Unrealistic Standard for Fourth-Grade MathTo the editors:
Wilfried Schmid (Opinion, "New Battles in the Math Wars," May 4) used student performance on a MCAS question--what is 256 x 98--as the basis to describe the ills of school mathematics programs and attempts at reform. Yet before we go too far with this, let's take a closer look.
When is multiplication of this type introduced in schools? For the past 60 years it has been introduced in the fourth grade in textbooks. Examples like 3 x 65 generally have appeared late in third grade, with a more extended treatment of multiplication the next year. Even there, the work tends to be at the level of "what is 56 x 98?" Since this material is introduced in fourth grade, did children have a chance to develop skill with this work at the time the test was given?
My hunch is that this may not be the case, in which the statistic of 34 percent choosing the correct answer is about right--a bit above the guessing level (25 percent) on a four-choice multiple-choice item.
Historically, double-digit multiplication has been a tough skill for students to learn--performance has been low and the literature, extending from the 1920s until now, is full of the many gross errors that students have made with it. For instance, the average 1977 scores from the California Achievement Test are not far from those of the Massachusetts students.
Multiplying 256 and 98 is a difficult example of 2-digit by 3-digit multiplication. Multiplying by 8 and 9 is tougher for kids, partially because of the "carried" amount that must be added in; to get the answer correct a child has to carry out 6 multiplications and 4 additions in producing the two partial products. In this case the addition of the partial products is simple, but typically there are 3 more additions to be performed.
So, to get an example correct a student has to correctly complete 10-14 mixed computations in addition to the correct placement of the partial products, which depend on place value knowledge or remembering the rote rule. This is a wonderful example of an exercise that is far more easily solved mentally. Perhaps we should be teaching this, too.
Whether this is really worth knowing in general, and at fourth grade in particular, depends on what each of us values. To me, for fourth grade, it would seem that this wouldn't rate high--not many 10-year-old children get summer jobs that require this skill. Certainly with the calls for more real mathematics for all students, we have to reconsider what levels of efficient computation are important, either mental or written.
Paul R. Trafton
Cedar Falls, IA
May 5, 2000
The writer is a professor of mathematics at the University of Northern Iowa.
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