Wednesday, November 30, 2005

Flexible inefficiency

The National Education Association helpfully offers A Parent's Guide to Helping Your Child with Today's Math but then only goes halfway as this example illustrates:

A farmer sends his daughter and son out into the barnyard to count the number of chickens and pigs. When they return the son says that he counted 200 legs and the daughter says she counted 70 heads. How many pigs and chickens does the farmer have?

A student well versed in algebra might do the following to set up the problem: p = pigs, c = chickens. p + c = 70 (heads) 4p + 2c = 200 (pigs have 4 legs and chickens have 2 legs). These two equations may be used to solve the problem. Students might solve this problem by "guessing and checking," or drawing pictures. Some methods of solving problems might be considered more "efficient." That may be true, but the correct answer can be found using multiple methods. Children think about mathematics in different ways depending on their prior experiences at home and school. By allowing students to think flexibly about numbers, we encourage them to "own" the math forever, instead of "borrowing" until class is over. (Answer: 40 chickens and 30 pigs)

Constructivism is rearing its ugly head again. With this kind of help, students are not going to "own" the algebraic method of solving this problems. They might not even "own" the inefficient guess-and-check method. If you are being taught properly and practice sufficiently to the point of overlearning, then you are more likely to "own" math knowledge than when left groping in the dark.


NYC Educator said...

I think you're missing the point, Instructivist. This sort of problem is to give our youngsters insight into their everyday chores.

How many times, for example, have you asked your kids to go out and count the chickens and pigs only to have them return with an inaccurate answer?

There's simply nothing more frustrating, if you ask me, particularly after having them assiduously record the number of legs and heads.

Tarnation, it gets me riled to my britches just thinking about it.

Barry G. said...

I find it interesting that this example is supposed to be a "real life" problem. NYC Educator's comments hit the nail on the head. It's ironic that the educationists who promote this type of problem as "real life" hold in disdain problems such as two trains travelling at different speeds, etc., as being irrelevant.

Also interesting that educators would not care to instruct students on what method is the most efficient. They are perfectly happy to have kids "discover" that there are many ways to solve a problem, including bad ways, which the goal of mathematics is to eliminate and replace with efficient ones. One goal of math is to develop procedures that eliminate solving by trial and error. Interestingly, Singapore's math program provides students with a powerful procedure, called bar modeling, which allows solving problems such as the chicken and pigs one, using arithmetic techniques. Such procedure ultimately provides a natural progression into algebra.

"Ms. Cornelius" said...

It is my sorrow-filled experience that kids will give up rather than flop around without guidance, especially in math. These kids won't evven do long division.

Word problems like that are part of the problem-- it's not relevant and it is currently modish to make everything in math relative.

These kids have no "number sense," and this type of stuff will never encourage them to devlop it. We want them to study math to encourage thinking. Thinking is usually a painful enterprise among adolescents. Let's not make it pointless too.

Catherine Johnson said...

I actually DID grow up on a farm, gosh darn it, with REAL farm animals, and no normal farm child would count heads and feet when asked to count sheep and pigs.

That's number one.

Number two is: what's the difference between efficient and "efficient."