Thursday, November 15, 2007

Language, division, and calculators.

Since several related issues are floating around on various math education lists I'm reading these days, I thought the following problem and what I observed recently with some African-American students would be worth sharing.

This problem appeared on an actual ACT exam; it should be noted that it was only #5 of 60 questions in a section of a test which allows 60 minutes total time for those problems:

The oxygen saturation level of a river is found by dividing the amount of dissolved oxygen the river water currently has per liter by the dissolved oxygen capacity per liter of the water and then converting to a percent. If the river currently 7.3 milligrams of dissolved oxygen per liter of water and the dissolved oxygen capacity is 9.8 milligrams per liter, what is the oxygen saturation level, to the nearest percent?

A) 34% B) 70% C) 73% D) 74% E) 98%

First, I want to observe that there are quite a lot of words in the above problem, most of which make it difficult to read and which don't flow terribly smoothly. The actual mathematics is hardly at the high end of difficulty for high school students, if they get to it, or at least we would hope that to be the case. But the language seems quite "high end" for where this problem appears in the section, and that would potentially be an obstacle for students who are not good readers, not native English speakers, or who may be thrown by a science situation with which they are unfamiliar. Is it a good idea to embed this particular mathematical task in language like this? What, exactly, is being tested? Here in Michigan, where the ACT is now the official "exit" exam for high school students, these concerns are not trivial.

That said, let's look first at how I went over this problem with some students, once they had it set up as 7.3/9.8 or agreed that such was a reasonable way to begin. Of course, the ACT allows calculators (but unlike the SAT, does not permit the TI-89, which has a computer algebra system included). But I always start with the assumption that it might be quicker and safer to do mental math, and that knowing how to do the problems several ways is worthwhile. (That's also a personal holdover from the period prior to 1995, when calculators first were allowed on the SAT. I learned how to excel on these timed tests by studying them starting in 1979. Mental math, estimation, and knowing that fractions are often the easiest form in which to do quick arithmetic has stayed with me and I still encourage students to think that way on timed tests. I rarely use calculators for doing them, though it's not a bad idea to have one, if you know how to use it intelligently.)

First, I pointed out that we could multiply by 10/10, getting the equivalent expression, 73/98. Then I asked about 34%, hoping that students would see that it was far too low, 73 being clearly less than 50% of 98. Most students agreed. I suggested further that 98% was unlikely, as 73 is not "almost all of" 98. Again, this line of reasoning was amenable to most of the students. I next asked about 73%. Some students saw that for that to be correct, the denominator would have to be 100. That left 70 or 74, and most (though not all, of course) students at this point saw that if the denominator was less than 100, then 73 would comprise more than 73%, so that 74% was more reasonable than 70% would be.

The advantage of this sort of estimating and "number sense" approach avoids some of the errors that I found occurred with alarming frequency among those who attacked the problem with more traditional methods or with calculators. One student in particular ran into interesting mistakes, not unique to him. Without my help, he correctly set up 73/98. He articulated that he should divide, but then proceeded to start dividing 98 by 73. Since this yields approximately 1.34, it's not a coincidence that one of the choices is 34%: students simply assume that they should "drop" the "1" and round. Rather than considering that they've converted a fraction considerably less than one half into a percentage that initially (before the groundless dropping of the 1) represented more than 100%. Would they be likely to do this with the mental math approach?

But this student took it further. He grabbed his low-end calculator (non-graphing), pressed some buttons, and told me that he had been right to begin with, because he still got "34%" as the answer. Of course, he'd failed to realize or recall that when entering 73/98 into a calculator, his "wrong" impulse in the hand calculation was now the correct order (and such would be the case as far as the numbers are concerned were he using a HP or other Reverse Polish Notation device).

What I find provocative is that students can wind up going wrong on the same problem in so many ways and for reasons that aren't strictly mathematical. Not only is the language in which the calculation is imbedded more dense than one might expect, but the language of division "73 divided by 98" is in the opposite order from the order in which students are taught to divide, leading many to then try to divide 98 by 73. And then, having made that error, they find their mistake reinforced by the fact that the calculator wants them to enter things exactly in the order they appear in print (at least in THIS example). Our way of expressing division in language is at least partly to blame for a lot of the confusion in even setting up a correct calculation, let alone carrying it out successfully.

There are other difficulties with division surrounding the conceptual basis for the algorithm, but here we see how students can be led astray regardless of whether they know how to do long division by hand on an exercise that simply offered 98, the side-ways "L" we put to its immediate right, and 34 placed above it. This student did the wrong problem correctly by hand. He then made precisely the error the test-makers set him up to make by selecting 34%. I would have been interested to see how he would have done with the calculator alone, or with the calculator first, at any rate, as far as the order of the numbers was concerned.

To conclude, I would hesitate to say that this student didn't know how to do long division. He could successfully carry out the algorithm. Where his problems lay had to do with issues that may have been more linguistic than anything else. How he said the problem to himself or heard me expressing it was not reliably leading him to set up the right calculation. And the answers were, of course, tempting him to make some unsound and unjustified moves to retrofit his results to the answer choice that seemed to fit. This occured both when he did the problem by hand AND with a calculator. The mistake had nothing to do with his relying on the calculator, in fact, since it was his choice initially to do the problem without it. He did the division he set up properly. Yet his answer was dead wrong, regardless of the technology or its absence. I don't claim this proves anything, but it is thought-provoking. And this incident does help emphasize the potential power of estimation and mental math.

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