What problems of the educational system are we solving with these books?
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 4 th grade?
Well, the 4 th 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 5 th 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 textbooks and some teachers nowadays (perhaps, under pressure from school administration or for other reasons) to occasionally show off via: extra-curricular 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 well-known, standard U.S. 6 th grade textbooks, alongside computer codes that intend to demonstrate the growth of the “partial sums of the series”.
Let us reason logically here.
- Question 1. 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.
- Question 2. 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 from 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, “Realism vs. Local Color in Mark Twain’s novels”…
Relationship with Mathematics
With some notable exceptions, especially in Geometry, the actual math content of these textbooks does not “look” much different from just about any standard U.S. high school textbook. After all, A linear equation 5x -711looks 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 a 2 + b 2 c 2 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 content, 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, in the difficulty and breath of math content; in the expectations of quality, depth, and rigor of student work; 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.
by a gradual climb:
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 and high 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 handouts from other curricula.
- Sacrificing 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.