High Stakes Examinations and Educational Foolishness

 

Every time I look at one of the exams being given by some state, or even the practice problems exhibited on a web site by some publisher in self-praise, I find something wrong – ignorant, askew, confusing, ambiguous, pretentious – something deleterious to the putative purpose of the exercise or exam problem.  There was a time, fifty years ago, when ignorance was evident, and surely boredom and tedium, but the intent was more direct and a student could learn to please the teacher without having to lie to himself or to outguess some deliberately evasive language of his teachers, making it hard to find a correct answer, or – in the other direction – some deliberately revealing language, that immediately advertises the correct answer without troubling the student with having to know anything.  Sometimes these faults are not even deliberate, but are the consequence of ignorance on the part of those who framed the questions.  We shall see examples of all four sorts of infelicity in our examinations.  It is true that mere facts and canned procedures, however definite and relevant to the desired instructional results they may be, are not an ultimate good in mathematics instruction, and are rightly condemned when they are the sole material of school examinations; but what has replaced these things is worse. 

 

Carelessness and ignorance are often present, but are  not the only distressing characteristics of the illness in mathematics instruction I see evidenced in the exercises and examinations that accompany it today.  More intrusive and abusive are some subtleties that are simply not noticed, or – if noticed – not believed by the testing professionals to be deleterious, and yet which I count a greater danger than to the instruction these tests are intended to measure than the general “dangers of high-stakes testing” so loudly and dishonestly averred by demagogic opponents of the public oversight of educational success and failure.  A more complete characterization of the evil I have in mind will be easier to explain after some examples have been probed at some length.

 

I shall therefore begin by describing in detail a few of the problems found in just a few minutes’ search in three locations that have had interest for me.  They do not represent an exhaustive search for the best examples, but they are more than enough, even though it was not my intention, when I first went to the three sites named below for other purposes, to write an analysis of their offerings.  I went to the first one, CMP (“Connected Mathematics Project”), to see for myself what improvements they had made on the first edition of their middle school program, for I had been told that CMP now claims to have answered some of my mathematician friends’ criticism of important missing content in their first edition, one of the neglected subjects having been the division of fractions. 

 

I went to the second web site, the one for the New York State Regents Examinations, because I live in (upstate) New York and have had some connection with persons associated with these tests in the past year, and with the controversy engendered by the 2003 Math A examination, which had caused an unusually large number of students to fail, and which turned out to have had some plainly faulty questions.  So embarrassing were the results that the state appointed a Commission to study the matter, this Commission comprising (among the certified experts on school mathematics and testing) a mathematician or two, which the committees that had composed the examination plainly had not.  In the wake of their report the entire examination system in New York is still under study, and indeed a new set of State Standards for mathematics is being written.  The succeeding (April, 2004) Regents’ “Math A” examination is now reported to have been posted on the Department of Mathematics web site, and I was interested to see if the Regents, or those they appointed to create examinations,  had learned something from the ambiguities that had been pointed out in earlier editions. 

 

And I went to the third web site, that for the Michigan statewide mathematics examinations called MAEP (Michigan Educational Assessment Program), because my grandchildren live in Ann Arbor and had been subjected to them or their immediate predecessors two or three years ago, and because Michigan middle schools are great consumers of the CMP materials, which were  produced by a team of five authors, three of them from Michigan State University.  Thus the MEAP would give me an insight into what the state imagined CMP had, or should have, provided the middle school children who had used that program.

 

 

I will admit at the outset that I did not expect to find a good result in any of these cases, but I insist that examples I present below are those that came up immediately upon my glancing through the material on offer in each of the web sites consulted.  It was not necessary to sift through all the newly posted CMP exercises, or all the Michigan or New York examination questions, to find the kind of thing that exemplifies, and implicitly defines, the defects in today’s school mathematics philosophy and instruction, for much of it is immediately visible even in questions that are correct enough and simple enough to be “correctly” answered by any student who has learned what the school has sought to teach.  But some of it is not even correct, as will appear.

 

First Example:  Connected Mathematics Project (CMP)

 

The Connected Mathematics Project web site has posted some new textual material that CMP now features, mathematical items, the division of fractions for example, that had somehow been omitted from its earlier vision of constructivist mathematics for middle school students.  At  http://www.phschool.com/math/cmp/new_student_pages/LB6_BPII.pdf>

one immediately finds, in Problem 1, a setting of the sort the authors regarded as a real-life situation appealing to middle-school students:  It is posited that a store offers its customers coupons, some of which when scratched reveal an entitlement to a discount.  Part (c) of the mathematics question  (rubric: 6th Grade skill with fractions and decimals) reads:

 

(c)  The store conducted a survey to find out whether the scratch-off coupons had influenced customers to buy.  At the end of the day they tallied the results:

 

          *  Would have purchased the items without the coupon: 556

          *  Were strongly influenced by the coupon:  378

          *  Were somewhat influenced by the coupon: 137

 

(i)      What percent of the customers was influenced by the coupon?

(ii)     Make a graph to show the percentage of customers in each category.

 

 

          I was initially unable to answer question (i), and had to look back a couple of times to see if I had read it right.  Yes, it wanted me to say what percent of the customers had been influenced; but it had not told me how many customers there were, nor whether those customers who had answered the poll and admitted influence had in fact been influenced as they had reported.  Upon reflection, and making use of my own childhood experience answering badly posed examination questions, I decided that the question must have presumed that all the customers had answered the survey question, and that they had all correctly recalled whether or not they had been influenced.  But nowhere did the question say so. 

 

I can, you know, compute the fraction of respondents who said that they had been influenced; but this was not what was asked.  Surely the authors could have written:

 

( i ) What percent of the people who answered the questionnaire said they had been influenced by the coupons?

 

and

 

( ii ) Draw a graph to show the percentage of respondents in each category.

 

Distinguishing between “customers” and “people who answered the survey” would not have been to introduce difficult notions beyond the capability of middle school students, nor would it have lengthened the text particularly.  And it would have named the population reported on with no possibility of misunderstanding. “Respondent” is no longer a word than “customer”, nor more difficult to understand.  It is possibly known to 6th grade students; if not, maybe it should be taught.  If the task is too daunting, the examiners can avoid it by repeating in (ii) the circumlocution I used above in ( i ).  The second failing, ignoring the possibility that the numbers tallied “at the end of the day” sometimes err because the respondent doesn’t really know what he would have done in the absence of coupons, is a bit more subtle; but it is certainly something that has been pointed out to anyone who has been given lessons in random sampling and statistical interpretation, such as CMP and all the other popular NCTM-inspired math programs have been advocating and including in their “exemplary” or “promising” textbooks for the past fifteen years.

