Introduction to the program

Archimath Blog

Origins

This middle 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 targeting specifically 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:

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.*

Purpose

The present math program extends the mathematics taught in elementary school and completes it to a comprehensive K-8 math pro- gram that prepares students:

to enter any high school with confidence in their
to be successful at any level of high school math classes;
to develop maturity and a life-long relationship with mathematics that will distinguish them among their peers; will serve them well in any profession and in any endeavor in math skills and knowledge; which they chose to engage.

The Temple of Mathematics

The math education of every child is like a building: every year we hope that a new floor emerges on top of the previous floor. Unfortunately, more often than not, our children come under damaging (even though, perhaps, well-intended) actions of math educators on any floor of this building. For example,

What happens when a teacher decides to spend 3 months reviewing single-digit multiplication in 4th grade? Well, the 4th graders become really skilled with their multiplication tables, and, say,
a 6-foot thick wall of “single-digit multiplication” is erected on the 4th floor of the building. Alas, time is not stretchable, and such educational decisions inevitably cause:
some other wall on that same floor to be thin or non-existent; e.g., a “wall” of word problems, logical thinking, or a rich geometric vocabulary.

As a result, when in 5th grade the next teacher tries to build “walls” upon such previously expected but non-existent or weak walls, not surprisingly,

the structure collapses! And in the ensuing debris, time is wasted figuring out what and why it happened. Worse, a student may lose confidence in his/her math skills, in the teacher’s abilities, in the integrity of the math program, and, sadly but not infrequently, in mathematics itself.

What may strike us as even more dangerous for the young developing mind (yet, still well- intended) is the desire of many text- books and some teachers nowadays (perhaps, under pressure from school administration or for other reasons) to occasionally show off via:

extracurricular activities that are un- supported by and incompatible with the math building erected up to that moment. A typical example is an “enrichment” section on:
Series! This topic, ordinarily covered in a college Calculus course, has made its debut in standard, well-known U.S. 6th grade textbooks, alongside computer codes that intend to demonstrate the “growth” of the “partial sums of the series”.

extracurricular activities that are un- supported by and incompatible with the math building erected up to that moment. A typical example is an “enrichment” section on:
Series! This topic, ordinarily covered in a college Calculus course, has made its debut in standard, well-known U.S. 6th grade textbooks, alongside computer codes that intend to demonstrate the “growth” of the “partial sums of the series”.

Let us reason logically here.

How can one truly understand why the sum 1 + 1/2 + 1/3 + 1/4 + 1/5 + … adds up to infinity, but the sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + … does not, before having mastered, say, the distributivity laws?! Although U.S. students may be able to recite from memory these distributive laws (e.g., a . (b + c) = a . b + a . c), by and large, they will have no clue when, why, and how to apply these laws, so much more efficiently.

Thus, we ask again: what effect will such a “fancy” and untimely introduction of high-level math ideas have on the budding young mind? To put it bluntly, the situation will be equivalent to hanging a fancy chandelier on a non-existent ceiling. Imagine the “aftermath” of such an action, the loss of time, energy, and confidence.

And we are not talking about the precocious youngster who can handle with ease and excitement any bombardment with advanced math material, whether on series, in combinatorics, abstract algebra, or other such topics! We have plenty of such young people at the Berkeley Math Circle and observe this phenomenon on a weekly basis all the time.

We are talking about the typical U.S. student, whose primary or only source of learning math is through math classes in school.
The math structure for this student will be jeopardized by such disorderly and out-of- context teaching.

Just like when learning:

To play the piano, one might first want to learn to read music before attempting Rachmaninoff’s Piano Concerto No 2’;
To skate, one ought to practice a figure-8 before performing an Axel jump;
To write well, one should first concentrate on sentences and paragraphs before producing a literary analysis on, say, “The Realism vs. Local Color in Mark Twain’s novels”…

… So should there be order, balance, and harmony in everything we do, teach, and learn. We cannot lower our standards when teaching mathematics to the young minds. On the contrary, we must be especially careful because of the hierarchical structure of mathematics. And thus, the present textbook series has found the golden ratios between:

mathematically sound
and pedagogically sound

and

the practical and the inspiring

The textbook series promotes at all times

First and foremost:

Learning to Think Deeply:
Make connections with previously learned material
Diversity of Ideas:

– Via multiple solutions to the same problem.

