Nature of Math Anxiety: Mapping the Terrain Math Anxiet… · The Nature of Math Anxiety: Mapping the Terrain - knows how a cold man feels. —Alexander Solzhenitsyn Symptoms of Math

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The Nature of Math Anxiety:Mapping the Terrain

- knows how a cold man feels.—Alexander Solzhenitsyn

Symptoms of Math Anxiety

The first thing people remember about failing at mathis that it felt like sudden death. Whether it happenedwhile learning word problems in sixth grade, copingwith equations in high school, or first confronting calcu-lus and statistics in college, failure was sudden and veryfrightening. An idea or a new operation was not justdifficult, it was impossible! And instead of asking ques-tions or taking the lesson slowly, assuming that in amonth or so they would be able to digest it, peopleremember the feeling, as certain as it was sudden, thatthey would never go any further in mathematics. If weassume, as we must, that the curriculum was reasonableand that the new idea was merely the next in a seriesof learnable concepts, that feeling of utter defeat was

THE NATURE OF MATH ANXIETY 45

simply not rational; and in fact, the autobiographies ofmath anxious college students and adults reveal that nomatter how much the teacher reassured them, theysensed that from that moment on, as far as math wasconcerned, they were through.

The sameness of that sudden death experience is evi-dent in the very metaphors people use to describe it.Whether it occurred in elementary school, high school,or college, victims felt that a curtain had been drawn,one they would never iee behind; or that there was animpenetrable wall ahead; or that they were at the edgeof a cliff, ready to fall off. The most extreme reactioncame from a math graduate student. Beginning herdissertation research, she suddenly felt that not onlycould she never solve her research problem (notunusual in higher mathematics), but that she had neverunderstood advanced math at all. She, too, felt her fail-ure as sudden death.

Paranoia comes quickly on the heels of the anxietyattack. "Everyone knows," the victim believes, "that Idon't understand this. The teacher knows. Friendsknow. I'd better not make it worse by asking questions.Then everyone will find out how dumb I really am."This paranoid reaction is particularly disabling becausefear of exposure keeps us from constructive action. Wefeel guilty and ashamed, not only because our mindsseem to have deserted us but because we believe thatour failure to comprehend this one new idea is proofthat we have been "faking math" for years.

In a fine analysis of mathophobia, Mitchell Lazarusexplains why we feel like frauds. Math failure, he says,passes through a "latency stage" before becoming obvi-ous either to our teachers or to us. It may in fact takesome time for us to realize that we have been left be-

46 OVERCOMING MATH ANXIETY

hind. Lazarus outlines the plight of the high schoolstudent who has always relied on the memorize-what-to-do approach. "Because his grades have been satisfac-tory, his problem may not be apparent to anyone, in-cluding himself. But when his grades finally drop, asthey must, even his teachers are unlikely to realize thathis problem is not something new, but has been in themaking for years."1

It is not hard to figure out why failure to understandmathematics can be hidden for so long. Math is usuallytaught in discrete bits by teachers who were them-selves taught this way; students are tested, bit by bit, asthey go along. Some of us never get a chance to inte-grate all these pieces of information, or even to realizewhat we are not able to do. We are aware of a lack, butthough the problem has been building up for years, thefirst time we are asked to use our knowledge in a newway, it feels like sudden death. It is not so easy to ex-plain, however, why we take such personal responsibil-ity for having "cheated" our teachers and why so manyof us believe that we are frauds. Would we feel thesame way if we were floored by irregular verbs inFrench?

One thing that may contribute to a student's passivityis a common myth about mathematical ability. Most ofus believe that people either have or do not have amathematical mind. It may well be that mathematicalimagination and some kind of special intuitive grasp ofmathematical principles are needed for advanced re-search, but surely people who can do college-level workin other subjects should be able to do college-level mathas well. Rates of learning may vary. Competence undertime pressure may differ. Certainly low self-esteem willinterfere. But is there any evidence that a student


needs to have a mathematical mind in order to succeedat learning math?

