriculum; who is ready to face the mathematical and cultural difficulties that will inevitably arise in introducing such a program. A teacher who will persevere.

**The approach and methodology**

If it were up to us, up through 11th grade high school students would study a good balance of Algebra and Geometry topics, mixed with some Number Theory, Combinatorics, Probability, and Statistics. And only in 12th grade, with a solid preparation and math maturity under their belts, would they venture into advanced concepts such as limits, continuity, and derivatives, which comprise about half of Calculus BC. Thus, during their senior year, they would study thoroughly and deeply:

- sophisticated definitions like the e-d definition of limit,
- and powerful statements like the Intermediate and Mean Value Theorems, and the occasionally “dangerous” L’Hôpital’s Rule,
- along with their proofs and relevant problem-solving techniques. A rigorous treatment of this material would take time and sustained effort from both teachers and students, and a whole year may be barely enough.

But this would be “the right prequel to college math courses” and “the right way to learn Calculus.”

Our students would thus march into college with fortified math knowledge and skills, and a clearer understanding of how mathematics works, better prepared for success in their college freshman math courses.

## Why us?

## Benefits

### Еach lesson includes:

• Conceptual examples that illustrate new concepts and demonstrate solutions of related problems.

• A clear summary of new theories, formulas and algorithms.

• Answer key at the back of the textbook.

### Optimized for Independent Learning

Our program is comprehensive and easy to understand, written in a straightforward and simple manner. Suitable for students who are motivated learners and are ready to explore new material, as well as providing a resource for kids working with parents or tutors. The lessons are written so that children can read them.

### Straight to the Point and Self-Contained

Everything is crystal clear: each lesson is on exactly 2 pages. No need to search for other sources. The textbooks have all of the background material that one may need.

### Plenty of exercises

Each textbook lesson includes up to ten exercises to strengthen understanding and develop skills. Problems always reinforce previous knowledge without needless repetition, by introducing new applications of familiar topics. "Spiral" curriculum where each new grade revisits topics that extend the students understanding.

### Learning to Think Deeply

This program supports learners' deep conceptual understanding, and develops the ability to transfer learned skills to new mathematical contexts.

### Logic, Proofs and Problem-solving education

A key strand of the curriculum is learning to distinguish correct and flawed reasoning, and to express one’s mathematical ideas smoothly, convincingly and logically, using numbers, drawings, and narratives.