Head First, Calculator Second

Wow, did I locate an excellent page on "fuzzy math" procedures and thoughts, entitled "Everyday (Fuzzy) Math is Dumbing Down our Children", written by Ian Shapira. You people in Virginia may have already read Ian Shapira's writings.

Some of his observations about Everyday Mathematics follow:

"There is a 23-page chapter that teaches nothing but how to use a calculator." Shapira goes on to explain his own take on why so much space is used for teaching how to use a calculator: ". . . the odd algorithmic methods taught in the book for solving math problems are so confusing and unworkable that the students must resort to using a calculator in order to solve math problems."

"The most surprising thing is that the total number of pages in the book devoted to teaching algorithms using whole numbers is 11 pages! That's correct! There are only 11 pages in a 400 page book devoted to explaining algorithms using whole numbers. Only 3 of those pages offer instruction on standard algorithms. . . . On page 50 the book states: 'Finding a percent of a number is the same as multiplying the number by the percent. Usually, it's easiest to change the percent to a decimal and use a calculator.'"

"The preferred Everyday Math are crutches. The crutches are needed because the students are not taught the standard algorithms. The lack of skill in standard algorithms ends up crippling their ability to solve math problems without their crutches. The EDM crutches become cumbersome and hold children back when they are later exposed to to more advanced math problems. Their crippled minds are unable to sprint ahead in math, because they trip all over the crutches imposed upon them by EDM."

"One of the alternative algorithms that is a standard method taught in Everyday Math was authored by a first grader!"

". . . students are expected to invent their own algorithms. Adding to the silliness, the authors of Everyday Math expect the children to invent their own algorithms before they are taught any standard algorithms."

". . . The Everyday Mathematics advocates admit that the standard algorithms used for the past 100 years are 'highly efficient'. One might ask: If the standard algorithms are 'highly efficient,' why replace them with invented and other non-traditional algorithms? The reason is that the Everyday Math advocates are not satisfied with a 'highly efficient' method. They want the 'most efficient' method. In their view the most efficient method is 'mental arithmetic or a calculator.'"

There is so, so much here to read. This article helps all of us identify "fuzzy math" programs, whatever they may be named. Everyday Math is only one, and they all do their damage. The sad thing is, once the damage is done, it's very difficult to go back and redo 4,5,8 years of damage.

Knowing Leads to Understanding

You have to "know" math before you can "understand" math; you have to "know" math before you can "do" math; you have to "know" math before you can "solve math problems".

The above is a paraphrase from a long article by William G. Quirk, Ph. D. in Mathematics.

In this article, Quirk explains that even though the National Council of Teachers of Mathematics has toned down their push for "new math", "constructivist" math, fuzzy math, (probably in an attempt to calm the uproar caused by their original 1989 Standards), they still push student-centered "discovery learning"

[On April 12,2000, The National Council of Teachers of Mathematics (NCTM) released Principles and Standards for School Mathematics (PSSM), a 402 page revision of the NCTM Standards.]

Translation: NCTM sees the error of their ways in the 1989 Standards and so here comes the 2000 Standards in which they pretend to drop all of this "fuzzy" stuff, such as emphasizing calculator skills or student-invented procedures and now appear to emphasize mastery of basic facts.

I think the NCTM must be more patient that the opponents are persistent. They know that if they just give the appearance of change, the "math wars" will subside, we will drop our guard, and then they will be able to come back with terms redefined and no one will notice. They they will be free to unleash, once again, their confusing, dumbed-down ideas on a new and unsuspecting group of children, families, and school districts.

Oh, yes, they say, we are all for "putting arithmetic back into mathematics." We are all for "teachers emphasizing the fundamentals of computation." And the public, with a sigh of relief, smiles and thinks "Oh, isn't that wonderful! The "new math" is gone! Traditional math will return at last! Our children are now safe!"

And we fail to read the fine print:

"When calculators can do multidigit long division in a microsecond, graph complicated functions at the push of a button, and instantaneously calculate derivatives and integrals, serious questions arise about what is important in the mathematics curriculum and what it means to learn mathematics. More than ever, mathematics must include the mastery of concepts instead of mere memorization and the following of procedures. More than ever, school mathematics must include an understanding of how to use technology to arrive meaningfully at solutions to problems instead of endless attention to increasingly outdated computational tedium." --NCTM

Quirk helps us understand that although NCTM says they want to emphasize "understanding", they fail to understand how the brain works. Says Quirk,

". . . they still fail to recognize that specific math content must first be stored in the brain as a necessary precondition for understanding to occur. Although rarely the preferred method, intentional memorization is sometimes the most efficient approach. The first objective is to get it into the brain! Then newly remembered math knowledge can be connected to previously remembered math knowledge and understanding becomes possible. You have to "know math" before you can "understand math", "do math", or "solve math problems.

"Similar to the orignial NCTM Standards, PSSM fails to clearly acknowledge that the abililty to instantly recall basic number facts is an essential preskill, necessary to free up the mind, first for mastery of the standard algorithms of multidigit computation, and next for mastery of fractions. Then, once this knowledge is also instantly available in memory, the mind is again free to focus on the next task level, algebra."

. . . . . .

OK, it boils down to what you want your student to be able to do:

Master the basic facts or Derive basic facts when needed

If basic facts are mastered, students can proceed quickly through multidigit computations, particularly when learning a new concept. Their minds are free to focus on what is new, rather than having to also repeatedly derive methods and facts to help them get through the steps.


. . . . .

The entire article really upset me because it makes statements about students' inability to learn how to properly use division of fractions. I've taught division of fractions for 15 years very successfully using Saxon Math. I strongly refute the statement by PSSM that the "process can seem very remote and mysterious to many students."

When presented properly and taught properly, students are indeed able to grasp the meaning of "invert and multiply". Students are indeed able to learn and understand the concept of dividing 1/2 by 1/4, or 1/3 by 1/6 and "the reasoning" of "How many 1/4's are in 1/2?" And students are indeed able to identify the types of story problems where division of fractions is the preferred method to find the solution.

Read the entire article here.

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The article concludes with some excerpts from Roger Howe, Professor of Mathematics at Yale University, several of which will be quoted here:

"An important feature of algorithms is that they are automatic and do not require thought once mastered. Thus learning algorithms frees up the brain to struggle with higher level tasks."

". . . we suspect it is impractical to ask all children personally to devise an accurate, efficient, and general method for dealing with addition of any numbers -- even more so with the other operations. Therefore, we hope that experimental periods during which private algorithms may be developed would be brought to closure with the presentation of and practice with standard algorithms."

"We do not think it wise for students to be left with untested private algorithms for arithmetic operations -- such algorithms may only be valid for some subclass of problems. The virtue of standard algorithms -- that they are guaranteed to work for all problems of the types they deal with -- deserves emphasis."