Leaving aside for the moment the sources of thismyth, consider its effects. Since only a few people aresupposed to have this mathematical mind, part of ourpassive reaction to difficulties in learning mathematicsis that we suspect we may not be one of "them" and arewaiting for our nonmathematical mind to be exposed.It is only a matter of time before our limit will bereached, so there is not much point in our being me-thodical or in attending to detail. We are grateful whenwe survive fractions, word problems, or geometry. Ifthat certain moment of failure hasn't struck yet, thenit is only temporarily postponed.

Sometimes the math teacher contributes to thismyth. If the teacher claims an entirely happy history oflearning mathematics, she may contribute to the ideathat some people—specifically her—are gifted in math-ematics and others—the students—are not. A goodteacher, to allay this myth, brings in the scratch paperhe used in working out the problem to share with theclass the many false starts he had to make before solvingit.

Parents, especially parents of girls, often expect theirchildren to be nonmathematical. If the parents are poorat math, they had their own sudden death experience;if math was easy for them, they do not know how it feelsto be slow. In either case, they will unwittingly fosterthe idea that a mathematical mind is something oneeither has or does not have.

Interestingly, the myth is peculiar to math. A teacherof history, for example, is not very likely to tell studentsthat they write poor exams or do badly on papers be-cause they do not have a historical mind. Although we


might say that some people have a "feel" for history,the notion that one is either historical or nonhistoricalis patently absurd. Yet, because even the experts still donot know how mathematics is learned, we tend to thinkof math ability as mystical and to attribute the talent forit to genetic factors. This belief, though undemonstra-ble, is very clearly communicated to us all.

These considerations help explain why failure tocomprehend a difficult concept may seem like suddendeath. We were kept alive so long only by good fortune.Since we were never truly mathematical, we had tomemorize things we could not understand, and bymemorizing we got through. Since we obviously do nothave a mathematical mind, we will make no progress,ever. Our act is over. The curtain down-

Ambiguity, Real and Imagined

What is a satisfactory definition? For the philos-opher or the scholar, a definition is satisfactoryif it applies to those things and only thosethings that are being defined; this is what logicdemands. But in teaching, this will not do: adefinition is satisfactory only if the students un-derstand it.

—H. Poincare

Mathematics autobiographies show that for the be-ginning student the language of mathematics is full ofambiguity. Though mathematics is supposed to have avery precise language, more precise than our everydayuse (this is why math uses symbols), it is true that mathe-matical terms are never wholly free of the connotationswe bring to words, and these layers of meaning may get

THE NATURE OF MATH ANXIETY

in the way. The problem is not that there is anythingwrong with math; it is that we are not properly initiatedinto its vocabulary and rules of grammar.

Some math disabled adults will remember, afterfifteen to thirty years, that the word "multiply" as usedfor fractions never made sense to them. "Multiply,"they remember wistfully, always meant "to increase."That is the way the word was used in the Bible, in othercontexts, and surely the way it worked with wholenumbers. (Three times six always produced somethinglarger than either three or six.) But with fractions (ex-cept the improper fractions), multiplication always re-sults in something of smaller value. One-third timesone-fourth equals one-twelfth, and one-twelfth is con-siderably smaller than either one-third or one-fourth.

Many words like "multiply" mean one thing (like"increase rapidly") when first introduced. But in thelarger context (in this case all rational numbers), theapparently simple meaning becomes confusing. Sincestudents are not warned that "multiplying" has verydifferent effects on fractions, they find themselvessearching among the meanings of the word to find outwhat to do. Simple logic, corresponding to the wordsthey know and trust, seems not to apply.

A related difficulty for many math anxious peopleis the word "of as applied to fractions. In generalusage, "of can imply division, as in "a portion of."Yet, with fractions, one-third of one-fourth requiresmultiplication. We can only remember this by sus-pending our prior associations with the word "of," orby memorizing the rule. Or, take the word "cancel"as used carelessly with fractions. We were told to"cancel numerators and denominators" of fractions.Yet nothing is being "cancelled" in the sense of

OVERCOMING MATH ANXIETY

being removed for all time. The same holds true fornegative numbers. Once we have learned to associ-ate the minus sign with subtraction, it takes an expli-cit lesson to unlearn the old meaning of minus; or, asa mathematician would put it, to learn its meaningas applied to a new kind of number.

