A stone falls to the ground faster and faster. Its vertical trajectory basically resembles the equation: y = -x^2, x>0, where x is time.1
The question is: How should a mathematician describe its height above the ground over time? Is it:
(a) decreasing at an increasing rate? or (b) decreasing at a decreasing rate?
Mathematicians are actually divided over the answer to this question.
POLLING THE PROFESSORS
I took a poll of mathematicians in 2008.2 Most were from U. C. Berkeley, the rest were from other universities and math forums. The results were as follows:
40 math professors responded to the poll; 31 (or 77.5%) agreed with (a) decreasing at an increasing rate; 2 (or 5.0%) agreed with (b) decreasing at a decreasing rate; and 7 (or 17.5%) refused to answer; they considered the question “ambiguous” or “pedantic,” as did 9 of the 33 who did choose either (a) or (b).3
Eight of these math professors also indicated, “This is an English question, not a math question.”
Speaking of English, I also polled the U. C. Berkeley English Department:
14 English professors responded to the poll; 14 agreed with (a) decreasing at an increasing rate.
Six of these suggested that the question was more a matter of physics, math, or logic.
Then a physics professor got informed about my poll, and volunteered his own opinion in support of (a), along with his concerns about the ambiguity of the question.
REASONS FOR EACH ANSWER
Answer (a)
The phrase “decreasing at an increasing rate” is favored by 77.5% of the above math professors, 100% of the above English Professors, and the lone physics professor who gave an opinion. Some of its proponents have referred to it as “the common sense English description.”
Everybody agrees that the stone is “falling faster and faster,” which naturally translates into “decreasing at an increasing rate.” There is a certain logic to it. For one thing, “decreasing” is analogous to “going south.” If someone is “going south at an increasing rate,” then it is understood that one is going south at an increasing rate of going south; one is accelerating, i.e. going faster. The first phrase, “going south,” establishes the direction of interest and the effective orientation of the y-axis. Whatever it is doing, whether going south or decreasing in altitude, it is doing so at an increasing rate of change of doing so over time.
More mathematically, consider the curve y = -x^2, x>0. Its y value (altitude) is “decreasing” because its slope, or “rate of change of y with respect to x,” is negative. Now for a postulate: Change = Increase. For if the change is positive, then the increase is positive. If the change is negative, then the increase is negative and the decrease is positive (though it is more common and preferable simply to say the curve is decreasing). Likewise, rate of change = rate of increase, not rate of decrease. Since the slope is negative and getting more negative, the slope is mathematically decreasing. Since slope = rate of change = rate of increase (of y with respect to x), it must be true that the function’s rate of increase is also negative.
Now, if the rate of increase is negative, then the rate of decrease must be positive. Blasphemous as the phrase “rate of decrease” may be to pure mathematicians, that is how it is understood; for there were three professors who independently and without being asked about it volunteered reasons for (a) by saying verbatim, “The rate of decrease is increasing.” The curve is decreasing at a positive rate that is getting more and more positive. Hence, y = -x^2, x>0 is decreasing at an increasing rate [of decrease], just as someone accelerating south would be going south at an increasing rate [of going south].
The physics professor made this point:
“My view (as just one physicist) is that once you say 'the altitude is getting smaller at,' then the 'rate' in 'an increasing/decreasing rate' refers to how fast it's getting smaller. So I think that the correct statement is 'the altitude is getting smaller at an increasing rate.' ”
Answer (b)
The logic for opinion (b) starts with established mathematical truths, which are then downloaded into English expression. Its proponents sometimes call it “the mathematical description.”
The rate of change of y with respect to x is negative (i.e. it’s slope is negative). Also the value of the slope is getting more negative, or decreasing. Therefore it would be correct to say that the function is “decreasing with a decreasing slope,” or “decreasing with a decreasing rate of change of y with respect to x.” If you abbreviate this long expression by omitting the phrase “of y with respect to x,” it becomes “decreasing with a decreasing rate of change,” which is correct, as long as you understand "rate of change" to be slope. Finally, if you delete the “of change” and substitute “at” for “with,” you arrive at opinion (b), “decreasing at a decreasing rate.”
One professor argues the mathematical basis for answer (b) as follows:
“2nd deriv negative means the 1st deriv is decreasing in values. Since the 1st deriv is also negative, those values are negative. When negative values "decrease" that means they stay negative but get bigger in magnitude (ie -2 goes to -3). So the function is decreasing at a decreasing rate (slope -2 changes to slope -3; that is a decrease). But the magnitude of the slopes (ie |-2| to |-3|) are changing at an increasing rate (2 to 3).”