 

Is it pedantic of me to wish for the more accurate wording, and to suggest that there is some pedagogical evil in the manner of its original presentation?  Well, I have never heard of a survey that obtains 100% response, or for which a reasonable pollster assumes 100% truthful or comprehending response from those who do respond.  In reading the introduction to the coupon survey questions I took at face value the notion that something about polling was intended, among other things.  But in fact this question had been placed in the program in answer to a complaint about a paucity in the first edition of CMP of questions (and instruction, or discovery rituals) concerning “fractions and percentages”.  The  writer of this particular question was trying to ask whether the student understood “percentage”, and could divide each of the three numbers of responses by their sum, and could graph something about the results.  (It is not clear to me what a graph “showing the percentages” is, by the way.)

 

But the author of this question was not intending something about statistics at all.  He could have asked a mathematically equivalent question in a direct way, without inventing stores, coupons and surveys, but the demand for “relevance” and “math across the curriculum” required him to get into foreign territory, where his expertise in quotients was insufficient.  He evidently forgot that in another part of his own CMP curriculum the subject of “statistics”, imagined to be a real-life interest of students, includes lessons in sampling bias that should address directly the confusion he was here making between respondents and customers, among other things.  Study of bias in sampling consumes a lot of “reform math” class time, in fact, time that could be used for learning arithmetic and geometry (or music).  Suddenly, here, in this chapter, all that is forgotten.  Problem 1(c) is about “fractions and percentages”, you see, so that paying attention to accuracy in the statistics part of this problem, to be sure to get it right, was not his duty at the moment.  It was as if he had invented a math problem set in the time of Charles I of England, and had the Duke of Buckingham flying an airplane at various speeds for graphing purposes.  To the reader distracted by the anachronism he might say,  “Why quarrel about dates?  Today’s subject is graphs, not memorization of history.”

 

It wasn’t until I returned to the CMP web site for a second look that I found out that the rubric for this example had been “fractions and percentage”, not statistics, and thus understood the reason for my having been put off the track.  In all truth, the first thing I had looked for when reading the problem, foolishly thinking it was about a poll of customers, was the total number of customers, as against the numbers whose answers were tabulated.  And if this had been a real store with real coupons I’m quite sure the manager who paid real money for the real survey would also have noticed the omission, for a self-selected sample is never random.  In short, this problem simply did not give enough information, unless the student is expected to know that this is a “fractions and percentages” problem, and that the polling part is a fraud; and that he should in the process command himself to forget the lessons he learned three weeks earlier in the chapter on probability and sampling.  He should know, too, that by “customers” the examining authority meant “respondents”, and that what respondents said they would have done was (in textbooks and exams) what respondents would have done, in the absence of coupons.

 

That makes a goodly budget of missing information that the student is to intuit before beginning to exhibit his mathematical skill.  Yet “not enough information” is not the whole story here; there is also too much information, words added to make the problem richer, more vital, not the same old gray lists of sums that disfigured the textbooks of yesteryear.  Vital?  What could be more tedious and unnecessary than the information that the results were tabulated "at the end of the day"?  Is this a bedtime story?  Should it perhaps, to make it more relevant to a youngster’s world, have begun with “Once upon a time”?   On the other hand, the problem’s description of the coupon scheme itself, which I have not quoted in full above, was woefully verbose, including unnecessary (and sometimes questionable) information about the way sales taxes are computed when discounts are given, and the exact sorts of discounts the winning scratched coupons provided, whatever a scratched coupon might be.  Students not conversant with up-to-the minute forms of merchandising at American shopping malls would have trouble understanding the setting of this problem to begin with.

 

In past years, before “reform programs” were composed to make mathematics more meaningful, this problem, or a problem assessing the student’s understanding of the arithmetic involved, would have been stated something like this: 

 

(i)  Given the three numbers 556, 378, and 137, what percentage of the total is each? 

(ii) Draw a bar graph plotting the percentages on the vertical axis and identifying the bars along the horizontal.

 

(At least, this was my interpretation of what the original question meant by its graphing demand, before I found out from a correspondent that CMP has published suggested answers to its problems, and in this case suggested a “pie chart”, with colored sectors representing the three percentages, along with very detailed instructions on the calculation of the angles subtended by each of the three sectors.)

 

A student who can answer this question will certainly know what to do when he grows up to be a storekeeper and counts the customers as they come in and (some of them) elect to enter the scratch-test sweepstakes.  Moreover, the examination will be shorter.  And it will be correct, for among other things it will have explained the sort of graph it expects the student to know how to plot. 

 

          A few words are worth writing about that graph at this point.  The CMP example asked the graph part in these words:

 

(ii)      Make a graph to show the percentage of customers in each category.

 

          This phrasing is entirely obscure to a mathematician, let alone the man in the street.  A regular reader of the newspapers sees graphic displays of all sorts of things every day, and especially on Sundays, and they come in a hundred styles.  What they have in common is that they illustrate with graphic art something that when given in tabular form or in words is less easily assimilated, or less decorative.  In the present case a “tabular” form might be:

 

Not affected by coupon promotion:             51.9%

Strongly affected by coupon promotion:      35.3%

Somewhat affected by coupon promotion:   12.8%

 

(Here I wondered in passing why the authors of the problem had ordered the three categories in so strange a way.  I, at least, would have found more illuminating the listing in the order “strongly, somewhat, not,” or “not, somewhat, strongly”, that is, in order of strength of influence exerted by the coupon scheme.  Either of the two orderings I have just suggested would make the display, whatever it turned out to be, more memorable to the reader than a listing placing “strongly” between “somewhat” and “not at all”.  But then, this is a question about percentage and graphs, not about illuminating displays.  Can you imagine the New York Times printing a graph or similar tabular display concerning family income and giving the percentage of families having income

 

(1) Less than $20,000 per year;

(2) Between $50,000 and $100,000 per year

(3) Between $20,000 and 50,000 per year

(4) Over $100,000 per year

 

and listed in that order?  Not in a million years.  But CMP, obsessed with fractions and percentages, was only playing with polls and graphs for window-dressing, and it shows.