Knowles Dougherty, a skilled teacher of mathemat-ics, notes:

It is no wonder that children have trouble learning arithme-tic. If you ask an obedient child in first grade, "What is Zero,"the child will call out loudly and with certainty, "Zero isnothing." By third grade, he had better have memorized that"Zero is a place-holder." And by fifth grade, if he believesthat zero is a number that can be added, subtracted, multi-plied by and divided by, he is in for trouble.2

People also recall having problems with shapes,never being sure for example whether the word "cir-cle" meant the line around the circle or the spacewithin. Students who had such difficulties felt theywere just dumber than everyone else, but in fact theword "circle" needs a far more precise definition. It isin fact neither the circumference nor the area butrather "the locus of points in the plane equidistant froma center" (Fig. II-l).*

A mind that is bothered by ambiguity—actual or per-ceived—is not usually a weak mind, but a strong one.This point is important because mathematicians arguethat it is not the subject that is fuzzy but the learnerwho is imprecise. This may be, but as mathematics isoften taught to amateurs differences in meaning be-

•One student, learning to find the "least common denominator," took the

-most unusual denominator" she could find. Instead of finding the smallestcommon denominator, then, she found a very large one and was appropri-ately chastised by her teacher for misunderstanding the question.


** •**

tween common language and mathematical languageneed to be discussed. Besides, even if mathematicallanguage is unambiguous, there is no way into it exceptthrough our spoken language, in which words areloaded with content and associations. We cannot helpbut think "increase" when we hear the word "multi-ply" because of all the other times we have used thatword. We have been coloring circles for years beforewe get to one we have to measure. No wonder we areunsure of what "circle" means. People who do a littlebetter in mathematics than the rest of us are not asbothered by all this. We shall consider the possible rea-sons for this later on.

Meanwhile, the mathematicians withhold informa-tion. Mathematicians depend heavily upon customarynotation. They have a prior association with almostevery letter in the Roman and Greek alphabets, which

gz OVERCOMING MATH ANXIETY

they don't always tell us about. We think that our teach-ers are choosing X or a or delta (A) arbitrarily. Not so.Ever since Descartes, the letters at the end of the al-phabet have been used to designate unknowns, the let-ters at the beginning of the alphabet usually to signifyconstants, and in math, economics, and physics gener-ally A means "change" or "difference." Though thesesymbols appear to us to be chosen randomly, the lettersare loaded with meaning for "them."

In more advanced algebra, the student's search formeaning is made even more difficult because it is al-most impossible to visualize complex mathematical re-lationships. For me, the fateful moment struck when Iwas confronted by an operation I could neither visual-ize nor translate into meaningful words. The expressionX~z=\2 did me in. I had dutifully learned that expo-nents such as 2 and 3 were shorthand notations for mul-tiplication: a number or a letter squared or cubed wassimply multiplied by itself twice or three times. Tryingto translate math into words, I considered the possibil-ity that X~2 meant something like "X not multiplied byitself or "multiplied by not-itself." What words or im-ages could convey to me what X~* really meant? To allthese questions—and I have asked them many timessince—the answer is that X~* = ja is a definition consist-ent with what has gone before. I have been shownseveral demonstrations that this definition is indeedconsistent with what has gone before.* But at the time

•While interviewing for this book, I have finally found out that negativetwo is a different kind of number from positive two and that it was naive of

e to think that it would have the same or similar effect on X. And it does

up with |I. Sedivide] end up w i. See the following:

v-2 — _iL - XXX - _1_ - JLA '-= x* ~ xxxxx ~ xx ~ x*


I did not want a demonstration or a proof. I wanted anexplanation!

I dwell on the X-2 example because I have oftenasked competent mathematicians to recall for me howthey felt the first time they were told X~z = J,- Manyremember merely believing what they were told inmath class, or that they soon found the equivalencyuseful. Unlike me, they were satisfied with a definitionand an illustration that the system works. Why somepeople should be more distrustful about such mattersand less willing to play games of internal consistencythan others is a question we shall return to later.