While only 2 out of the 40 polled math professors (or 5%) supported answer (b) “decreasing at a decreasing rate,” there seems to be stronger support for it among teachers affiliated with one or more college placement testing agencies. A few months after my above polling of professors I briefly engaged some officials from one of these agencies in a chatroom to discuss the issue. Though the dialogue was private, I will share that about 8 people responded to my poll, and they were pretty evenly split over the issue, and many expressed concerns about the ambiguity of the question. A couple of participants assured me that they are all striving to avoid writing any test questions that contain such ambiguities.
ANALYSIS
So which is right? Well, if there is a right answer - and that's a big "if" - I think it would have to be (a) decreasing at an increasing rate. The polling percentages were 77.5% to 5% in favor of it, and the English phrase has a seemingly established meaning that orients the “rate” with the word before the “at,” so that “rate” is contextually understood to mean “rate of decrease.” This is precisely how the three professors above understood it when they said, “The rate of decrease is increasing.”
The mathematical basis for (b) is without fault, but its translation into English is not. If they had stopped at "decreasing with a decreasing slope," no problem. But they casually and without explanation substituted "at" for "with," and left "rate" without an "of" statement after it, creating a problem that one professor pointed out by asking, "Rate of what?" Perhaps the phrase "decreasing at a decreasing rate" was such a familiar English phrase that it seemed convenient to seize upon it to express the (b) answer. But this use of "at" implies that the "rate" with no "of" after it is describing the "decreasing" that precedes the "at," as the physics professor argued above.
Furthermore, after finishing my polling and discussing this matter with various math minds, I happened to meet another math professor on the (b) side of the issue. He defended the (b) phraseology by saying basically:
“Mathematicians do not speak in English; they invent their own language for mathematical communication.”
The whole debate depends on whether this claim is right or wrong. If nothing else it proves that answer (b) is admittedly not compatible with the common language of the test takers who would naturally assume the right to interpret a seemingly clear question, worded in English, in the most commonly natural English way. Such students must never be marked down for failure to know another "language" that is so precisely opposite of their own, unless great pains are taken by the teacher to reeducate the students on the new math lingo, a process that will certainly aggravate students determined to uphold the integrity of their language.
Likewise there are some students who have learned to interpret the phrase in the (b) way, and they also do not deserve to be caught off guard by test questions graded by an (a) grader after being taught by a (b) teacher. Neither side deserves to be punished at grading time because of a linguistic conflict like this.
Therefore it seems that the math professors who found the question too ambiguous to answer may have the best solution. A total of (7+9=) 16 out of 40 (that's 40% of math professors) volunteered these concerns without even being specifically asked about them. They recommend using terminology that is understood by all, like “decreasing and concave down” or “negative first and second derivatives.”
It may seem an unfortunate sacrifice of one useful form of English expression, but I believe that in order to do the most justice to all students, both answers (a) and (b) should be avoided.
(1) The actual equation would be y = -0.5gt^2, but I'm trying to keep things simple here.
(2) The first question I e-mailed to professors was:
“If a stone is thrown sideways off a cliff (under normal circumstances), which of the following would you say about its altitude above the ground? Is its altitude: (A) decreasing at an increasing rate? or (B) decreasing at a decreasing rate? Thank you for your time. Further explanation in not necessary, but it would be all the more valuable.”
This question received 12 responses. After 1 of these professors refused to take sides, saying that the question lacked mathematical precision, I changed the question and resumed e-mailing professors with the following question:
“I have a 20-second question about math terminology: If a function has negative first & second derivatives (i.e. y = -e^x), is the function: (A) decreasing at an increasing rate? or (B) decreasing at a decreasing rate? Thank you for your time. Further explanation in not necessary, but it would be all the more valuable.”
While y = -e^x is a far cry from a gravitational equation, it is equally relevant to the "decreasing at" debate because, like y = -x^2, x>0, it is always decreasing and concave down. I chose originally it because it has the advantage of not requiring the qualification x>0.
As for names of participants, I am leaving them anonymous for the time being, since I did not exactly secure their permission to use their names (must I?), nor do I want to appear to be "using" their names and opinions as authoritatively decisive. Of course, this may run the risk of plagiarism or the appearance of forging evidence on my part. I'll just take the heat so they won't have to.
(3) The first question I e-mailed to professors received 12 responses, 11 of which supported (a), and 1 of which refused to take sides. The second question received 28 responses, 20 of which supported (a), 2 of which supported (b), and 6 of which refused to answer. If one counts results only from math professors who were asked the second, more mathematical question, the proportions change to:
20/28 (71%) for (a); 2/28 (7%) for (b); and 6/28 (21%) refusing to answer.
A change to "71% against 7%" emerges in favor of (a), but this is not a significant change from the previous results, "77.5% against 5%."