A difficult child



I have been going to this periodontist for many years; I get my teeth

cleaned there and he measures the gaps in the gums, and he has by

now also made three (expensive) implants, saving me from having to have a

partial plate. His surgical assistant, very competent, named Evelyn, has

been all this time been pridefully telling me about her daughter

-- who isn't much good at math. That is, she is proud of the rest, but

worried about the math. The daughter, Sarah, did somehow pass Math I (the

standard 9th grade math for "accelerated" students, but 10th grade for the

others), but is now having trouble in Math II. Non-accelerated.


The new Regents standard for graduation will be an exam called

Math A, which will cover Math I and about half of Math II. I have been

getting reports on the kid (Sarah) for several years now, but the other

day, rather shyly, Evelyn asked me, when my mouth didn't have some

machinery in it, whether I did any tutoring. I said no, so she asked me

if I knew anyone who did. Sarah was in math trouble and she was willing

to pay for maybe a lesson a week. I told her I'd think about finding

someone, but most of our graduate students, I told her, were really not

the sort that ... well, I'd see.


After a day of considering our graduate students, I decided they

were absolutely not the type, except maybe for a couple who were already

TAs and working hard on their studies, too. Furthermore, the university

is a good 15 miles from the village Evelyn lives in, which is not quite a

contiguous suburb of Rochester, the trip being at least a half-hour each

way. And I didn't know how bad off Sarah was. In short, I told Evelyn

to come to my house and I'd examine Sarah and see where she stood.

Once I knew her math situation I'd know if this or that person might be

useful. So tonight I spent 45 minutes talking to Sarah, sometimes

writing on paper and or having her do so, and sometimes having her use a

calculator, but mostly conversation and mental calculation when necessary.


Her current assignment concerned finding where a point like (7,9)

goes under a given translation, e.g. by (3,-5). Last week her assign-

ments concerned simplifying certain expressions, e.g.,




In the latter case she cancelled the 9s and went on from there.

It was marked wrong when it came back, but she didn't know why. On the

other hand, her "transformational geometry" was not giving her any



I had asked her mother to have her bring her current textbook and

some homeworks, but the book she brought was actually for last year's

course (Math I). She explained that her teacher didn't use books, only

handouts, but that she (Sarah) had borrowed this one from a friend to

bring along since she couldn't find the one for this year's course. I

asked how close this year's book would be if they did use it, but she

didn't know. I asked if the book she brought fairly represented what she

had done last year in her bookless class but she didn't know, as she

hadn't looked inside it before coming to me.


I looked briefly at the Math I book, which was, except for the

fragmentation into algebra, geometry, logic (truth tables), etc., a very

traditional set of chapters with no reasons whatever given for why one

would want to move a point by (3,-5) or add the fractions cited. (This

being the beginning of the following year it seemed to me that what her

present homework handouts concerned were all more or less in last

year's book.)


In the course of discussing the fractions I asked her to evaluate

x/(x2-9) when x is 4, or 7, or -3. She wasn't used to this sort of thing,

or the word "evaluate", but I explained, and when I showed what to do when

x is 4 she was able to do the equivalent when x was 7, and was pleased at

how easy it was, she still was not in the least interested in why one would want to

do that. She stopped with the answer in the form "7/(49-9)", and I didn't

press her further at the time. For x= -3 she got 3/(9-9) sure enough, and

wanted to stop there, but I wanted to know what number that *was*. She

was annoyed, but finally said "3". We talked that over for a while and I

asked her a couple of other questions about fractions. She was quite dim

about how to handle zeros in arithmetic of any kind, and confused zero

with one from time to time, as above, though when I reminded her that

division by zero was undefined she recognized having heard that before.

That this sentiment implied that "x/(x2-9) is meaningless for x=-3", as I

put it, did not register with her.


She had never heard of "the number line", and when I suggested a

yardstick she said she had never used one. She didn't know what numbers

appeared on a yardstick, or what I was talking about when I wanted to know

the position of the markings between inch or foot labels. So I drew a

picture and got a reluctant agreement that there must be such things. I

asked her what the point midway between 1 and 2 might be and she said

"1.5", pronounced "one point five". Did she know any other names for that

number? Silence. I suggested "one and a half" and she agreed. Any other

names? She didn't know. I suggested "three halves". She was dubious.