Courage:
Not being afraid to make mistakes and to learn from them.
Simplicity and Elegance:
In anything we do, especially in mathematics.

These are not just catchy, “politically correct” phrases intended to sell more copies of this textbook. They truthfully describe this math program and help build a solid math structure, with ad- joining “walls, windows, and doors,” through which to move from topic to topic and make connections between various concepts, formulas, and ideas.

For someone who has not worked deeply with mathematics before, the words love, creativity, and excitement are usually light years away from having anything to do with math. Yet, here they all are, in a testimony of a pilot 6th grade parent, leading us to the next topic how a student interacts with and feels about mathematics.

Relationship with Mathematics

With some notable exceptions, especially in Geometry, the actual math contents in these textbooks does not “look” much different from just about any standard U.S. middle school textbook. After all,

A linear equation 5x- 7=11 looks just like a linear equation, no matter if it is displayed in a U.S., Uruguayan, or Bulgarian textbook.
The Pythagorean Theorem will still say a2 + b2 = c2 for the two legs and the hypotenuse of a right triangle.
A shortcut to finding out if a number can be divided by 3 (without remainder) will still be to check if the sum of its digits is itself divisible by 3.

… because mathematical truth is universal.

Beyond the contents, two fundamental questions that a good teacher must ask about any textbook are:

Question 1: How are the math facts presented? In what order and what style? Is there a balance between computational (algorithmic) parts and theoretical (abstract) parts?
Question 2: What relationship between the student and the material is developed? The deeper the relationship, the more lasting and more useful the math knowledge.

This textbook series creates the environment for a strong bond between the student and the material,

by building a robust Temple of Mathematics:
always depending on previous knowledge and reinforcing it via new applications and without needless repetition;
always looking ahead and linking current material to new things to discover;
and by a gradual climb:
in the expectations of quality, depth, and rigor of student work;
in the difficulty and breath of math content.

At any time throughout the school year the place and importance of the concepts, ideas, and techniques studied are clear within the “bigger picture” of Mathematics. Unfortunately, just as described on the previous pages, meddling with textbooks and curricula is a wide-spread phenomenon in the U.S. middle school math education! Here are four sure ways to ruin the structure and purpose of any math textbook, whether good or bad:

Omitting sections.
Changing the order of the material in the textbook.
Cluttering lessons with hand- outs from other curricula.
Sacrificing the depth and teaching to the answers!…

Remember that no one wants to climb up a flimsy structure! As a result, over time the top floors of the Math Temple will become uninhabitable if the actions in the red box run amok in math lessons. Instead, the deep thought and math wisdom put into creating these textbooks must be respected, or there is no point in starting with this program. Thus, the present textbooks must be followed just as systematically as one studies a new language, violin, or chess.

A Cultural Shift

Several features of the textbooks, which are typically ignored or de-emphasized in standard U.S. middle school curricula, represent a cultural shift in how mathematics is viewed and studied in the U.S., but which are very well-known around the world and have been adopted by many countries for decades, if not centuries.

Reading Mathematics.

Every math problem has “words” – these are concepts expressed in different ways, whether as numbers, standard everyday words, or by visual aids such as diagrams and figures. Emphasizing the different forms of “words” as a regular part of math language and math communication and learning to read, interpret, and write in “words” is a major part of the new curriculum.

Logic and Communication.

Learning to know correct from flawed reasoning and being able to ex- plain one’s mathematical ideas smoothly and convincingly to others will represent a gradual move towards a more mature relationship with mathematics.

Writing Mathematics.

“Showing your work” is only the beginning to understanding how math really works. All problems have answers in the back of the textbook. Bringing back just an answer on a homework problem from the textbook will be worth no credit. The explanation that leads to the answer will be what counts in our study of middle school mathematics. Even harder than learning to correctly interpret problems is learning to consistently write solutions in a correct, complete, and clear way.

Multiple Solutions.