 

Actually, I cannot imagine that a “bar graph” representation shows anything this “tabular” representation  (“not affected…, somewhat affected…,  greatly affected…”) does not do as well or better.  I wonder if the display I might myself have made in answering this question would have received full marks.  Long after having made these reflections to myself I discovered that the new CMP materials do include not only a list of problems, of which this one about the scratched coupons was a part, but also a later section, “Explain Your Reasoning”, with space for the student to answer some questions concerning the rationale for the numerical calculation or graphing he had just exhibited.

 

          In the case of the graphical display, the question under the “Explain Your Reasoning” rubric was worded, “How did you find the percent of customers in each category in Problem 1c?  Explain why you chose the particular type of graphic display that you used.  Explain how you constructed that display.”  Taking a stab at an answer I would have given had this been an examination of importance to my own future, I could only think that I chose the graphic display I used (the bar graph as described above) because I had to choose something.  How I constructed it (yes, they asked that, too) should  be evident from the diagram itself, which I won’t reproduce here.  I wonder, would it really please my teacher if in this part of my homework I wrote, “I took a pencil and marked the horizontal axis with three points labeled “not affected”, “strongly affected”, and “somewhat affected” and then drew a vertical bar above each of the three points with lengths 51.9 mm, 35.3 mm, and  12.8 mm.”?  This would convey no more information than the graph itself, were it properly marked, and I believe a teacher’s requirement that a graph be self-evident is a better lesson than the requirement for “explanation” of something that itself is intended to be an explanation.

 

          (A more amusing question appears in this “explain your reasoning” section of the CMP “Bits and Pieces” addenda to the earlier versions of its program, this one connected with a problem in division of fractions. It is associated with Problem 5, which I will not reproduce here, and asks: 

 

“Do you agree with these computations:  4 ⁄ (1/3) = 12 and 4 ⁄ (2/3) = 6?  If so, why is the second answer half of the first?”

 

My own view, which I won’t spend much time elucidating, is that the second answer, 6, is half of the first answer, 12, because twice 6 is 12.  What on earth does the book mean by that question?  Probably that when you divide by 1/3 you don’t reduce things as much as when you divide by 2/3.  No, it can’t be “reduce”, can it?  Well, something like that, or so I thought before I was led to the part of the CMP web page containing model “answers” for its examples.  I recommend that the really interested reader go there, for a quite opaque discussion.  Altogether, I think this emphasis on ‘why’ instead of ‘how’ is a pernicious way to teach about fractions -- and a good bit else -- especially with subject matter quite difficult for the very authors and teachers of the material at hand.  Very often the only satisfactory answers to “why” questions are mathematical theorems of considerable subtlety, while other “why” questions, such as this one about 12 and 6, are either not answerable at all or are fully answered by the very computation being asked about.)

 

The casual assumptions these CMP authors made about the imagined customer survey conveyed more than one unintended false lesson to the child test-taker.  The worst such lesson is that mathematics is a game of mind-reading.  I will describe other examples below.  I see such things all the time -- en passant -- even in otherwise good exam questions.  The question just described was not very good in any way at all; it was a simple arithmetic problem of Grade 6 level housed in an unnecessary setting commanded by the current ideology of having all school mathematics “situated”. The second false lesson conveyed by this problem is that mathematics is more complicated than it is. Asking for a graph of some undefined and thoroughly unnecessary sort, following an unlikely story of a scratch-test discount scheme and survey studied at the end of the day,  sounds like a  probing demand, but it serves only to confuse.

 

There is also a false lesson given the people who use the results of an examination composed of such problems. (The CMP questions were from a book, not an examination, but are typical of examination questions as well.) The state of New York, for example, wishes to use the results of its statewide examinations to determine not only how each student performed, but by looking at the results on a statewide level to discover what part of its mathematics program is being successful and what part needs improvement.  Were this one of New York’s examination questions, expressed as it is in a farrago of commercial and polling language, the resulting statewide tabulations would not give it such information as whether it was misunderstanding of the idea of “percentage” that produced the wrong answers, or whether is was inability to read a rather complicated bit of English prose.  Even in the CMP setting, as evidenced by the carelessness of the description of the survey, the question, intended only as an exercise in percentages and graphs, failed to address its own intent sharply enough. 

 

In the contested 2003 New York State Regents’ examination the complaint, in the case of a similarly tangled tale posing as a mathematics question, was that it had been a test of English, and that Spanish-speaking students not yet skilled in English were being graded on that which was not mathematics.  Of course Spanish-speaking immigrants should learn English, and even be examined in the schools on their acquisition of the language, along with the Chinese and all the others, but to paint this difficulty of the mathematics examination as an “equity” question (which is a popular objection, though it confuses a poor examination score with some sort of punishment) ignores the very purpose of the examination.  A valid test is one that reports what it intended to report, that tests what it claims to be testing.  Questions of so mixed a nature, even were the questions good ones, and even if the questions had been translated into everyone’s native language, would only provide comparative (as between one student and another) results relating to a mixture of mathematics learning and a few other things, some influenced by home environment.  The State Education Department would be unable to determine from these results just which parts of the mathematics curriculum were being done well, or badly.

 

          On past occasions when I have made complaints resembling those above, concerning the wording of examination questions, I have been countered with the scornful,  “Well, everyone knows what we mean, don’t they?”  I have to admit, sorrowfully, that upon reflection (but by no means initially) I did indeed come to understand what they meant in most such cases; but such a defense is inadequate just the same.  Authors of teaching programs owe it to the public to say what they mean, do they not?  And they owe it to the children to teach them, too, by example much more than mere precept, to say what they mean.

 

Second Example:  The New York Regents’ Examinations

 

 

On the most recent NY Math A exam, given in January of 2004 (see http://www.nysedregents.org/testing/mathre/mathatestja04.pdf ), the following appears:

 

Now the indicated construction is an arc apparently centered at P and intersecting AB in an unlabeled chord, and then two more small arcs below the chord (and below AB), evidently with centers in the two points where the first arc had intersected AB, so that connecting P with the intersection of the two final arcs would, if my interpretation of the diagram is correct, produce a line perpendicular to AB from P.  The point C, pictured between A and B on AB, is the base of this apparent perpendicular.

 

The four choices are that the construction is that of:

 

(1)              an altitude drawn to AB;

(2)              a median drawn to AB;

(3)              the bisector of angle APB;

(4)              the perpendicular bisector of AB;

 

and clearly (1) is the desired answer.  What makes the question a bad one is that the correct answer is visible to anyone with an understanding of the dictionary meaning of the word “altitude”, someone who needs absolutely no knowledge of Euclidean constructions, of triangles, or the nature of circular arcs.  A student can see with his eyes that PC is an “altitude” as (1) describes, even by the incorrect reasoning that it is vertical.  If the writer of this question had sincerely wished to test understanding of the construction of an altitude he should have diagrammed an altitude constructed to a non-horizontal side; lots of correct answers would have been lost this way. From the mere dictionary definition of “altitude” the student gets full marks, yet this question is intended to test Euclidean constructions, not appearances. 

