I've been following the links the new edition of the Carnival of Mathematics sponsored by the Wild About Math! blog. This is an issue #66, and it appears that I have learned about this undertaking relatively late, of which I regret. It is a curious collection of online articles. The next submissions deadline is set for July 2. But I got a little diverted. So, I've been following the links, and arrived at a great post Word Problems in Russia and America which was based on an online book of 159 pages by Andrei Toom (via a comment from Maria Miller. The book is an expansion of a 2005 talk and is dated 2007. This, too, I regret not having seen earlier. The book makes an absorbing reading.
Here's just my first (and incomplete) impression. One point is that the word problems offered to the American kids in 7-8 grades are often tackled by their Russian or Singaporean counterparts 4 (and even more) grades earlier. There is also a difference between how the problems are supposed to be treated. In the US, the problems are offered in Algebra I and, consequently, are expected to be solved algebraically. Problems of the comparable difficulty in Russia and Singapore, are treated arithmetically, with great benefit to students.
Toom complains that, although math educators in America are wont of referring to the problem solving methodology of G. Polya who placed great value on solving word problems, there are practically no word problems in the American grade school.
At one point, Toom quotes Alan Schoenfeld on the results of one of the NAEP (National Assessment of Educational Progress) secondary mathematics exams. Specifically, the Schoenfeld commented on the solutions to the following problem:
An army bus holds 36 soldiers. If 1128 soldiers are being bused to their
training site, how many buses are needed?
According to Schoenfeld,
Seventy percent of the students who took the exam set up the correct long division and performed it correctly. However, the following are the answers those students gave to the question of "how many buses are needed?": 29% said... "31 remainder 12"; 18% said... "31"; 23% said... "32", which is correct. (30% did not do the computation correctly).
Toom then brings up a problem from a Russian Grade 4 textbook:
Each box can contain 20 kg of carrot. How many boxes are necessary to transport 675 kg of carrot?
He then observes:
I have no doubt that Russian children, when meeting with such problems first time in their life, made the same ridiculous mistakes as mentioned by Schoenfeld. But pay attention to the difference of ages: Russian children encounter such problems already in the 4-th grade, when they are ten years old. At that innocent age it is not fatal to make ridiculous mistakes.
In this connection let us consider also the following problem: A company packs carrots into boxes. Each box must contain 20 kg of carrots. The company has 675 kg of carrots. How many boxes can it fill with carrots?
Note that semantic of the problem changed, although the numbers have remained the same. The answer to the first problem is 32, to the second is 31.
I believe that having a problem and its modification (and the more of these the better) side by side is a great educational device. G. Polya's problem solving methodology consists of four steps. Quite frequently the last one (Looking Back) is interpreted as an advice to check the solution while the intention is much deeper. In Polya's words
Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. By looking back at the completed solution, by reconsidering and reexamining the result and the path that led to it, they could consolidate their knowledge and develop their ability to solve problems. A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution.
I'd add "... and also our understanding of the problem." Tweaking the problem parameters goes a long way to improving our understanding.