A letter to a school headmaster about math programs

Dear Headmaster,

Thank you for the opportunity to review the books, especially Saxon Math - I didn't have a chance to look at it before. The notes below are much longer than what you expected to see - I am trying not to just state an opinion but also to motivate it, and with some examples at hands.

So, here is my attempt to compare the US math curricula you are using or considering, or better to say, to identify their common fundamental flaws. These seem to be attributable to the whole US culture of math education, rather than to the particular group of curricula. It would be useful therefore, before entering the discussion of any specifics, to overview the pertinent features of this culture.

There is an enormous trench, of the size of the Grand Canyon, between professional training of mathematicians, physicist and engineers, worldwide, and the practice of general math education in the US in the K-12 framework (and up to junior college). Many textbooks accepted at leading US universities for upper-division and graduate level math courses are recognized masterpieces written by American authors, translated to foreign languages and used in many countries such as England, France, Germany, Hungary, Italy, Japan, Korea, the Netherlands, Romania, Russia for similar purposes. At the same time, it is most likely that none of the commonly used K-12 US curricula or typical lower-division college math textbooks have ever been translated to foreign languages or used in those countries (and, help them God, will never be). The cause is a basic disagreement between the two genres of literature on the issue of what mathematics is and what it isn't.

Mathematicians, the authors and readers of the advanced mathematics literature, would applaud the following description of the essence of math education written by Madge Goldman, President of the Gabriela and Paul Rosenbaum Foundation, in her Preface to the US edition of Teacher's Guide for "Primary Mathematics" (Singapore):

"The central idea of all of mathematics is to discover how it is that knowing some few things will, via reasoning, permit us to know much else - without having to commit the new information to memory as separate facts. Mathematics is economy of information, not its unnecessary proliferation. Basic mathematics properly presented conveys this lesson. It is the connections, the reasoned, logical connections, that make mathematics manageable. Understanding the structure of mathematics is the key to success."

Examining typical math curricula and elementary textbooks shows that general mathematics education in the US relies on values quite opposite to those just quoted. Here is my attempt to classify the key differences into four categories.

I. Science begins with the realization that naming stars does not produce new knowledge. Mathematicians consider their subject as a science which discovers and exploits deep and non-trivial relationships between seemingly unrelated matters. Basic math education in the US relies on the opposite premise: giving names to objects without pointing out any non-tautological relationships among them is an honorable and useful activity on its own.

II. As a form of practice, K-12 mathematics in the US consists in memorizing and executing algorithms which allow one to answer questions stereotyped into hundreds of patterns practically unrelated to each other.

III. Low or false expectations. Students are not required to achieve and retain mastery before progressing to the next level. Students are not expected to acquire comprehensive understanding of the logic and structure of the core mathematical knowledge. Students are never exposed to genuine challenges of elementary mathematics. The whole idea of mathematics as an intellectual quest and adventure is severely compromised by attributing "challenge" to repetitive and trivial tasks.

IV. The goals of acquiring (for the student) and disseminating (for the teacher) mathematical knowledge are replaced with the goal (for both) of practicing educational routines. In the following discussion of particularities I will refer in parentheses to the items I,II,III,IV from my list.

SAXON MATH 7/8

To start with a few good words, the style of the book (except for its size) resembles "normal" scientific or mathematical literature. The text is focused on the material. Occasionally it gives clear and meaningful explanations (for instance, see division of fractions), and the language seems usually correct.

Yet, both philosophy and execution of Saxon Math are in strong opposition to what people with professional competence in mathematics usually think of their subject.

From the opening letter of the author Stephen Hake to the Student:

" ... similar problems are presented over and over again. Solving each problem day after day is the secret to success." (II,III,IV)

Note how far this is from Madge Goldman's "Understanding the structure of mathematics is the key to success."