 

Furthermore, the drawing shows a triangle which is apparently not isosceles, so that the other three choices are apparently wrong: the very picture shows AB is not bisected by the vertical line constructed, nor the angle APB bisected.  But if the triangle were isosceles, with sides PA and PB equal to one another, all four answers would be correct, both visibly and by construction.  Yet the problem does not state that the triangle fails to be isosceles, something that must be stated if the mathematical falsity of the other three choices is to be guaranteed.

 

I would call this question ambiguous on the grounds that the indicated construction could be what is described in (2), (3), or (4) if the indicated triangle happens to be isosceles.  The author of the item would probably argue with me, saying that the picture clearly shows a non-isosceles triangle.  To him I would reply that Euclidean constructions are not proved by using the “obvious” properties of the special figure that happens to have been drawn to illustrate the data.  It is the duty of the problem author to give all the needed data, including – in the present case – the statement that the triangle is not isosceles, or at least that it is intended to represent a generic triangle (which at the middle school level might be referred to as “any triangle”).  The language used, “in the accompanying diagram”, invites the student to make his own interpretation of how general or special a case is intended, since every diagram is unique, special.  (Euclid would write “a triangle” when he meant a generic triangle.  When he did not, as in the Pythagorean Theorem (I;47 of Euclid’s Book I), he explicitly referred to a right triangle, and not to a triangle “as pictured”.)  Without precision there is no mathematics, and no way to apply what one thinks the theorems say.  Even the pictured arcs of the NY Regents problem are not described in words, which they should be; the author simply knew that students taking the New York Math A examination had by state standards been exposed to this particular construction, and so didn’t think he had to ask a complete question.  “Well, everyone knows what we mean, don’t they?”

 

          In sum, this question really didn’t test whether the student had remembered this construction or why it worked; all it did was offer, by means of its presentation, a multitude of mathematically irrelevant clues to the correct answer.  A serious question about this construction would have been to ask the student to use straight-edge and compass to construct an altitude from P to side AB?  The Math A examination does after all have a place for “extended response” questions, and that would have been the place for this one. 

 

It is my belief that those who presented this problem in this (multiple-choice) way did not want to take the chance that anyone would get it wrong.  This anxiety for high (or “passing”) scores on statewide examinations is one of the notable features of today’s “accountability” pressures now emanating even from national legislation appropriating money for “education” based on demonstrated success of the programs being subsidized.  The construction of examination questions that appear to examine knowledge of things listed in the state standards as published these days in almost every state in the country, but which in fact “give away” the answer, is a prominent form of cheating on the part of the educational authorities, and thus one source of bad exam questions.  There are other sources for the bad examples under examination here, however, and they are to me more disturbing for the long run than this sort of “pretend problem”, since they so often reveal mathematical ignorance, and ignorance of good English usage, both probably harder to conquer in public affairs than mere guile.

 

The web site of the New York Education Department contains copies of all recent Regents’ Math A and Math B examinations (Math B being voluntary and more than the minimal needed for a high school diploma) at http://www.nysedregents.org/testing/hsregents.html,

and there are sample problems from tests at or below the Grade 8 level on other parts of the site.  At http://www.emsc.nysed.gov/osa/testsample.html    

the menu offers a link to “High school math” sample problems which turn out to be samples illustrating the intended level of questioning for the new Math B examination, which at the time of posting (1999) had not yet been given.  These particular examples were never used on the actual Math B examinations, which commenced in the following year.  It was fortunate for the state that they were not used, or the state would have found the public response to Problem 1 as embarrassing as the response to the ambiguities of the Math A examination given in 2003, which received such coverage in the newspapers that a Commission of enquiry had to be established.  If there is no public outcry, however, the state is hard to shame, as I discovered from my mainly one-sided correspondence concerning Problem 1 of the Math B sampler.

 

The State’s answer to my first letter didn’t address the question of whether the problem was correct or not, but stated that the example as printed could not be withdrawn or changed because the scaling of scores on later Math B examinations depended on the recent use of these sample problems in trial runs, and if any question were now changed the scaling would have to be done again.  To my second protest, that the problem was wrong, egregiously wrong, and a disservice to the public whatever scaling system adjustment they might have to make upon withdrawing it, got no answer.  The State is presumably still using these sample questions, and the statistical analysis of responses, for some purpose I can only guess at. 

 

Here is the problem, exactly as found at the “samples” website mentioned earlier:  http://www.emsc.nysed.gov/osa/testsample.html . 

 

A thoughtless glance at the graph will suggest the familiar contours of a logarithmic function graph, assuming it were plotted on a Cartesian plane with equal spacing in the vertical and horizontal scale markings.  The composer of the problem (not the composer of the graph, which is clearly reproduced from a scientific source) was thoughtless indeed, for the axes are marked carefully:  the stream velocities are 0, 100, 200, ... , 800, equally spaced horizontally; but the vertical axis has particle sizes .00001, .0001, .001, .01, .1, 1.0, 10.0, and 100.0 (mm) also spaced equally.  This is not a linear scale; it is a semi-logarithmic graph, as is useful in engineering applications.  In such a graph a straight line would represent the exponential function, and the graph actually shown is the graph of something more like a linear function, as one can see by tabulating some paired values, i.e. coordinates of points on the graph..  Not exactly, as I discovered by taking actual measurements off the page, but a great deal closer than logarithmic. Were I taking the examination would I answer “linear”, which is choice (1), closer to the truth than the “correct” answer (3)?  No, on second thought I would realize the examination-maker’s error and give him the answer he wanted.  Why make trouble?

 

It is my belief that the person or persons proposing this problem were not only careless, but were actually deeply ignorant, not to have noticed that the non-linear scaling of an axis on a graph makes a difference in the shape of the graph of a function.  Apparently all they knew about “logarithmic” graphs was the shape, and this is all they expected of the students, too.  The fact that a function is defined by the functional values, not the picture, never entered their minds, though I suppose close questioning might have taught them something they hadn’t been thinking of before.  One could, to clinch the point, make a polar plot of the logarithm function, via the equation r = log(θ), and the result would be a spiral of a rather unfamiliar kind.  Would they say this was not the graph of the logarithm function because it had the “wrong shape”?