Although there is usually (but not always!) only one correct answer to a math problem, the beauty of mathematics is that there may be different solutions leading to that answer. These are not some “subjective opinions”; rather, they are objective mathematical creations that obey the laws of Logic. Students will learn that each solution (and even each incorrect attempt) has its value and usefulness in the long run and that one needs to open up to others’ ideas and ways of thinking as a path to enriching oneself, to becoming more proficient and, ultimately, wiser.

Efficient Solutions.

Yet, among “all the roads [that] lead to Rome,” there might be a shortest or an easiest to follow. The ability to see “the big picture” and pick out the most efficient solution is a skill developed over a lifetime. And we start on it in this very textbook series.

The Pillars of Mathematics

There are three pillars of mathematics: Algebra, Geometry, and Logic/Problem Solving. They must be present at any stage of pre-college math education.

Algebra

There is so much fuss to distinguish between Pre-Algebra, Algebra 1, and Algebra 2… when, in fact, there is no such thing as “Pre-Algebra” as a math subject, nor are there “Algebra 1 or 2”! In a well-designed and well- executed K-12 program

Arithmetic gradually turns into Algebra naturally leads to Real Analysis

Alas, the typical U.S. student:

– hops along uneven and vaguely defined passages between Pre-Algebra, Algebra 1, Algebra 2, Pre-Calculus, and Calculus;

– experiences a chop-chop approach of poorly matched Algebra curricula from grade to grade;

– is taught to be more concerned about memorizing algorithms and passing test benchmarks, than what is truly important:

– making connections and fitting the many pieces together in a giant “jig-saw puzzle” to see the “big picture”; and little by little, building a solid Pillar of Algebra in his/her Temple of Mathematics.

The Missing Algebraic Link

Why is this happening?

The major Algebra mishaps in U.S. middle schools are due to events that have already occurred in elementary school, and NOT necessarily due to poor arithmetic skills. Expecting that a middle school student will leap from arithmetic operations with numbers to algebraic operations with symbols is asking for trouble. There is a whole array of missing intermediate steps, to which we refer as “Algebra with Numbers” and which can and should be started very early in the learning process.

The Geometry Neverland

In K-8 grades Geometry appears sporadically here and there, with few (if any) connections to the rest of the math curriculum.

Later, in U.S. high school curriculum, there is one intense year of Geometry, during which:

No other math courses are studied. This in itself is a recipe for trouble: the few connections between Geometry and the rest of mathematics that were hinted before in the math curriculum are mostly forgotten during that year. Geometry is studied in isolation!

4 years worth of Geometry are compressed in one year and they are presented to, by and large, students unprepared for such a huge single “dose” of Geometry. Unlike making bread, there is no time for Geometry knowledge to “rise.

Proofs are introduced through this Geometry course, often in the notorious “2-column format”. Truly powerful Geometry theorems are not reached. As a result, many students not only do not learn how to do proofs, but also… They end up, by association, disliking Geometry itself!

Question: Why talk about a high school Geometry course in the midst of a middle Because everything starts at an early age and builds up over the years, regardless of whether it is Algebra, Geometry, Logic, playing an instrument, or learning to write excellent essays. And if we do not have the vision of what awaits the young learner in a few years from now, we will be doing a big disservice to young learner, and we will be cutting off his/her chances to succeed in the future school curriculum?

Knowledge and skills in Geometry cannot be amassed “miraculously” in a year. Geometric intuition develops over time, not instantaneously. To package and present several years worth of Geometry in one course to the untrained mind, and to top it off with introducing formal proofs (a hard topic even in a college curriculum) inevitably leads to a poor outcome.
Worst of all, by the time the high school Geometry course comes, “the train is already gone”: it is too late to repair the damage that was systematically done over the K-8 years to the Geometry Pillar in the Temple of Mathematics. Consequently, the young budding math mind has missed the creative and formative time in middle school for a fundamental geometric development.

The truth again: Geometry has been sacrificed in favor of

Arithmetic in elementary school;
Algebra in middle school, and
Pre-Calculus/Calculus in high school.

Insufficient Geometry knowledge and skills will reverberate through- out a person’s whole life, including college and post-college career.