Willing suspension of disbelief is a phrase thatcomes not from mathematics or science but from lit-erature. A reader must give the narrator an opportu-nity to create images and associations and to "enter"these into our mind (the way we "enter" informationinto a computer) in order to carry us along in thestory or poem. The very student who can accept thesymbolic use of language in poetry where "birds arehushed by the moon," or the disorienting treatmentof time in books by Thomas Mann and James Joyce,may balk when mathematics employs familiar wordsin an unfamiliar way. If willingness to suspend disbe-lief is specific to some tasks and not to others, per-haps it is related to trust. One counsellor explainsmath phobia by saying, "If you don't feel safe, youwon't take risks." People who don't trust math maybe too wary of math to take risks.

A person's ability to accept the counter-intuitive useof time in Thomas Mann's work and not the new mean-ing of the negative exponent does not imply that thereare two kinds of minds, the verbal and the mathemati-cal. I do not subscribe to the simple-minded notion thatwe are one or the other and that ability in one area


leads inevitably to disability in the other. Rather, Ithink that verbal people feel comfortable with lan-guage early in life, perhaps because they enjoyed suc-cess at talking and reading. When mathematics contra-dicts assumptions acquired in other subjects, suchpeople need special reassurance before they will ven-ture on.

Conflicts between mathematical language and com-mon language may also account for students* distrust oftheir intuition. If several associated meanings are float-ing around in someone's head and the text considersonly one, the learner will, at the very least, feel alone.Until someone tries to get inside the learner's head orthe learner figures out a way to search among the vari-ous meanings of the word for the one that is called for,communication will break down, too. This problem isnot unique to mathematics, but when people alreadyfeel insecure about math, linguistic confusion increasestheir sense of being out of control. And so long as teach-ers continue to argue, as they have to me, that wordslike "multiply" and "of," the negative exponents, andthe "circles" or "disks" are not ambiguous at all butperfectly consistent with their definitions, then stu-dents will continue to feel that math is simply not forthem.

Some mathematics texts solve the problem of ambi-guity by virtually eliminating language. College-levelmath textbooks are even more laconic than elementarytexts. One reason may be the difficulty of expressingmathematical ideas in language that is easily agreedupon. Another is the assumption that by the time stu-dents get to college they should be able to read sym-bols. But for some number of students (we cannot knowhow many since they do not take college-level math)


proofs, symbolic formulations, and examples are notenough. After I had finally learned that X~* must equali2 because it was consistent with the rule that whendividing numbers with exponents we subtract the ex-ponents, I looked up "negative exponents" in a newhigh school algebra text. There I found the followingparagraph.

Negative and Zero ExponentsThe set of numbers used as exponents in our discussion so farhas been the set of positive integers. This is the only setwhich can be used when exponents are defined as they werein Chapter One. In this section, however, we would like toexpand this set to include all integers (positive, negative andzero) as exponents. This will, of course, require further defi-nitions. These new definitions must be consistent with thesystem and we will expect all of the laws of exponents as wellas all previously known facts to still be true.3

Although this paragraph is very clear in setting thestage to explain negative exponents through definitionswhich are presumably forthcoming, it does not providea lot of explanation. No wonder people who need wordsto make sense of things give up.

The Dropped Stitch

"The day they introduced fractions, I had themeasles." Or the teacher was out for a month, thefamily moved, there were more snow days that yearthan ever before (or since). People who use eventslike these to account for their failure at math did,nevertheless, learn how to spell. True, math is espe-cially cumulative. A missing link can damage under-


standing much as a dropped stitch ruins a knittedsleeve. But being sick or in transit or just too far be-hind to learn the next new idea is not reason enoughfor doing poorly at math forever after. It is unlikelythat one missing link can abort the whole process oflearning elementary arithmetic.

In fact, mathematical ideas that are rather difficultto learn at age seven or eight are much easier to com-prehend one, two, or five years later if we try again.As we grow older, our facility with language im-proves; we have many more mathematical concepts inour minds, developed from everyday living; we canask more and better questions. Why, then, do we letourselves remain permanently ignorant of fractions ordecimals or graphs? Something more is at work than amissed class.