 

A year or two later, with this erroneous example still on the web site, I revisited the New York Regents examinations and looked up the Math B examinations as they had actually been given statewide, to see if at least in the serious business of grading children they had finally got the lesson straight.  Well, they had cured the problem, in  a way.  In one exam they exhibited pictures of four graphs (see below), no longer real-life graphs taken from science somewhere but a made-up graphs such as one sees in textbooks, with the same four familiar shapes.  But they took no chances on scaling them.

 

 

 

 

 

“The cells … increase logarithmically” is obscure to me; do they mean the number of cells?  And what is “increase logarithmically”?   Is a formula of the sort “y = 3 + log (x+1)” a formula of logarithmic increase?  If it is,  (1) and (4) would be acceptable answers if the scales were interpreted with suitable guile. Whatever they mean by cells growing logarithmically, the absence of printed coordinated scales to trip them up saves the day; it is clear that, by New York State Regents standards, (3) is the logarithmic graph, and must be the answer.

 

          In general, the New York Regents Math B examinations that have actually been given do not have any graph recognition problems as egregiously incorrect as the sample question about the velocity of particulates in running water; the “logarithmic growth of cells” problem just under discussion being ambiguous rather than wrong.  But graphs without indication of scale are common, not only in New York, but in the Michigan examinations called MEAP, some examples from which are discussed below. While it can be found at the MEAP web site mentioned below, where other MEAP questions are discussed at greater length, it seems appropriate to insert this MEAP “high school” level problem here:

 

 

And every year now, year after year, a version of this problem occurs on examinations all across the country, with no coordinates marked.  Teachers have come to take such “shorthand” for granted; absence of a coordinate indication has a schoolbook interpretation of “imagine Cartesian coordinates with equally spaced integers marked in as a scale indication.”  A mathematician might accept such a convention when writing something on the back of an envelope in discussion with a colleague or student, if the point was to illustrate the way linear functions behave (straight line graph in the usual Cartesian scheme) as against the cosine, say, which has a familiar periodic oscillation, up and down; but he would know immediately how the graph would have to be modified if some other scaling were used, while the school exam-maker has nothing else in mind than to repeat the tired questions of yesteryear and wait for retirement. 

 

“Save one who, stout as Julius Caesar…”   Yes, one courageous New York education official, taking too seriously the NCTM dicta about making mathematics relevant, vibrant and true to life, went out and found a real graph about real particles and stream velocities, a graph that looked somewhat like what he had been calling “logarithmic” all his life for schoolroom purposes.  Though I cannot be sure my letters to Albany convinced him, or them, that they had been wrong to do so, someone seems to have noticed and not repeated that particular error, but I cannot say I am pleased to see the way the actual  Math B examinations have repaired the unintended consequence of going to real-life for “real-life” problems.  They have instead (I should have known!) returned to old-fashioned ways, and are again judging students on how well they know how to produce “correct” answers to literally nonsensical questions, using visible graphs of “functions” having secret inputs and outputs. Even if the exact scales are not marked on a problem of this sort, it should be indicated by unlabeled marks that the x and y axes have identical scales if that is the intention.  To put actual numbers on those marks might give away the answer to any student astute enough to check some values with a calculator (these are often permitted on examinations), in order to eliminate the wrong graphs, but this maneuver would be made more difficult with the compromise convention of indicating comparable scales for the axes.  All this assumes that the intent of the question is to know if the student recognizes the behavior of the common functions appearing in elementary applications.  If the intent is to make it easy for students to score points, however, the present system suffices.

 

Returning to New York now, but to the web site http://www.emsc.nysed.gov/osa/mathei/matheiarch/gr8bk1math.pdf

we find at the Grade 8 level questions of such simplicity that the state should be shamed, not only for its high rates of failure, but for setting its standards so low that the public believes it is -- even with its high failure rate – more successful in its mathematics programs than it really is. (This examinations is footnoted as having been created by the publishing company McGraw-Hill, though © 2002 NY State Department of Education.) Samples:

 

4.  What rational number is the multiplicative inverse of 3½?

                   F  2/7

                   G  - 2/7

                   H  7/2

                   I  - 7/2

 

This question is mere vocabulary, a fossil from the days of The New Math, and certainly not indicative of any ability or skill in mathematics.  It separates the children who went to class at all from those who didn’t, but no more so than a question about the name of the war going on during the presidency of Franklin Delano Roosevelt.  As for the vocabulary itself, “multiplicative inverse” and “rational number” can have no function beyond making the question sound profound.  Well, it has another function, which is to give warrant to the wasting of class time during the semester making sure children can pronounce those words and use them correctly.

 

If Question 4 were rephrased as follows it might have some point:  “For what number A is the product of A and 3½ equal to 1?”  My belief is that the question is not put this way because it would then be seen to be a Grade 6 problem rather than the advertised Grade 8. Indeed, I suspect fewer people will get this correct than the question as posed, concerning “multiplicative inverse”.  Indeed, the rigidity and futility of traditional mathematics instruction is evidenced by the fact that the some students are able to answer a question about “multiplicative inverse” without really understanding what it is all about, which my rephrased version would uncover – should anyone really be interested in uncovering it.

 

 

The following question on that same New York examination is:

 

 5.  Which number is equivalent to the expression below?

                   | -15+4 |

 

A  -19

                   B  -11 

                   C  11

                   D  19

 

          This one is equally trivial, though it does ask for the meaning of “absolute value”, and the recognition of the symbol denoting it; these are sometimes thought to be difficult for students.  Upon examination, however, any difficulty a student might have with the idea as here illustrated can only be due to the pedagogical ineptness of his teacher, or (much more likely) the unnecessary elaboration of the definition of “absolute value” as given in traditional textbooks, themselves often victims of “new math” elaboration from the 1960s, where the definition

 

 |x| = x if x 0, and |x| = -x if x 0

 

was popular, though indeed hard to understand at first reading.  At the introductory

level it is much more illuminating to say |x| means the (positive) distance of x from zero.  With such teaching, along with the interpretation “|x-y| is therefore the (positive) distance between x and y on a number line”, from which one obtains that |x-y|=|y-x| for any two numbers x and y, all difficulties with “absolute value” tend to evaporate, and yet these definitions are quite as mathematical, once numbers are identified with distances, as the technical one exhibited above, whose value only appears in more advanced contexts anyhow.  Not that the story about |x-y| is needed in answering this particular question. With the distance interpretation it is clear that the answer to problem 5 is C (i.e. |-15+4| = |-11| which is 11), while for some reason the bipartite definition still commands a following among textbook writers, and exam writers, who should long ago taken this particular sad lesson from the troubles of the 1960s more seriously.

