When you go out and buy a math textbook or workbook series, you trust it to be designed to teach math, especially if it "meets state standards" or "meets NCTM (National Council of Teachers of Mathematics) guidelines." You might think that all modern materials are based on the best math teaching theory drawn from the combined successes of generations of dedicated, brilliant mathematics teachers, and that "state standards" and "NCTM guidelines" would be drawn from the same research.

Would you be right? Not necessarily.

For the past several decades, a quiet war has been fought between those who believe in the traditional methods of teaching arithmetic by rote and those who favor teaching kids in preschool to grade 6 how to "think like a mathematician." The fallacy in teaching this way is that all the mathematicians I know learned to add and subtract, without using their fingers or a calculator, before they started "thinking like mathematicians."

**Math as a Foreign Language**

The problem with modern elementary math education is that the math programs aren't written by mathematicians. The motto of mathematicians is, "Can you prove that?" From the mathematician's point of view, every new advance in math has to be logically tight and thoroughly justifiable from the math that went before. The mathematician's goal for elementary math is that the student become comfortable with numbers and the operations that are done on them and is prepared to move on to higher math such as Algebra, Geometry, Calculus, etc.

In contrast, check out this goals statement from the Connected Mathematics Project out of Michigan State University as reported on their site at connectedmath.msu.edu:

*The Overarching Goal of the Connected Mathematics Project:*
All students should be able to reason and communicate proficiently in mathematics. They should have knowledge of and skill in the use of the vocabulary, forms of representation, materials, tools, techniques, and intellectual methods of the discipline of mathematics, including the ability to define and solve problems with reason, insight, inventiveness and proficiency.

That's it - mathematics as a foreign language. (Substitute "Russian" in place of "mathematics" in the above definition and it makes more sense.)

**Objective v. Subjective**

The Wikipedia article on "Math Wars" defines the traditional method of math instruction as: "skills based on formulas or algorithms (fixed, step-by-step procedures for solving math problems)." You can evaluate the results of this method, because the student works the problem correctly and gets the correct answer or he doesn't. (Partial credit can be given for correctly doing the steps of the problem, but getting a wrong answer due to an arithmetic mistake.)

The "reform" method, on the other hand, is subjective in the way it's taught and graded. This makes it easy for its proponents to delude themselves about how their students are doing. As Wikipedia says about the reform method, "In this latter approach, computational skills and correct answers are not the primary goals of instruction." The student gets points by how creative (not necessarily how valid) his approach is to a problem. Grading a math problem becomes as subjective as grading an English essay.

**By Their Many Names Shall Ye Know Them**

How can you spot one of these "progressive" math programs? Is it really true that "reform math" = "new math" = "new new math" = "fuzzy math" = "constructivist math" = "PC Math" = "Rainforest Math"?

Yes, these types of programs are different in subtle ways, but all have in common that they strive to teach the student to "understand" math, rather than teach him how to do math.

"Reform"/"new math" programs can be identified by the unusual topics they introduce into the elementary curriculum. The two most common ones of these are set theory and different bases.

The trademark of set theory is the Venn diagram (see below). If you see lots of Venn diagrams in your elementary math textbook, you are holding a "new math" program. Set theory is one of the most abstract branches of mathematics - but elementary school students are concrete thinkers. I wonder what Einstein came up with this idea as a good topic to add to the curriculum.

Working in different bases requires an understanding of place value in our own base-ten number system. No one -illustrates the joys of teaching arithmetic, while trying to include an understanding of place value, better than Tom Lehrer in his song "New Math." A lipsynched version of this amusing song can be found at Youtube. The last line says it all: "It's so simple, so very simple, that only a child can do it."

"New new math" or "fuzzy" math says the process is more important than the result. The correct answer is the one that demonstrates the most ingenuity, not necessarily the one that results in the correct answer.

"PC Math" (politically correct math), also dubbed "Rainforest Math," is the same stuff, with the added wrinkle that now word problems have to teach moral lessons that are socially acceptable to the largely left-leaning educational establishment - and that have practically no actual math in them.

**Too Young to Derive**

Which brings us to constructivism. Constructivists want children to learn math through discovery on the theory that if you figure something out for yourself, rather than having it fed to you, it will be yours forever. The problems with this are obvious. To develop math to its present form took thousands of adult mathematicians a millennium and a half. One kid can't be expected to work it out in six years of elementary school.

The claim that the child is discovering math for himself is disingenuous anyway. The process of "discovery" is carefully guided so that ideally every student will "discover" what the teacher wants him to discover.

**Are the Math Wars Over?**

A *Wall Street Journal* article of March 5 this year claimed that there was now a "truce" in the math wars, thanks to the newly appointed National Mathematics Advisory Panel.

This followed a September 2006 report by the National Council of Teachers of Mathematics, which finally put in a good word for teaching the basics of arithmetic after years of the NCTM exclusively promoting "think like a mathematician" approaches.

A special ed teacher made this comment about the WSJ article: "for the past 12 years I have been doing Educational Evaluations on high school students, and a consistent pattern has emerged. While our students may be capable of higher order thinking regarding math concepts such as the application of the Pythagorean Theorem, they cannot do multi-digit subtraction or long division, nor can they manipulate fractions. They never really mastered these concepts in elementary school so they have "forgotten" how to do them as adolescents... but they definitely know how to do them on the calculator."

Another WSJ reader said: "I judge high and middle school science fairs. I routinely see the following:

- "No understanding of significant digits: students measure something to one significant digit and report the average results to five decimal places, well beyond the measuring capacity of the equipment they used.

- "Meaningless graphs: five-color graphs and/or bar charts that have nothing to do with the data that were collected, no labels on the axis, and no clue what the data mean. It came from Excel, so it must be right and relevant.

- "Undetected gross calculation errors: averages that are higher than the highest number in the observed data or lower than the lowest number in the observed data. Because these students cannot estimate the average value of their data in their heads, they do not realize that they keyed the wrong data into the calculator."

A retired teacher who is "still active in the school system" added this. "Think about this the next time you hear someone being snobbish about 'drill and kill' software: Skill repetition or reinforcement is considered 'busy work,' therefore, I have seen schools where the majority of students in the 8th grade don't know their times tables!"

**Develop This!**

It's not just students in the eighth grade who don't know their times tables, either. When I began looking into math instructor positions in local community colleges, I discovered a course called "Developmental Mathematics" (not to be confused with the homeschool curriculum of the same name). Developmental Math is just remedial grade school math with a fancier name. Students entering college who can't demonstrate proficiency in basic arithmetic, including fractions, are assigned this course before they are allowed to tackle the normal college math sequence (assuming they ever do). We are talking about millions of college students who can't do fractions!

One last WSJ reader put it best: "There is a place for 'drill and kill' and for 'exploration and discovery'; one needs to know the basics before venturing into the unknown."