The Geometry Pillar will never be built with the strength and height as it should be in order to hold up a robust and elegant Temple of Mathematics.

The lack of serious and consistent Geometry in early grades makes it impossible to overturn the “algebraic dominance’’ in the K-12 curriculum. Can this be changed? Can Geometry “Neverland” be replaced by a healthy symbiosis between Geometry and the rest of mathematics? Let us look at what could have been.

A Personal Lamentation

Or … the Golden Geometry Days of My Childhood

From my 25-years of experience teaching just about every undergraduate math course in college, the most mystifying formulas to my students, the least well understood, and (it goes without saying) the least frequently proven in pre-college years are the trigonometric ones. During the socialist times in Bulgaria, trigonometry was part of the standard 8th grade Geometry curriculum; the theory behind such formulas was systematically studied, and the formulas themselves were rigorously proven via geometric or, as needed, algebraic methods and applied to various problems for years in a row. In fact,

Starting in 6th grade we had two separate textbooks: one in Algebra and one in Geometry. We had separate home- work, tests, and two grades: one in Algebra and one in Geometry. During the week, we alternated studying: Algebra on MWF and Geometry on TTh.

Most importantly, we had a day between lessons in the same subject, which allowed us:

to think deeply about what we studied in class;
to try various
approaches on the homework problems;
and ultimately, to make the Algebra and the Geometry material “our own”.

Although in two formally separate subjects, our studies of Algebra and Geometry were synchronized and mutually enriching. A constant and deep relationship between Geometry and Algebra was reinforced at all stages of pre-college education: by using geometric facts to supply algebraic reasoning, and by using algebraic formulas to solve Geometry problems.

The Geometry Pillar of Mathematics was gradually built upwards throughout every single grade. There was no week 10th grade Solid Geometry advanced topic without Geometry, especially from middle school onward. To this day, I distinctly remember the yearly Geometry themes that we studied, starting in 6th grade:

6th grade Plane Geometry start of proofs; 7th grade Circle Geometry with proofs; 8^th grade Trigonometry with proofs; 9^th grade Vector Geometry with proofs with proofs; 10^th grade Solid Geometry with proofs; 11^th grade Elective Geometry advanced topic

Former Bulgarian

Geometry Curriculum in Middle and High School

Every theme automatically included applications of the Geometry theory and techniques to problem solving and contributed to a balanced and rich math curriculum. This is the ideal symbiotic model of Geo-Algebra that I would like for my children and grandchildren. Unfortunately, high school math curriculum in the U.S. is driven, by and large, by college requirements and, let’s face it, earning “bonus entrance points” to the more competitive universities. Hence, it is ultimately a Calculus-driven curriculum, regardless of whether students complete Calculus courses in high school, or not. In turn, middle school math curriculum aligns itself with high school priorities. Backtracking, the pre-requisite to Calculus is Pre-Calculus, and the pre-requisite to Pre-Calculus is Algebra. Thus, Algebra “wins” by default, due to an established Algebra-driven and Algebra-dominated environment.

A Reality Check: Geo-Algebra in the U.S.?!

It is unrealistic (although not entirely impossible) to imagine that the pre-college curriculum in the U.S. can quickly change so much that it will embrace Geometry as an equal counterpart of Algebra, will split textbooks and classes into Algebra and Geometry ones, and will employ the Geo-Algebra symbiotic model.

The present textbook series capitalizes on a compromise between Eastern European and U.S. math educational systems and realities.

On the other hand, times have changed. The Bulgarian curriculum has moved part-way towards the U.S. one; e.g., newer Bulgarian textbooks feature both Algebra and Geometry, while preserving the mathematical integrity and pedagogical insights of a tremendously successful curriculum from the near past.

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.

Can a U.S. Middle School be Successful with this New Math Curriculum?

What kind of question is this?! Let us analyze the situation as a mathematician would, putting aside our own ambitions and de-politicizing math education.