 

Like its predecessor, this “problem” (Problem 5 above) is made more impressive than it deserves by its use of the word “equivalent” in place of “equal” or “the same as”.  That, too, is a hangover from “The New Math” of the 1960s, when “equivalent” had a meaning in connection with the sometimes necessary distinction between symbols and what they represent, a distinction that has now been totally lost in what now remains of the school mathematics of that time.  If the question had begun, “Which numeral …”, and if the distinction between numeral and number, so popular and unnecessary in those days were still part of the curriculum today, this verbiage would have some point to it, but its use in the present context illustrates only that the writer was searching for ways to sound deep while asking nothing worthwhile.  A problem in which the absolute value symbol was actually of use would tell us something; this one does not, except as a ghastly reminder, once the state grades are tabulated, of how badly our children are being taught.

 

                   A good example of another sort of problem, very common in a time when everyone must seem to succeed,  one so trivial as to be a “give-away”, is # 25:

                               

 

          Clearly the question can be answered (9 inches) by anyone with half an eye, whether or not he knows what similarity is.  All he has to do is to believe the picture.  If the state really wanted to know whether its 8th grade (Eighth grade!) students understood similarity and ratio, and how to handle rational numbers, it could have made the three given lengths  2, 5.7, and 3, instead of 2, 6, and 3, and offered as its choices labeled A, B, C, and D four numbers closer to each other than a look at the diagram could distinguish – and the problem would still be simple enough for examination purposes.  A syllabus that claims to include ratios and fractions is surely entitled  to consider even more complicated fractions than these. No child can be a carpenter until he learns that much, at least; and this is 8th grade arithmetic, not quantum theory.

 

Third Example:  The Michigan MEAP

 

          The State of Michigan has an examination system called the Michigan Educational Assessment Program (MEAP).  Released questions, with answering rubrics in the case of extended response problems, are found for recent examinations at

http://treas-secure.state.mi.us/meritaward/mma/released.htm,

 

the “merit award” part of the URL for this site referring to the fact that good scores on these examinations lead to state awards of money for college tuition.

 

          I consulted, at this site, the sampling of MEAP Grade 8 math examination questions administered in April of 2001 and the sampling of “high school” level MAEP examinations questions for 2002, which were among those  provided there as a public service, aid to children and teachers who wish to know what to expect in subsequent testing. I made notes on the first five problems of the 2001 test, which were multiple choice, and Problems 15 and 18 of the 2002 test, for the discussion I give below.

 

[Added April 29, 2004:  The Michigan web site referred to above now no longer contains all the information it had when this paper was written, and might well contain nothing at all when the reader of this paper gets to it.  In particular, Problems 1-5 of the April, 2001 MEAP Grade 8 level examination, which I will later be using as examples, are no longer on exhibit.  The other two I shall exhibit,  Problems 15 and 18 of the 2002 “high school” mathematics test, are still there.]

 

From the 2001 Grade 8 Examination:

 

Problem 1.  In which set are the numbers equivalent?

 

          A. 1/3, 3/27, 33%

          B.  0.090, 90%, 0.90

          C.  88%, 88/100, 22/25

          D.  0.66%, 2/3, 66.7%

 

          This question actually does measure something (C is correct), though it is made unnecessarily easy to answer in that each of the wrong answers contains either a downright howler of an error or a superabundance of errors:  In A it should be known (by the 8th Grade – when a good school program is already into algebra) that 1/3 is not precisely 33%, for example, this alone is enough to eliminate A as the answer.  But 1/3 is also not equal to 3/27 which is 1/9, and 1/9 is surely not 33%.  The student thus gets three chances to see that A is not the answer, so that even if he knows nothing of the reduction of fractions to lowest terms, or has no understanding of “percent”, he can still eliminate A.  D also has all three entries different from one another, more than needed for elimination.  B, which requires no knowledge of fractional notation at all, is the best “distractor”, for it isolates a common misreading of decimally expressed numbers among young children and illiterates, but  while two of its entries are the same the elimination of B requires only understanding of decimal notation and no knowledge whatever of  “%”, let alone fractions.  C is therefore all that remains.

 

          Well, it is an easy problem, though it belongs at the 6th grade level rather than at the 8th (for which it was intended), and can be correctly answered by students having no understanding of fractions or of the word “percent”; but this is not all that one can criticize in the text.  Why the words “set” and “equivalent”?  Well, “set” is a pretty ordinary English word, and each choice features a set of three numbers; but there is here another disturbing echo of the language of “the new math” of the 1960s, where “set”, “number”, “numeral”, “equal” and “equivalent” were the meat and potatoes of advanced pedagogy in the schools – and usually, as in the present case, garbled. 

 

The “new math” distinguished carefully between “numeral” and “number”, a numeral being a symbol representing a number, while a number was an ideal object in one’s mind.  The distinction is necessary for fully logical discourse, in serious linguistic and philosophical discussion, but even among mathematicians only observed when the distinction itself is the subject of the discussion, and the attempt to be rigorous in such matters is often to tedious to bother with when numbers are being spoken of.  The number “three” is a mathematical abstraction, and there is only one such object in the universe.  Everyone knows what three is, though the writing of it might be “III”  in Roman numerals, “5-2” in the middle of some calculation, and perhaps “300%” where percentages are construed as numbers.  The three examples I have just given are examples of different numerals denoting the idea of a triple or triad.  Moreover, they are equivalent numerals.  As numerals they cannot be equal, for they are of different appearance, size and perhaps color, since equality means identity in arithmetic; therefore in the context of the sort of precise language promulgated by the logicians of “new math”  the numerals “5-2” and III were called “equivalent”.  Two numbers are either the same or not the same; there is no other “equivalence” in ordinary arithmetic, and mathematicians freely say things like “1/3 = 2/6” when they mean that the two different numerals just written represent the same number. 

 

The idea of equivalence (not equality) of numerals takes on particular importance when fractions are in question, for as fractions, the symbols 1/3 and 2/6 are surely quite different, though they are equivalent numerals in that they represent the same number.  Recognizing and creating equivalent fractions is a necessary skill in the arithmetic of rational numbers, for it is at the bottom of all practical calculations concerning fractions.  In adding fractions, especially, one looks for a “common denominator”, that is, one looks for two fractions equivalent to the original two being added, but with the replacement fractions having the same denominator; only after these have been found is there an easy calculation for their sum.