I tried to show on my drawing how 1.5 was situated three jumps, each of

length 1/2, from the end, but she was not following me. She thought she

had come to have me help with her homework, and I was talking about



I asked her what number was midway between, 1.5 and 2, pointing at

my drawing of a yardstick, and she said "1.8?" She explained that she

had forgotten all about fractions, since the calculator gave all the

answers anyone needed. I asked her to calculate x/(x2-9) for x=6, using

a calculator. I gave her my calculator, and she entered "6 divided by,

let's see, 27" and got the correct answer to 8 places. I asked how she

got that 27. She didn't use the calculator for that because she didn't

know how to chain the calculations, i.e. didn't understand the use of

parentheses on the calculator, but she was able to do the denominator

mentally. I asked her explicitly about the parentheses, and she said she

didn't understand how to use them on the calculator, though earlier she

had explained that "the distributive law" produced the fact, that she had

used earlier, that (x-3)(x+3) = x2-9. She told me that at school they

had $100 calculators, but "nobody knows how to use them."


I asked her to add the fractions 1/2 and 1/3 mentally, and she

answered without much hesitation, 2/6. We discussed that for a while, but

she had no interest in the meaning of that sum and only wanted to know

what she “had done wrong". I had wanted to use this question as a lead-in

to something algebraic we were looking at, and couldn't explain it without

departing from that program, but I did. Her answer of 2/6 had another

name? Yes, she knew it and told me when I asked: 1/3. So how could 1/2

be added to 1/3 and still come out 2/6, i.e. 1/3? She had no difficulty

following this question mentally. She realized then that her answer was

wrong, but still was mainly concerned with what she had done that was

wrong. I told her that one cannot add fractions by adding numerators

unless the denominators were the same. She knew this, to be sure, and

said so, but her cure was to multiply the 3 and the 2 to get the 'common

denominator' 6. Wasn't that right? So why was 1/2 + 1/3 not 2/6, since

6 was the common denominator? She didn't say this in quite those words

-- in fact she was pretty sullen about the whole thing -- but that was

clearly her method; and she felt misused to be told (in effect) that the

common denominator method didn't always work.


Sarah told me she didn't like math and didn't see why she had to

learn the things she was now going through. She was pleased that she got

a 71 on the last exam, and thought her mother was making an unnecessary

fuss based on an earlier examination, in which her score had been worse.

She thought that if someone would show her what was wrong with her

homework problems (the teacher evidently didn't, or at least not

individually) she would get through the year all right.


Xerox pages from a book is the way her teacher hands out

assignments, but the pages contain no explanations. One of last week's

homework problems was to solve the equation [8/(x+2)] = x. This one came

back to her unmarked, meaning she had got it right. Her answers were

squeezed into the margins and other spaces of the Xerox page, that being

the way homework seems to be done at her school. No English words

appeared either on the sheet or in her responses. In this case, her

answer was very crowded, but evidently included a sufficient sequence of

equations, more than necessary actually, eventuating in the factorization

of the quadratic in question, and with boxes around the two answers "x=2"

and "x= -4".


I asked Sarah how she knew 2 was an answer, and she said (with

some annoyance) that the teacher hadn't marked it wrong, had she? I

again tried to get her to substitute 2 in the original equation, and with

some effort under my coaching she went through the steps of verifying

the equation in this case, but she didn't see what that had to do with any-

thing; and the case of -4 was worse, as the arithmetic of the substitution

was more difficult. She thought what I was asking her to do was un-

necessary, as the teacher had already said it was right. I didn't want to

badger her or incur further ill-will and so didn't try very hard to explain

my purpose, or hers, in substituting "2" in the given equation.


Indeed, at an earlier time, when I asked her to tell me what was

being "asked" in a certain homework problem, she hadn't understood. She

didn't regard homework problems as questions, but as things, to be

converted to other things. In the case referred to I then gave a gentler

explanation of what I meant by "what was being asked", by asking her to

disentangle the part that had come to her on Xerox from the marks she had

added in solving it, since the crowding was such that I wasn't sure, I

said, where the Xerox part left off and her writing began. Had I not

switched the meaning of my question about what was being asked I foresaw

that I would have to get into things she didn't want to hear just then.