It is of course comforting to have an excuse fordoing poorly at math, better than having to concedethat one does not have a mathematical mind. Still,the dropped stitch concept is often used by mathanxious people to excuse their failure. It does not ex-plain, however, why in later years they did not takethe trouble to unravel the sweater and pick upwhere they left off.

Say they did try a review book. Chances are it wouldnot be helpful. Few texts on arithmetic are written foradults.* How insulting to go back to a "Run, Sport, run!"level of elementary arithmetic, when arithmetic can beinfinitely clearer and more interesting if it is discussedat an adult level.

Moreover, when most of us learned math we learned

•Deborah Hughes-Halle te is \id college students that starts

•iting a book (W.W. Norton, 1978) for adult'ith arithmetic and brings the reader up t(


dependence as well. We needed the teacher to explain,the textbook to drill us, the back of the book to tell usthe right answers. Many people say that they nevermastered the multiplication table, but I have encoun-tered only one person so far who carries a multiplica-tion table in his wallet. He may have no more skills thanthe others, but at least he is trying to make himselfautonomous. The greatest value of using simple calcula-tors in elementary school may, in the end, be to freepupils from dependence on something or someone be-yond their control.

Adults can easily pick up those dropped stitches oncethey decide to do something about them. In one mathcounselling session for educators and psychologists, thefollowing arithmetic bugbears were exposed:

How do you get a percentage out of a fraction like 7/ie?Where does "pi" come from?How do you do a problem like: Two men are painters. Eachpaints a room in a different time. How long does it take themto paint the room together?*

The issues were taken care of within half an hgur.This leads me to believe that people are anxious not

because they dropped a stitch long ago but rather be-cause they accepted an ideology that we must reject:that if we haven't learned something so far it is proba-bly because we can't.

'See Chapter Six for a discussion of fractions and percents; see ChaptiFive for a discussion of the Painting-the-Hoom Problem. Pi can be deriviby drawing many-sided polygons (like squares, pentagons, hexagons, ehand measuring the ratio of their perimeters to their diameters. Everi ifdo this roughly, the ratios will approach 3.H.

derivedetc.)


Fear of Being Too Dumb or Too Smart

© 1966 United Features Syndicate, Inc.

One of the reasons we did not ask enough questionswhen we were younger is that many of us werecaught in a double bind between a fear of appearingtoo dumb in class and a fear of being too smart. Whyanyone should be afraid of being too smart in mathis hard to understand except for the prevailing no-tion that math whizzes are not normal. Boys whowant to be popular can be hurt by this label. But itis even more difficult for girls to be smart in math.Matina Horner, in her survey of high-achieving col-lege women's attitudes toward academic success,found that such women are especially nervous about


competing with men on what they think of as men'sturf.4 Since many people perceive ability in mathe-matics as unfeminine, fear of success may well inter-fere with ability to learn math.

The young woman who is frightened of seeming toosmart in math must be very careful about asking ques-tions in class because she never knows when a questionis a really good one. "My nightmare," one womanremembers, "was that one day in math class I wouldinnocently ask a question and the teacher would say,'Now that's a fascinating issue, one that mathematiciansspent years trying to figure out.' And if that happened,I would surely have had to leave town, because mysocial life would have been ruined." This is an extremecase, probably exaggerated, but the feeling is typical.Mathematical precocity, asking interesting questions,meant risking exposure as someone unlike the rest ofthe gang.

It is not even so difficult to ask questions that gave theancients trouble. When we remember that the Greekshad no notation for multi-digit numbers and that evenNewton, the inventor of the calculus, would have beenhard pressed to solve some of the equations given tobeginning calculus students today, we can appreciatethat young woman's trauma.

At the same time, a student who is too inhibited toask questions may never get the clarification neededto go on. We will never know how many students de-veloped fear of math and loss of self-confidence be-cause they could not ask questions in class. But themath anxious often refer to this kind of inhibition. Inone case, a counsellor in a math clinic spent almost asemester persuading a student to ask her mathteacher a question after class. She was a middling


math student, with a B in linear algebra. She askedquestions in her other courses, but could not or wouldnot ask them in math. She did not entirely understandher inhibition, but with the aid of the counsellor, shecame to believe it had something to do with a fear ofappearing too smart.