 

Now, “rational number” and “fraction” are often used interchangeably in daily conversation, and this is not harmful, and  it is true that too much was made of all this nomenclature during the 1960s.  Unfortunately, many a child (and teacher, and writer of so-called “new math” textbooks), unnecessarily drilled in it learned it incorrectly, so much so that the whole movement towards precise language in the formulation of mathematical statements got a bad name. I would not wish to include exercises in “number” vs “numeral” in today’s 8th grade curriculum, but people who do have occasion to use such words should understand them and use them correctly.  The author of Problem 1 of the MEAP 2001 8th Grade examination was evidently one of those who did not.  The word “set” must have frightened him into recollection of the days of “equivalence” and “numeral”, but not enough so that he got it all right in the present case.  The four sets which made up the choices in this problem were in this case really sets of numerals to be tested for equivalence, in one of the few contexts in school mathematics where the distinction between number and numeral makes a difference; but after all this preparation he called them “numbers” anyhow!  Sad.  With a little less pretension the author of this simple problem should have written, a bit colloquially but quite correctly in today’s mathematicians’ lexicon as well as the language of commerce and engineering,

 

 “Which of the four groupings below exhibits three equal numbers?”

 

          The MEAP Problem 2 is mathematically entirely trivial, though its statement fills the page with diagrams and symbols.  It is typical of such exams to do this.  Here is depicted a coordinate system with a circle inscribed in the square of side length 6 nestled in the first quadrant, the circle therefore tangent to the  sides of the square at (0,3), (3,6), (6,3), and (3,0), with a (horizontal) diameter drawn through the points (0,3) and 6,3).  The picture is given, with all the points mentioned signaled by heavy black dots as well as the coordinates written out.    The text reads:

 

Problem 2.  Which coordinate point satisfies the following requirements:  It serves as the endpoint of the given diameter, and does NOT lie on the y-axis?

 

and the four exhibited points of tangency, (0,3), (3,6), (6,3), and (3,0) as listed above, are given as the four choices for this multiple-choice question. So roundabout a method of determining whether the student knows the point notation for the Cartesian coordinate system, is able to recognize the words “diameter”, “endpoint”, and can discern when a point is or is not on the y-axis is embarrassing.  The problem might confuse someone new to the English language, but even this is doubtful.  It serves only to divide the totally illiterate into two groups:  the 25% who guess right, and the 75% who don’t.  Maybe there is some value in finding this out.  Let us go on to the next one.

 

          Problem 3.  Winona took five chapter tests.  The table shows her scores:

 

Test             1                 2                 3                 4                 5

Score          86               90               80               79               70

 

If Winona retakes Test 5, what score would she need to have a mean score of 84?

 

This problem is common and all students are drilled on it these days, though it is generally given in a slightly different form.  The score Winona already achieved on Test # 5 is totally irrelevant here, since that score is to be replaced by her “retake” score, so that the question could well be reduced to the following:  “Given the four numbers 86, 90, 80, and 79, what fifth number would make the mean of the five numbers equal to 84?”  More often the problem of this type merely doesn’t give the original score for Problem # 5, but offers the first four scores and asks what score is needed on a fifth exam to make the average (or mean) score 84. I believe the words “Winona”, “chapter tests”, and “retake” are confusing elements in the statement of the problem, which trumps up the story for “real-live” flavor only, and that some students will be sufficiently distracted to make some error they would not make in the simplified re-wording I have just offered.  Probably the most common error would be in misconstruing the phrase “retakes Test 5” to mean “Take a sixth test to make up for Test 5, by averaging it in with the others”.  One could even make a case for such a technical use of the word “retest”, and for such a use of a sixth test as a compromise between giving a chance at replacing Test 5 and not giving a second chance at all.

 

Unlike the composers of several of the earlier examples, whoever made this problem was not trying to make it easy to score points, but was asking a good question.  However, he should have thought more carefully about what is really being looked for in asking this question in this form, for it offers several avenues to error, most of them having little to do with mathematical competence.  As with the CMP example of the scratch test coupons and the polling described earlier, teachers and others who wish to evaluate school programs by making an “item analysis” of the errors on a given administration of an examination will not learn much from trying to classify the students getting this one wrong.  It would simply not be clear from the mere erroneous answer just what it was that is lacking in the student’s understanding.  On the other hand, it might be that the examiner really wanted to be sure the student was able to handle a multistep problem and wise enough to ignore the irrelevant data; therefore I am not here condemning the problem.  It is probable that the examiner intended the solution to be written as follows:

 

(1/5)( 86+90+80+79+x)=84; solve for x.

 

This would exhibit an early use of algebraic notation and its value in finding an unknown quantity specified by the conditions.  The problem, by the way, can also make a good mental exercise for classroom use:  One notices that the total deviation from the desired average (84) is 2+6-4-5, or –1 points on the first four examinations; hence an extra point (above the desired average) is needed for the fifth examination, i.e. the fifth score must be 85.  Check: 86+90+80+79+85 = 5X84, sure enough. 

 

          However, the problem was multiple-choice!  If CMP is the expected middle school program behind the syllabus for this particular MEAP problem, calculators will be used for this examination, and then, of course no algebra or ingenuity will be needed for this problem.  The student has only to compute the average, using each of the offered choices; if it is not 84, change the guess.  If Michigan were really serious in its enthusiasm for calculators when used “appropriately”, as the phrasing of today’s state standards almost invariably has it, it would ask a question of the sort: 


”What score would Winona need on her fifth examination to end the term with an average of exactly 85.5?”

 

and require a written answer, perhaps warning the student that the required score might not be an integer.

 

Problem 4.  As a reward for good behavior, Mrs. Rafferty writes student names on tickets and stores them in a container.  At the end of the week, she draws a ticket for a reward.  Hans has 6 tickets in this container.  The container has a total of 48 tickets for this week.  What is the probability that one of his tickets will be drawn?

          A. 1 out of 6

          B. 1 out of 8

          C. 1 out of 48

          D. 1 out of 54

 

          This one is quite trivial, though not incorrect except for some stickler who might (like me) prefer the language of probability (“1/8”, rather than “1 out of 8”, in the correct answer).  Even so, the list of answers should have been improved to include the incorrect, though attractive, choices “1/9” and “1/7”.  If students at this level are to be taught this little about probability, they should at least be taught the difference between “probability” and “odds”, a distinction important to daily life.  The odds against Hans, in race-track language, are 42 to 6, or 7 to 1.  I believe the really significant “distractor” would be “one out of seven”, though “one out of nine” would have its proponents, but again this would be to make this problem too difficult to guess.  Of course it would not make the problem a whit different. Again we have a case of window-dressing complication covering extreme guessability.