She livened up considerably when we talked about things which were

not mathematics. She wants to grow up to be an emergency room nurse.

She likes her biology class and her "history and music" class, where she

learns about "classical" and "baroque" music. She says she can't take

chemistry, which I had suggested as useful for a nurse, because she wasn't

going to take Math III. There was lots of math needed for the chem course,

she said, more than for the physics. (Yet she had asked irritably several

times during our interview "what good was all this math" for her.)


I asked her about geometry, had she learned about circles and

tangents, and she said she was unacquainted with the words "secant" and

"tangent" in connection with circles. She knew "chord" and "radius" and

"center". I told her circles were quite interesting, and drew one with a

central angle marked in of approximately 90 degrees. Then I sketched an

angle inscribed in the same arc and asked her for an estimate of its size,

and she suggested 25 degrees. I told her it was exactly half the size of

the central angle, i.e. about 45 degrees, no matter where on the circle

the vertex was. I drew a new picture showing several such angles

inscribed in the same arc, claiming their equality, and she found that

interesting. She had never heard that before. She didn't mind my telling

her things, only asking.


Sarah had a sharp cough throughout the interview, which had been

quite distracting. I took her downstairs and told her mother I would be

calling later with some advice. That I would have stayed upstairs longer

with Sarah except that she had a terrible cough and it seemed to me she

would be better off going home for a cup of tea. I said (in Sarah's

presence) that it seemed to me that there would have to be a lot of work

done (to improve her math), but I'd save the details for the moment. I

felt as if I had diagnosed someone with cancer but was offering the hope

that it would go away if she took enough aspirin. We all parted amicably

enough, though Sarah clearly had felt badly used to have been

brought to my house in the first place.


Sarah doesn't want to learn mathematics -- any mathematics. I

think a proper program could in fact interest her, but she has no time to

begin one, no person to provide it, and no visible use for it. Meanwhile,

she has these homeworks to turn in, and these exams to pass. I'm not sure

that nurse training requires algebra or transformational geometry a la

Math II. But there is absolutely no practical possibility of a cure for

her present troubles. Two types of advice are possible, as it seems to



1. At considerable expense she could get a tutor to go over her

homework with her twice a week or so, preferably daily, by which she

would learn all the finger placements necessary to continue to get 71 on

her exams until she is finally able to stop math. Or


2. She could stop right here.


She is scheduled to take a non pre-college math course next year (some-

thing like statistics one term and business math the other, but I've

forgotten what she told me on that), and she could probably pass that one

now. It is cruel and inhuman to push algebra and trig on her this year, and truth

tables forsooth, given her background and her school's attitude

towards textbooks and other such unnecessary explanations. Can I recommend

she spend extra time on math, besides the torture of a meaningless class

every day? But I haven't found out if it is even legal for her to stop

her present course at all. I'll have to ask around.


If I were asked what seriously could be done to teach something

useful in the name of math to this kid, I would advise starting with the

arithmetic of fractions, i.e. what she failed to learn in the 5th and 6th

grades and since, and their applications and meaning of course. I believe

this could be made interesting to her once she knew she didn't have to

learn all those symbol manipulations she has been plagued with these last

five years. But there is nobody to do this for her, and there is no clear

incentive, since all she thinks she needs is to pass the next few exams.


Even with time and a knowledgeable teacher as private tutor,

fractions might not make it past the starting gate, since she has been

persuaded that her calculator has rendered them unnecessary. (That things

like "x/(x2-9)" are fractions has not really registered with her.)

Someone has obviously told her that fractions were obsolete, since in fact

she is not very good with the calculator, and has some difficulty

associating fractional thoughts with decimal language. She knows

enough to divide any two integers (or other decimally expressed numbers, I

suppose) and get a decimally expressed result on the calculator, but

that's about it.


Whose breath blew out the light within this brain?


November, 1999