Next, the text is almost 1000 pages long. The author's suggestion to "solve every problem in every practice set" is unrealistic, one simply cannot digest this much information in one year. Thus, in spite of the author's declaration, the student has no hope for comprehensive understanding of the subject. (III)

By the same token, the student is not really expected to be on the top of the material from the previous years (III). Instead, the entire curriculum is repeated in the book "from scratch". Yet most of the material is names and examples of mathematical concepts with no non-trivial relationships among them (I), and in those rare occasions when a non-trivial relation is in presence, it is almost never explained *why* it is true, and even the very fact that it is there can be suppressed. (I)

The list of contents has no structure, and consecutive topics seem to have little in common. A closer look shows that the ordering of lessons is not random, namely each next lesson usually uses some terminology described in the previous one. However this relationship is typically superficial, as no particular knowledge or skill from the previous lesson is used. (I,II)

To illustrate all the features, let's take a sample: Lessons 17 through 20 followed by Investigation 2.

Lesson 17 "Measuring angles with a protractor" hardly qualifies as a new prealgebra material (III). No genuine knowledge is represented in this section - only names of angles: acute, right, obtuse, straight; degrees, protractor (I).

Lesson 18. "Polygons. Similar and congruent." Names of polygons are discussed, and the definition of regular polygons and their examples are given. The (new!) concepts of congruent and similar polygons are introduced (here angles from lesson 17 are mentioned), and the terms "corresponding parts" are illustrated. No relationships of any kind is mentioned (I), not even the fact that similarity of a pair of triangles is established by identifying two pairs of angles.

Lesson 19. "Perimeter." (Of a polygon - that's why it comes after the previous lesson). There is nothing in this section beyond the material of grades 3 or 4 where the sum (called "perimeter") of numbers (called "lengths") written on sides of a polygon is computed. (I,III)

Lesson 20. "Exponents. Rectangular area, part 1. Square root." The lesson on area is here because the perimeter was just mentioned (no matter that no relationship between these concepts is studied in school). The material here does not differ from that of grade 2 (III). The exponent notation is suddenly introduced just to explain where the name "square units" for areas comes from. The square root concept is suddenly introduced as the relationship between the area of a square and its side. No real use, or any properties of areas, exponential notation, or square roots is mentioned.

Investigation 2. "Using a compass and straightedge, Part 1." The terms "compass, center, radius, concentric circles, inscribed polygons, inscribed angle, chord, diameter, circumference, semicircles, central angle" are introduced; needless to say, no relationships of any kind is mentioned. The construction of the regular hexagon and triangle using straightedge and compass is described. There is some issue to discuss here, because for the first time in these lessons we encounter a bit of mathematical knowledge.

The key fact in the construction is that a 60 degree arc on a circle is subtended by a radius-long chord.

This is not obvious and needs a proof. The latter could be easily given were the following simple results properly presented in Lessons 17 and 18:
(1) the angle sum of a triangle is 180 degrees (which is essentially an experimental fact, not a theorem)
(2) an isosceles triangle has equal angles at the base, and vice versa

When the fact about the 60 degree arc is established, one uses the compass to draw a circle and to divide it into 6 equal arcs of 60 degrees (each subtended therefore by a radius-long chord). Connecting the consecutive division marks by straight segments, one obtains a regular hexagon inscribed into the circle, and connecting consecutively each second mark - a regular triangle inscribed into it.

Not only the textbook gives a lengthy description of the construction that has been just stated in a few lines (so much for expected reading skills of the students, III), and gives no proof of the non-trivial property of the 60-degree arc (II), but it does not even mention that any non-obvious fact is involved (I).

To get further illustrations of (II), it suffices to look at the "mixed practice" section of each lesson. It contains several groups of exercises typically unrelated to the subject of the lesson or to each other. It is expected therefore that only constant drill of each and every skill can provide mastery. In Singapore math (to take an example of the opposite philosophy), the skills are layered in a way prescribed by the logic of mathematics; the higher layer skills rely on - and therefore absorb - those of the lower levels, thus providing further training of the latter and removing the need for repetitive drill.

To see how the principle (IV) manifests in Saxon Math, let us examine the titles of problem-solving lessons:

Lesson 11: Problems about combining. Problems about separating.
Lesson 12: Problems about comparing. Elapsed-time problems.
Lesson 13: Problems about equal groups.
Lesson 14: Problems about parts of a whole.
Lesson 22: Problems about a fraction of a group.
Lesson 79: Insufficient information. Quantitative comparisons.
Lesson 92: Percent of change.
Lesson 103: Translating expressions into equations.