 

The hypothesis that all tickets have an equal chance of being drawn is not mentioned, and in fact it is the sort of thing that is taken for granted in problems of this sort.  If one looks at the state standards for most states, however, one will find under the heading of statistics and probability the demand that students be on the lookout for sources of bias in polling, and sources of inequality in the probabilities of disjoint outcomes of some experiment.  Such instructions are seldom followed up, as this example shows; still, if the subject is mentioned at all it ought to be demanded that students learn at least the language of equiprobability, and that an urn problem such as this one include the phrase “where all tickets have equal probability of being chosen”, or “chosen at random”.  Why should students be expected to understand scratch coupons and not “at random”?  The school should teach mathematical language when it can, or at least use it.  Eighth grade students need not learn about conditional probability, but it can’t hurt to point out that there are other possibilities than equiprobability in games of chance.

 

 

          Problem 5.  A triangle has 0 diagonals, a quadrilateral has two, a pentagon has five, and a hexagon has 9.  If the pattern continues, how many diagonals must an octagon have?

 

          Sides                    3        4        5        6       

          Diagonals            0        2        5        9       

         

          Now I happen to know, and can prove, that an octagon (convex, by the way, since a non-convex polygon might have strange looking “diagonals”) has 20 diagonals, whether or not the pattern exhibited with the question “continues”.  The composer of this problem knows (or should know) that the usual proof of the formula for the number of diagonals depends on a recursive analysis which shows that the first differences in the sequence of numbers of diagonals, {0,2,5,9,…}, continue to increase by one.  The reasoning is intricate, however, so that the test-maker, in his role as point-maker for students, makes it easy for the student to answer the question by recognizing the partial sequence he has given as data as being of that nature.  With that reassurance from the test-maker, the student can forget everything he ever knew about polygons and still answer the question, for this problem is only illusorily about polygons and the recursive reasoning involved in counting their diagonals.

 

          But bad as the problem is for its announced purpose, it is worse than that, for it says, “If the pattern continues …”  It might look obvious that the pattern continues with first differences increasing by one, but it isn’t; no finite sequence without a stated rule has an obvious continuation.  The second error is semantic:  The number of diagonals of an octagon is 20 whatever the sequence does.  What would one answer to this question, for example, “If the pattern does not continue, how many diagonals does an octagon have?”  An octagon has 20 diagonals no matter what.  From a technical point of view, taken from the first-order prepositional calculus, an irrelevant or incorrect hypothesis can yield a correct conclusion, but the conclusion in real-live terms derives from something other than the putative hypothesis.  “If the sun goes around the earth,” one might ask, “ then how much is 2+ 2?”  Question 5 is just another in the link of semantic confusions that beset the teaching of elementary mathematics and reinforce every day that popular view that the study of math is a can of worms.  That the injury accomplished by this wording of this question is subliminal – hardly anybody would notice this one, after all – doesn’t make its damage any the less.

 

Turning to the “high school” level Michigan problems cited as samples by the state, Number 15 has already been reproduced above as a footnote, as it were, to the New York Regents’ suppression of embarrassing inclusion of coordinate scales in a problem asking for recognition of the graphs of common functions.  The other one I happened on  also serves as a footnote to an earlier complaint concerning a middle school problem about probability:

 

          My complaint was that a certain middle school problem failed to mention an equiprobability hypothesis.  There were 48 tickets in a box, of which 6 were winners, and the question was, what is the probability of choosing a winner?  My colleagues sometimes say that one ought not, in such examinations, say such things as that the probability of choosing one ticket is equal to the probability of choosing any other, but I say that the schools do claim to teach the necessity of such mention (to ordinary students) even if it does not get brought up systematically, and that it behooves the state to make mention, at least of such possibilities as soon as probability as such gets mentioned.  As evidence I cite the question above, where the student is asked to imagine reasons why one sample of 20 customers might not be a random choice.  While a full-scale treatment of the statistics of sampling is not intended by Michigan even at the high school level, surely this question is evidence that the state wishes students to take cognizance of non-random possibilities.

 

Unfortunately, the answer rubric is typically poor.  Examples are given of actual student responses, with the deserved score (0, 1, 2, 3, or 4 points, 4 being full marks) for each.  The first response is handwritten and not reproduced below, but was scored 4, with the Michigan explanation of the reason for the full marks printed as follows:

 

 

           

          The flat-out statement that a sample of 20 from a population of 750 is necessarily an insufficient number because that leaves “the other 730 unaccounted for” is very bad statistics.  I would much prefer that the schools omit statistics from the ordinary, non “Advanced Placement” mathematics sequence, and include more arithmetic, algebra and geometry.

 

Final Example:  Let Everyone Pass

 

One last typical example, from NYS Regents web page:  Sample Grade 8 problems.  Grade 8; imagine! Problems like this one are called “guess and check”, and the method is taught by every teacher who is compelled by the Principal to avoid mathematics and have the children get good test scores.  Sometimes the last three weeks of the year are spent reviewing point-getters like this one, and the Education Department in Albany makes sure that every child gets enough of them to succeed in school, whether or not he knows anything about mathematics.  Here it is:

 

Problem # 13.

 

Bill picked 1/2 of the apples on his grandmother’s tree. After Bill finished,Sally picked 1/3 of the apples that were left on the tree. After Sally finished,there were 40 apples left on the tree. How many apples were on the tree before they picked apples?

 

A 40

B 60

C 80

D 120

 

How much arithmetic would a third-grade child of a hundred years ago have needed to know to be able to reject A, B and C?  No more than average, if that. There were forty apples left after more than half had been picked!  Again, the provision of obviously inappropriate distractors makes the answering easy.  As a real problem, with the student asked to calculate the answer without being given strong hints for guessing, this would not have been a bad one, even though the algebraic formulation should not be beyond the power of an 8th grader.  All he has to do, even without the wrong choices being as wrong as they are, is to try them out.  Start with 40:  Take away half, then take away a third of the result; can there be 40 left on the tree?  Start with 60, and go through the same ritual.  And so on.  Even with the worst luck, using 120 after failing to succeed with all three other choices, the answer is soon arrived at, with nothing but the most elementary arithmetic.

 

          Ralph A. Raimi,     June 3, 2004