There is much more to be said about women andmathematics. The subject will be discussed in detail inChapter Three. At this point it is enough to note thatsome teachers and most pupils of both sexes believethat boys naturally do better in math than girls. Evenbright girls believe this. When boys fail a math quiztheir excuse is that they did not work hard enough.Girls who fail are three times more likely to attributetheir lack of success to the belief that they "simplycannot do math."5 Ironically, fear of being too smartmay lead to such passivity in math class that eventuallythese girls also develop a feeling that they are dumb. Itmay also be that these women are not as low in self-esteem as they seem, but by failing at mathematics theyresolve a conflict between the need to be competentand the need to be liked. The important thing is thatuntil young women are encouraged to believe that theyhave the right to be smart in mathematics, no amountof supportive, nurturant teaching is likely to makemuch difference.

Distrust of Intuition

Mathematicians use intuition, conjecture andguesswork all the time except when they are inthe classroom.

—Joseph Warren, Mathematician


Thou shalt not guess.—Sign in a high school math classroom

At the Math Clinic at Wesleyan University, there isalways a word problem to be solved. As soon as one issolved, another is put in its place. Everyone who walksinto the clinic, whether a teacher, a math anxious per-son, a staff member, or just a visitor, has to give theword problem a try. Thus, we have stimulated numer-ous experiences with a variety of word problems and bydebriefing both people who have solved these prob-lems and people who have given up on them, we gainanother insight into the nature of math anxiety.

One of the arithmetic word problems that was on theboard for a long time is the Tire Problem:

A car goes 20,000 miles on a long trip. To save wear, thefive tires are rotated regularly. How many miles willeach tire have gone by the end of the trip?

Most people readily acknowledge that a car has fivetires and that four are in use at any one time. Poor mathstudents who are not anxious or blocked will pokearound at the problem for a while and then come upwith the idea that four-fifths of 20,000, which is 16,000miles, is the answer. They don't always know exactlywhy they decided to take four-fifths of 20,000. Theysometimes say it "came" to them as they were thinkingabout the tires on the car and the tire in the trunk. Theimportant thing is that they tried it and when it re-sulted hi 16,000 miles, they gave 16,000 a "reasonable-ness test." Since 16,000 seemed reasonable (that is, lessthan 20,000 miles but not a whole lot less), they werepretty sure they were right.

The math anxious student responds very differently.


o

The problem is beyond her (or him). She cannot beginto fathom the information. She cannot even imaginehow the 6ve tires are used (See FIG. II-2.) She cannotcome up with any strategy for solving it. She gives up.Later in the debriefing session, the counsellor may askwhether the fraction four-fifths occurred to her at allwhile she was thinking about the problem. Sometimesthe answer will be yes. But if she is asked why she didnot try out four-fifths of 20,000 (the only other numberin the problem), the response will be—and we haveheard this often enough to take it very seriously "Ifigured that if it was in my head it had to be wrong."


The assumption that if it is in one's head it has to bewrong or, as others put it, "If it's easy for me, it can'tbe math," is a revealing statement about the self. Mathanxious people seem to have little or no faith in theirown intuition. If an idea comes into their heads or astrategy appears to them in a flash they will assume itis wrong. They do not trust their intuition. Either theyremember the "right formula" immediately or theygive up.

Mathematicians, on the other hand, trust their intui-tion in solving problems and readily admit that withoutit they would not be able to do much mathematics. Thedifference in attitude toward intuition, then, seems tobe another tangible distinction between the math anx-ious and people who do well in math.

The distrust of intuition gives the math counsellora place to begin to ask questions: Why does intuitionappear to us to be untrustworthy? When has it failedus in the past? How might we improve our intuitivegrasp of mathematical principles? Has anyone evertried to "educate" our intuition, improve our reper-toire of ideas by teaching us strategies for solvingproblems? Math anxious people usually reply that in-tuition was not allowed as a tool in problem solving.Only the rational, computational parts of their brainbelonged in math class. If a teacher or parent usedintuition at all in solving problems he rarely admit-ted it, and when the student on occasion did guessright in class he was punished for not being able toreconstruct his method. Yet people who trust theirintuition do not see it as "irrational" or "emotional"at all. They perceive intuition as flashes of insightinto the rational mind. Victims of math anxiety needto understand this, too.