In Singapore math (for example), one-step word problems are constantly used to establish the relationship of mathematical operations with their meaningful interpretations; thus a student is expected to demonstrate that math makes sense for him by instantly figuring out how to solve such problems.

In Saxon Math, the author tries to classify these meaningful interpretations into recognizable patterns. This is a task valuable for curriculum development: the teacher should consciously choose problems to cover evenly all aspects of student's understanding. However, there is no mathematical value or meaning in this classification. Besides, the student, whose common math sense is tested, is not supposed to rely on knowledge of teacher's intentions. Here studying mathematics is substituted with tutoring for an easier way to pass teacher's tests.

STRUCTURE AND METHOD by Dolciani et al.

The main problem I have with the books of this series, which has the reputation of one of the best curricula, is that they are potentially lovable. The books are as twice as shorter than Saxon Math, are better structured, contain many examples, display key points in easily found frames, and by and large they resemble calculus-level college textbooks. In fact many college-level math instructors will testify that if only a person comes to loving this kind of literature, any sensible math education of that person becomes virtually impossible. The problem is that together with the affection for the particular style of exposition the readers acquire the point of view on mathematics outlined above as I,II,III,IV. The sample text from the series, namely Mathematics, course 1, is good enough to illustrate this.

What is the content of the 30-page long Chapter 1 "Operations with whole numbers"? We learn formal properties (such as commutativity, associativity, distributivity) of the four arithmetic operations which are stated in 10 blue frames in the abstract form. Each property is illustrated with several examples creating the illusion that several examples suffice to justify a general rule. The question *why* the properties hold true in general (are they theorems? axioms?) is never discussed. (In contrast, Gelfand-Shen's "Algebra" discusses the whys on p. 2 before stating the rules on p.11-12, and Euclid's way to prove (a+b)^2=a^2+2ab+b^2 is found on the front cover.) Even the obvious logical relationships between already stated properties are not mentioned. For instance, the distributive properties of addition and subtraction are stated in different frames without noting that they logically follow from each other. Thus, this chapter merely states some properties considered self-obvious (I), exhibits no "reasoned logical connections" (I) between them and expects the students to "commit them to memory as separate facts" (I,II,III).

Chapter 2 "Using variables", 25 pages. We learn how to write and solve an equation "with x" for questions which we knew how to answer "without x" in grade 3, e.g. "seventeen less than a number is fifty-six; find the number". Thus algebra is presented not as aid but as burden. (In contrast, the role of letters is explained in Gelfand-Shen's "Algebra" on p. 16-17 with 2 much less trivial examples, and they - the letters - are immediately put to serious use.)

Chapter 3 "The decimal system", 40 pages. This time there is a non-trivial mathematical content here, namely the decimal representation of numbers and arithmetic operations with them. Yet the key theoretical question (explaining advantages of the positional number system) "why each number has a unique decimal representation, and vice versa" is not addressed. (In contrast, in Gelfand-Shen's "Algebra" the subject is treated in depth on p. 8-10 as the reader is guided through the construction of the binary number system.) Furthermore, the operations with decimals are presented as framed "Rules" describing in the authoritative manner manipulations with strings of decimal symbols ("To multiply by n-th power of 10 move the decimal point n places to the right"), and with no reference to the number sense, meaning of the operations, or the reasons for the rules to hold true. (I,II) Besides, the bulk of the text in this and many other chapters consists of series of exercises following and stereotyped by Examples (II,III). The way students read such texts is well-known: they search for samples of homework problems among the examples and plug-in different numbers. (II,III,IV)

Chapter 4 "Geometric figures", 45 pages. Most of the chapter has absolutely no content and is pure "star naming" (I). A typical problem is "classify each triangle", i.e. determine how it is *called*: scalene, isosceles, acute, obtuse, etc. There are three exceptions.
1: A blue frame in section 4 declares the angle sum of the triangle to be 180 degrees. Yet on p. 127 we are told - as a separate fact, although it follows from the previous one - that the angle sum of a hexagon 720 degrees (I).
2: A blue frame in section 6 tells us four "Formulas" about circles. Although two of them follow immediately from the other two, apparently the authors find exhibiting all four handy for students to plug-in their numbers (II,III).
3: A really nice section 8 about construction with straightedge and compass. Unfortunately most exercises here are supplied with hints which give away the solution, leaving it to the student only to physically perform the construction (III). So much for high learning expectations and challenge.

