By Sean Freeman,2014-05-24 15:02
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    This web page describes the errors that I have seen most frequently in

    undergraduate mathematics, the likely causes of those errors, and their

    remedies. I am tired of seeing these same old errors over and over again. (I

    would rather see new, original errors!) I caution my undergraduate students

    about these errors at the beginning of each semester. Outline of this web page:

    ? ERRORS IN COMMUNICATION, including teacher hostility or

    arrogance, student shyness, unclear wording, bad handwriting, not

    reading directions,loss of invisible parentheses, terms lost inside an


    ? ALGEBRA ERRORS, including sign errors, everything is

    additive, everything is commutative, undistributed

    cancellations, dimensional errors

    ? CONFUSION ABOUT NOTATION, including idiosyncratic

    inverses, square roots, order of operations, ambiguously written

    fractions, stream-of-consciousness notations.

    ? ERRORS IN REASONING, including going over your

    work, overlooking irreversibility, not checking for extraneous

    roots, confusing a statement with its converse, working

    backward, difficulties with quantifiers, erroneous methods that

    work, unquestioning faith in calculators.

    ? UNWARRANTED GENERALIZATIONS, including Euler's xsquare root error, x.

    ? OTHER COMMON CALCULUS ERRORS, including jumping to

    conclusions about infinity, loss or misuse of constants of

    integration, loss of differentials.

    (There is some overlap among these topics, so I recommend reading the

    whole page.) ... Of related interest: Paul Cox's web page, and the books of Bradis, Minkovskii, and Kharcheva and E. A. Maxwell. Ultimately, what are the sources of errors and of misunderstanding? What

    kinds of biases and erroneous preconceptions do we have? Two of my

    favorite historic discoveries are Einstein's discovery of relativity and

    Cantor's discoveries of some of the most basic rules of infinities. These

    discoveries are remarkable in that neither involved long, involved,

    complicated computations. Both are fairly simple, in retrospect, to anyone

    who has studied them. But both involved "thinking outside the box" par

    excellence -- i.e., seeing past the assumptions that were inherent in our

    culture and our language. As philosopher John Culkin said, "We don't know

who discovered water, but we are certain it wasn't a fish." That certain

    mathematical errors are common among students may be partly a

    consequence of biases that are built into our language and culture, some of

    which we aren't even aware of.

Errors in Communication

    Some teachers are hostile to questions. That is an error made by teachers.

    Teachers, you will be more comfortable in your job if you try to do it well,

    and don't think of your students as the enemy. This means listening to your

    students and encouraging their questions. A teacher who only lectures, and

    does not encourage questions, might as well be replaced by a book or a

    movie. To teach effectively, you have to know when your students have

    understood something and when they haven't; the most efficient way to

    discover that is to listen to them and to watch their faces. Perhaps you

    identify with your brightest students, because they are most able to

    appreciate the beauty of the ideas you are teaching -- but the other students

    have greater need of your help, and they have a right to it.

    A variant of teacher hostility is teacher arrogance. In its mildest form, this

    may simply mean a teacher who, despite being polite and pleasant, is unable

    to conceive of the idea that he/she could have made an error, even when that

    error is brought directly to his/her attention. Actually, most of the errors

    listed below can be made by teachers, not just by students. (However, most

    teachers are right far more often than their students, so students should

    exercise great caution when considering whether their teachers could be in


If you're a student with a hostile teacher, then I'm afraid I don't know what

    advice to give you; transfer to a different section or drop the course

    altogether if that is feasible. The remarks on communication in the next few

    paragraphs are for students whose teachers are receptive to questions. For

    such students, a common error is that of not asking questions.