The Confinement of Exact Answers

"Computation involves going from a questionto an answer. Mathematics involves going from

—Peter Hilton, Mathematician

Another source of self-distrust is that mathematics istaught as an exact science. There is pressure to get anexact right answer, and when things do not turn outright, we panic. Yet people who regularly use mathe-matics in their work say that it is far more useful to beable to answer the question, "What is a little more thanfive multiplied by a little less than three?" than to knowonly that five times three equals 15.* Many math anx-ious adults recall with horror the timed tests they weresubjected to in elementary, junior and senior highschool with the emphasis on getting a unique right an-swer. They liked social studies and English better be-cause there were so many "right answers," not just one.Others were frustrated at not being able to have discus-sions in math class. Somewhere they or their teachersgot the wrong notion that there is an inherent contra-diction between rigor and debate.

This emphasis on right answers has many psychologi-cal benefits. It provides a way to do our own evaluationon the spot and to be judged fairly whether or not theteacher likes us. Emphasis on the right answer, how-ever, may result in panic when that answer is not at

*A little more than five multiplied by a little less than three will producea range between 12.5 and 16.5. Inequalities, of which this is one example,are common in more advanced math, as are equations that have more thanone solution.


hand and, even worse, lead to "premature closure"when it is. Consider the student who does get the rightanswer quickly and directly. If she closes the book anddoes not continue to reflect on the problem, she will notfind other ways of solving it, and she will miss an oppor-tunity to add to her array of problem-solving methods.In any case, getting the right answer does not necessar-ily imply that one has grasped the full significance ofthe problem. Thus, the right-answer emphasis may in-hibit the learning potential of good students and poorstudents alike.

In altering the learning atmosphere for the math anx-ious the the tutor or counsellor needs to talk franklyabout the difficulties of doing math. The tutor's scratchpaper might be more useful to the students than a per-fectly conceived solution. Doing problems afresh inclass at the risk of making errors publicly can also linkthe tutor with the student in the process of discovery.Inviting all students to put their answers, right orwrong, before the class will relieve some of the panicthat comes when students fail to get the answer theteacher wants. And, as most teachers know, lookingcarefully at wrong answers can give them good clues towhat is going on in students' heads.

Although an answer that checks can provide immedi-ate positive feedback, which aids in learning, the rightanswer may come to signify authoritarianism (on thepart of the teacher), competitiveness (with other stu-dents), and painful evaluation. None of these unpleas-ant experiences is usually intended, any more than thepremature closure or panic, but for some students whoare insecure about mathematics the right-answer em-phasis breeds hostility as well as anxiety. Worst of all,the "right answer" isn't always the right one at all. It is


only "right" in the context of the amount of mathemat-ics one has learned so far. First graders, who are work-ing only with whole numbers, are told tbey are "right"if they answer that five (apples) cannot be divided be-tween two (friends). But later, when they work withfractions, they will find out that five can be equallydivided by giving each friend two and one-half apples.In fact both answers are right. You cannot divide fiveone-dollar bills equally between two people withoutgetting change.

The search for the right answer soon evolves into thesearch for the right formula. Some students cannoteven put their minds to a complex problem or play withit for a while because they assume they are expected toknow something they have forgotten.

Take this problem for, example,

Amy Lowell goes out to buy cigars. She has 25 coins inher pocket, $7.15 in all. She has seven more dimes thannickels and she has quarters, too. How many dimes,nickels, and quarters does she have?

Most people who have done well in high school algebrawill begin to call the number of nickels X, the numberof dimes X + 7, and the number of quarters $7.15minus (5X + IQX + 70) without realizing that AmyLowell must have miscounted her change, becauseeven if all 25 coins in her pocket were quarters (thelargest coin she has), her change would total $6.25, not$7.15.*

This is a tricky problem, which is fair, as opposed toa trick problem which is not. But it also shows howsearching for the right formula can cause us to miss an

*I am indebted to Jean Smith for this example.