Some other books of the series feature "Challenge" activities which are typically trivial. To give an example, one of such "challenges" deals with triangular numbers 1+2+3+...+n. Is is a standard extra-curricular topic for ages 6-12 to figure out that the sum is n(n+1)/2. However the "challenge" in Dolciani et al is not to find the formula (it is provided there!), and not to prove it, but to evaluate it for n=1,2,3,4,5,6,7,8,9,10 (III).

Some other volume of the series supplies a bright example of (IV). As it is explained in Gelfand-Shen's "Algebra", factoring polynomials (e.g. ax^2+bx+q=a(x-u)(x-v)) is equivalent to finding their roots (x=u, x=v). Thus the famous "quadratic formula" for solutions of the equation ax^2+bx+c=0 solves both problems - solving equations and factoring. The skill of mental factoring is just a handy shortcut for ad-hoc treatment of simple examples. Dolciani et al managed to make factoring quadratic polynomials the subject of 3 (three!) chapters about 3 "mutually exclusive" cases: x^2+bx, x^2+bx+c, ax^2+bx+c. All three come before the quadratic formula and with no connection whatsoever with solving equations. Apparently the skill is developed for the sake of speedy performance on multiple choice tests, and with no mathematically meaningful aim in view.

EXCEL MATH, 3rd grade (?)

I will be brief here. The material within each daily lesson seems competent. Yet titles of the lessons are names of miniature tasks of purely educational nature (IV): "learning, recognizing, interpreting, fitting in missing numbers, solving story problems." Portioning the tasks (like "Regrouping using the addition facts of 11,12,13") has little mathematical meaning (I,IV). Ordering the tasks is deliberately chaotic (I,II,III). Support for conceptual understanding in each lesson is minimal (I,III). Constant drill on randomly chosen tasks (II,III) is a feature common with Saxon Math.

HOUGHTON-MIFFLIN's MATHEMATICS (I have the 5th grade, California edition)

I can refer you to an essay
http://Math.Berkeley.EDU/~giventh/m5.pdf
comparing one page of this book with the corresponding page in Singapore Math. The issues not stressed in the essay include the following.

Abundance of distracting "fun" features supports the math-phobic attitude: "math is inherently boring and needs to be spiced with side attractions"(III). The 13-page-long list of Contents does not aid in understanding the structure of the subject (I). A large portion of Lessons is dedicated to "star naming" - pointlessly introducing new terms (e.g. see Lesson 2 of Chapter 1). Most other sections (e.g. Lesson 3 of Chapter 1) describe "steps", i.e. algorithms (II), sometimes deficient (III), used for performing particular tasks (like rounding off a number) without resorting to the meaning of the operation (I,III). The books contains a substantial number of mathematically incompetent remarks and notations (III,IV). Most word problems contain redundant information, due to apparent misconception of the role of such problems in the subject (IV). The way challenging problem are presented and the use of "algebra" for solving trivial problems promote intellectual helplessness (III). However the difficulty level of routine exercises seems reasonable for the grade level, and some sections (e.g. in geometry part) seem nicely written.


I stop here. The above comments give reasons not to use such-and-such curriculum. The question what to use should better be addressed not universally but "school-wise" - much should depend on teachers, students, history, etc. Speaking abstractly, Singapore Math, at least in grades K-5 or 6, excellently fills-in the vacuum of mathematically meaningful programs in the US market.

H. Wu of UC Berkeley in his excellent review of Gelfand-Shen's "Algebra"
http://Math.Berkeley.EDU/~wu/Gelfand.pdf
actually misconstrues it as an Algebra-2 text missing many standard Algebra-2 topics. In fact it should better be thought of as the ultimate ideal for Algebra-1. It does not qualify on the role of a complete curriculum as it has to be supplemented with routine problems. An experienced teacher should be able to supply these (may be with the help of Gelfand's Mathematical School by Correspondence http://gcpm.rutgers.edu/ ). The key question is however if by the grades 7-8 the students (and the teacher) are strong enough to stomach it.

Best regards,

Alexander