In 1998 and 1999 I was engaged, as a member of team (called an ARG, for “Association Resource Group”) appointed by the American Mathematical Society at the request of NCTM (the National Council of Teachers of Mathematics) to write a critique of the draft version of its 1989 Standards revision that was to be published in 2000 as “PSSM”, the Principles and Standards for School Mathematics. In later years the NCTM made much of the fact that it had solicited input from the community of mathematicians for their new edition, for the Mathematics Association of America had also appointed an ARG, which also published rather detailed reports as well as commentary from individual members.
We were not paid for this service, except for our hope that what we wrote would be taken seriously and improve the result, which we knew from the reception of the 1989 edition was likely to have a lasting effect on the manner and content of school mathematics instruction in America. None of us expected all our comments to affect the resulting document, especially as our input was solicited late in the process; we could see that the NCTM was presenting us with a rather finished document. This “draft version”, called PSSM (for Principles and Standards for School Mathematics), represented something evidently close to what NCTM intended as final. Our ARGs were neither authors nor editors, and our efforts were confined to commentary, i.e., responses to a list of questions posed to us by the Editor-in-Chief, Joan Ferrini-Mundy.
The questions were good ones, however, and general enough to elicit pretty much all our thoughts on the successes and failures of what we had been given to read, even though any suggestions that the entire document be reformulated in any deep way, something most of us would have wanted, would clearly have been futile. The very timing of our last-minute attempt at intervention would have foiled that. There was, however, scope for objections to individual items in the draft PSSM, where amendment would not require more than local changes; and it seemed to me that much of the mistaken spirit of the document as a whole was contained in such particular clauses and recommendations.
One such item in my own list of suggested amendments was a negative comment on what many would say was “merely terminology”, but which to me was actually diagnostic of much else that was amiss in the draft document. My comment as sent to Ms Ferrini-Mundy, refers to a section on Measurement, for Grade levels 3-5. I will quote that section verbatim, then reproduce my objection as sent to the editor (Ferrini-Mundi), and then print the final version of that section as modified in the final PSSM, the document avowedly revised after due consideration of the contributions of the ARGs:
On page 178, then, of the Draft PSSM, lines 28-33 read:
Strategies for estimating measurements are varied and often depend on the particular situation. They provide a nice opportunity for students to share thinking and techniques. Students also need experience in judging what degree of accuracy is required in a given situation, and whether an underestimate or overestimate is desirable. If a person is buying carpet, error should be in the direction of an overestimate. However, an estimate of the amount of time to warm up a food item should probably be an underestimate.
As part of my Report for the AMS ARG I wrote, in reference to these lines:
I disagree, and every scientist would disagree. Even in approximate measurements one should never deliberately err in any direction, and it is a bad idea to suggest this to children. When one is ordering carpeting one might well want to order more than the best estimate, for safety's sake, but surely this is not making an error of measurement, as suggested. And the suggestion about food is prejudiced, even if taken as the author meant it. Perhaps we want to cook to destroy the salmonella? I would cook too long. But I would certainly want to estimate the correct time as well as possible, whatever use I later made of that estimate.
The final text appears on PSSM (2000) on page 174:
Strategies for estimating measurements are varied and often depend on the particular situation. By sharing strategies, students can compare and evaluate different approaches. Students also need experience in judging what degree of accuracy is required in a given situation, and whether an underestimate or overestimate is desirable. For example, in estimating the time needed to get up in the morning, eat breakfast, and walk or drive to school, an overestimate makes sense. However, an underestimate of the time needed to cook vegetables on the grill might be considered appropriate, since more time can always be added to the cooking process but not taken away from it.
The author of the final version used much of the original language, keeping his first and third sentences verbatim, and with minor corrections or improvements in the second one. The second sentence in the new version is a bit more forceful about the sharing feature of education at the Grade 3-5 level, I believe. It suggests, as the original version did, that students should "share strategies", but it adds, “ in order to “compare and evaluate different approaches”, apparently in response to some suggestion (not mine) that the sharing of strategies might have been recommended for some different, perhaps unhealthy, purpose.
As with most PSSM openings, the first sentence, which needed no amendment, is boilerplate "opening sentence" style, saying nothing but what is true of absolutely everything in human experience. It explains that "strategies are often varied" (what else is new? ) and "often depend on the particular situation" (ditto). I objected to a good many of such vacuous or vapid passages in my report, but now see that I didn't mention this particular one to the editors, probably because I had something more essential to object to, which was the notion that it was ever desirable to make an incorrect estimate.