1 THE NATURE OF MATH ANXIETY 67

obviously impossible situation. The right formula maybecome a substitute for thinking, just as the right an-swer may replace consideration of other possibilities.Somehow students of math should learn that the powerof mathematics lies not only in exactness but in theprocessing of information.

Self-Defeating Self-Talk

One way to show people what is going on in their headsis to have them keep a "math diary," a running com-mentary of their thoughts, both mathematical andemotional, as they do their homework or go about theirdaily lives. Sometimes a tape recorder can be used toget at the same thing. The goal is twofold: to show thestudent and the instructor the recurring mathematicalerrors that are getting in the way and to make thestudent hear his own "self-talk." "Self-talk" is what wesay to ourselves when we are in trouble. Do we eggourselves on with encouragement and suggestions? Ordo we engage in self-defeating behaviors that onlymake things worse?

Inability to handle frustration, contributes to mathanxiety. When a math anxious person sees that a prob-lem is not going to be easy to solve, he tends to quitright away, believing that no amount of time or reread-ing or reformulation of the problem will make it anyclearer. Freezing and quitting may be as much theresult of destructive self-talk as of unfamiliarity withthe problem. If we think we have no strategy withwhich to begin work, we may never find one. But if wecan talk ourselves into feeling comfortable and secure,we may let in a good idea.


To find out how much we are talking ourselves intofailure we have to begin to listen to ourselves doingmath. The tape recorder, the math diary, the self-moni-toring that some people can do silently are all tech-niques for tuning in to ourselves. Most of us who handlefrustration very poorly in math handle it very well inother subjects. It is useful to watch ourselves doingother things. What do we do there to keep going? Howcan these strategies be applied to math?

At the very minimum this kind of tuning in mayidentify the particular issue giving trouble. It is not veryhelpful to know that "math makes me feel nervous anduncomfortable" or that "numbers make me feel uneasyand confused," as some people say. But it may be quiteuseful to realize that one kind of problem is morethreatening than another. One excerpt from a mathdiary is a case in point;

Here I go again. I am always ready to give up when theequation looks as though it's too complicated to come outright. But the other week, an equation that started out look-ing like this one did turn out to be right, so I shouldn't be sodepressed about it.

This is constructive self-talk. By keeping a diary or talk-ing into a tape recorder we can begin to recognize ourown pattern of resistance and with luck we may soonlearn to control it. This particular person is beginningto understand how and why she jumps to negative con-clusions about her work. She is learning to sort out thefactual mistakes she makes from the logical and eventhe psychological errors. Soon she will be able to recog-nize the mistakes she makes only because she is anx-ious. Note that she has been encouraged to think andto talk about her feelings while doing mathematics. She


is not ashamed or guilty about the most irrational ofthoughts, not frightened to observe even the onset ofdepression in herself; she seems confident that hermind will not desert her.

The diary or tape recorder technique has onlybeen tried so far with college-age students andadults. So far as we can tell, it is effective only whenused in combination with other nonthreateningteaching devices, such as acceptance of discussion offeelings in class, psychological support outside ofclass, and an instructor willing to demystify mathe-matics. The goal in such a situation is not to get theright answer. The goal is to achieve mastery andabove all autonomy in doing math. In the end, wecan only learn when we feel in control.

References

This chapter is based primarily on interviews with andobservations of math anxious students and adults. Thesepeople are not typical of those who are math incompe-tent. Most of them are very bright and enjoy schoolsuccess in other subjects, but they avoid or openly fearmathematics.'Mitchell 1-azarus, "Mathophobia; Some Personal Speculations," The Princi-pal, January/February, 1974, p. IS.2Knowles Dougherty. Personal communication to the author.3J. Louis Nanney and John L. Cable, Elementary Algebra: A Skills Approach,Boston, Allyn and Bacon, Inc., 1974, p. 215.4Matina Horner, "Fear of Success," Psychnlogy Today, November, 1969, p.38ff.sSanford Dornbusch, as quoted in John Ernest, "Mathematics and Sex,"American Mathematics Monthly, Vol. 83, No. S, October, 1976, p. 599.

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