Friday, June 27, 2008

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of how to use technology to arrive meaningfully at solutions to problems instead of endless attention to increasingly outdated computational tedium."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.

. . . . . . . .

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."