The author seems to have thought, if he read my comment, that I didn't understand his purpose in encouraging children to make incorrect estimates, in that, e.g., I might not have understood why one might need to falsify the measurement of the time it takes to get to school in the morning. So he explains that "an overestimate makes sense" because of all the things one has to do, which he even enumerated for readers like me, breakfast included.
And my other suggestion, that maybe one wishes to cook something longer than the estimated time (as printed in a cookbook, say), to kill the bacteria, you see, was a bit of a joke, since I did understand that “warming up a food item” (probably a fully cooked leftover) doesn't always require killing bacteria; but the author of the revision evidently thought (in view of my apparent objection) that he hadn't made that point clear. Thus the last sentence explaining the circumstances and rationale for an "underestimate". He changed “warming up a food item" to the more definite example of "cooking vegetables on the grill", so that people like me would understand (salmonella no longer in question) why underestimates are better than overestimates.
I am conjecturing here, of course, but I believe the author missed my point entirely. Certainly his attempt at an improved version took no account of it. None. I was talking about mathematics and science. In daily speech, people often say, when using a measurement, that the number they are actually using, if a bit more than the center of the measurement's error interval, is "an underestimate". But that is only colloquial English. It is not the estimate that is "under"; it is the number ultimately used (as the time actually spent grilling the vegetables, say) that is “under”. It is the consequent action, based on a number greater than the true estimate which for PSSM reasons the person measuring is trying to forget or conceal.
If I had a patient listener he might finally understand the distinction between a false estimate and an action which prudently makes use of the (true) estimate to announce a number deliberately taken at some distance from it; and that listener might say that this is not the way the word "underestimate" is used in daily speech. My answer is that school is a place where the student is to learn precision, and that he is to make colloquial errors on his own time. Mathematics in particular should employ an accurate language, saying exactly what is meant. The vague or uncertain use of mathematical terms such as "estimate" will come back to plague the student in later years. Elementary school books and programs, and the education of elementary school teachers, should be firm on this matter. It is a great misfortune that NCTM seeks to insert the errors of everyday speech into mathematical speech and writing.
In particular, the “plus or minus …” that comes with all scientific measurements, apart from the counting of a finite number of discrete objects, has a meaning in statistics that even the non-scientist must learn to understand and use, for it announces the interval within which the true value is to be found, apart from a small (and specified!) probability. The word “overestimate” has no place in such statistical language, and the “confidence interval” is an idea not too hard for a schoolchild to understand, one that is important even to educational researchers who do not delve deeply into statistical theory.
If we are measuring a room for carpeting and arrive at 10 feet as the width, with an error of a few inches at most (one way or another), we do not announce or write down 11 feet as our estimate because we want to make sure there will be enough carpeting delivered for our purposes. We may order 11 feet from the carpeting store; we may call it an overestimate in talking to the salesman; but the 11 feet is an order in the shop, not our estimate, which was and remains 10 feet (plus or minus some few inches), no matter how many feet we had ordered measured out and delivered.
Even in a non-scientific application the PSSM prescription is mischievous: A child should know that if, perhaps many years later, the question of the width of that room comes up, when (say) the biographer of the person who once lived there is describing the house, the answer should be that it once had been measured to be “about 10 feet”, and certainly not “about 11 feet”, even if the purpose of that measurement had been an order of carpeting. If that biographer reads in a diary of some sort that the width was measured to be “about 11 feet”, he will have been lied to; he might in consequence make a serious error in his own estimate of (say) the prosperity of the subject of his book. (The fact that diaries often lie, in contexts more damaging than measurements of room size, does not affect the spirit of my objection.)
Science and the major applications of mathematics do not proceed in the manner that PSSM is suggesting here; if they did, as PSSM is encouraging our schools to do, our technological society would collapse. One can reason from approximate measurements, yes, but only if we call ten ten and not “eleven”, even if our number is only the width of a room which the famous artist who once lived there wanted to be sure was fully covered by the carpeting he purchased for it.
To say that the child in question will learn better measurement announcement behavior as he grows older is no excuse for teaching him to lie in Grades 3-5 via PSSM. What a child learns in early years is what is hardest to amend. This one learns to lie to himself. We have all seen his type:
His will be a house of many clocks, some fifteen minutes ahead, and some thirty, according to whether they are intended to get him to work on time or to the concert on time. When he gets too used to those spurs to promptitude, as people do, he then sets some of them ahead by another quarter-hour, maybe a half.
The day will come when his mind cannot hold all the meanings of the various clocks – none of them set to tell him what time it is – and he will find himself coming to a party a half-hour before the hostess, still in blue jeans, has finished setting the tables. “Oh, it’s no trouble,” she will say, for she, too, will have learned to lie, though (as a hostess) to better purpose.
Ralph A. Raimi
Revised 8 November 2008