The U.S. Gene Pool

Of course, U.S. students can do on the average just as well as (if not better than) their Bulgarian peers, and, for that matter, any peers around the world. As far as the “gene pool” of the U.S. is concerned, it is probably the best mixture in the world. The cream of the crop of the U.S. students have performed at the top places and occasionally have made the number 1 team in the world (!) at the International Mathematical Olympiads (IMO)*, the preeminent world math competition for pre- college students that tests not speed but depth and originality of mathematical thinking. The new math program emulates:

the IMO spirit of deep and rich math culture
mature attitude towards mathematics
dedication to long-term goals

It is true that very few students will end up being math Olympians, and we will not give false hopes that the current program is training students for the IMOs. However, the expectation is that:

Everyone who is touched by this program will build a solid Temple of Mathematics, where later, if desired, will hang the “fancy chandeliers” and “brilliant art pieces”. More importantly, the personal connection with mathematics will re- main all through life to be enjoyed and cherished.

Can girls succeed in math?

Upon coming to the U.S., I was shocked by this question. I was raised in a provincial town in Bulgaria and made it as a high school student to two IMOs, winning silver medals. There was another girl on the Bulgarian team for the 2 years I was there. It took the U.S. 25 years of participating at the IMOs before the first U.S. girl, Melanie Wood, qualified for the IMOs. I was privileged to train the U.S. national team when the Melanie competed at the IMOs, also earning two silver medals. Later, I participated in the training of the other two U.S. girls, who went on to win gold medals at the IMOs. To cut the long story short: anything is possible and gender does not matter as far as mathematical talent and success are concerned.

Gender apparently matters in the social and cultural aspects of education, and this severely handicaps many U.S. girls who might have otherwise become excellent mathematicians. Decades after competing at the IMOs, I learned that Bulgaria is the top country in the world in sending the largest number of girls, 21, to the IMOs. Germany and Russia are next, with 19 and 15 girls, respectively. USA has only 3 girls so far. Perhaps, it is time for the U.S. educational system to follow examples of other programs from around the world that have been hugely successful in raising generations of women who are highly educated in math and other STEM (Science, Technology, Engineering and Mathematics) disciplines, and who have been brought up to think of men and women as intellectually equal.

The Needed Cultural Shift

The current middle school math program originating in Bulgaria necessitates a cultural shift in a U.S. school community so that the program takes roots and yields results. For starters, the school must make math a top priority in its academic curriculum and follow unwaveringly in its determination to overturn the middle school math situation to the huge benefit of its students.

Continuity of the Curriculum

It is most important and advantageous for everyone that: The alternative has struck me as a very bad way to organize middle school math education. In- deed, when a person teaches the same grade year after year he/she:

gains no understanding of the “big picture” of pre-college math education;
shares no responsibility for the continuation of the program in the next grade (and a lot of time will be wasted in the beginning of each school year for the new teacher to test and get attuned to the new students);
has no incentive (or responsibility) for reaching the final goals of the program. Yes, I am acutely aware of the counterargument: “I do not want a bad math teacher to teach my kid for all of middle school. I would rather bite the bullet of a bad math teacher one year, so as to get the good math teacher the other year.”

In summary, many parents are (unwittingly) willing for their kids to endure on the average a mediocre middle school education. Can you imagine what kind of damage the “bad” teacher would do to the Temple of Mathematics during that, hope- fully, single year of “bad” math lessons? Would the next, “good” teacher have the opportunity, or even the incentive, to rebuild the Temple AND cover the new material on top of that? Probably not. This kind of “averaging math teaching” is an- other major reason for the tremendous difficulties in U.S. middle school math education. A teacher must be the asset of math education: equally important (if not more important) than the actual math curriculum! And so, the truth must be spoken:

a bad math teacher must be let go, to prevent more irreversible damage to our kids; • a good math teacher must be treasured and given opportunities to further develop.

Local Support

There is no easy remedy and no miracle solutions to math educational problems anywhere around the world. The new math Program will need all the support it can get from parents and students, in addition to teachers and administration. The Program will depend on the school community’s belief in it and commitment to it.

* I am a Professor of Mathematics at UC Berkeley. I am originally from Bulgaria, which has a rigorous mathematics curriculum. I founded BMC to help pass the math enthusiasm on to the younger generation in the San Francisco Bay Area.