    When your teacher says something that you don't understand, don't be shy

    about asking; that's why you're in class! If you've been listening but not

    understanding, then your question is not a "stupid question." Moreover, you

    probably aren't alone in your lack of understanding -- there are probably a

    dozen other students in your classroom who are confused about precisely the

same point, and are even more shy and inarticulate than you. Think of

    yourself as their spokesperson; you'll be doing them all a favor if you ask

    your question. You'll also be doing your teacher a favor -- your teacher

    doesn't always know which points have been explained clearly enough and

    which points have not; your questions provide the feedback that your

    teacher needs.

    If you think your teacher may have made a mistake on the chalkboard, you'd

    be doing the whole class a favor by asking about it. (To save face, just in

    case the error is your own, formulate it as a question rather than a statement.

    For instance, instead of saying "that 5 should be a 7", you can ask "should

    that 5 be a 7?")

    And try to ask your question as soon as possible after it comes up. Don't

    wait until the very end of the example, or until the end of class. As a teacher,

    I hate it when class has ended and students are leaving the room and some

    student comes up to me and says "shouldn't that 5 have been a 7?" Then I

    say "Yes, you're right, but I wish you had asked about it out sooner. Now all

    your classmates have an error in the notes that they took in class, and they

    may have trouble deciphering their notes later."

Marc Mims sent me this anecdote about unasked questions:

    In the early 1980s, I managed a computer retail store. Several of my employees were college students. One bright your man was having difficulty with his Freshman college algebra class. I tutored him and he did very well, but invariably, he would say, "the professor worked through this problem on the board, and it was nothing like this. I sure hope we got the correct answer."

    I accompanied him to class one morning and discovered the source of his frustration. The professor was from the music department, and didn't normally teach college algebra --- he had been pressed into duty when over enrollment forced the class to be split.

    During the class, he picked a problem from the assignment to work out on the board. Very early in the problem, he made an error. I don't recall the specifics, but I'm sure it was one of the many typical algebra errors you list.

    Because of the error, he eventually reached a point from which he could no longer proceed. Rather than admitting an error and going to work to find it, he paused staring at the board for several seconds, then turned to the class and said, "...and the rest, young people, should be obvious."

Unclear wording. The English language was not designed for mathematical

    clarity. Indeed, most of the English language was not really designed at all --

    it simply grew. It is not always perfectly clear. Mathematicians must build

    their communication on top of English [or replace English with whatever is

your native or local language], and so they must work to overcome the

    weaknesses of English. Communicating clearly is an art that takes great

    practice, and that can never be entirely perfected.Lack of clarity often comes in the form of ambiguity -- i.e., when a

    communication has more than one possible interpretation.

    Miscommunication can occur in several ways; here are two of them:

    ? One of the things that you've said has two or more possible meanings,

    and you're aware of that fact, but you're satisfied that it's clear which

    meaning you intended -- either because it's clear from the context, or

    because you've added some further, clarifying words. But your

    audience isn't as knowledgeable as you about this subject, and so the

    distinction was not clear to them from the context or from your

    further clarifying words. Or,

    ? One of the things that you've said has two or more possible meanings,

    and you're not aware of that fact, because you weren't watching your

    own choice of words carefully enough and/or because you're not

    knowledgeable enough about some of the other meanings that those

    words have to some people.

    One way that ambiguity can occur is when there are multiple conventions. A

    convention is an agreed-upon way of doing things. In some cases, one group

    of mathematicians has agreed upon one way of doing things, and another

    group of mathematicians has agreed upon another way, and the two groups

    are unaware of each other. The student who gets a teacher from one group

    and later gets another teacher from the other group is sure to end up

    confused. An example of this is given under "ambiguously written

    fractions," discussed later on this page.

    Choosing precise wording is a fine art, which can be improved with practice

    but never perfected. Each topic within math (or within any field) has its own

    tricky phrases; familiarity with that topic leads to eventually mastering those


    For instance, one student sent me this example from combinatorics, a topic that requires somewhat

    awkward English:

    How many different words of five letters can be formed from seven different consonants and four

    different vowels if no two consonants and vowels can come together and no repetitions are allowed?

    How many can be formed if each letter could be repeated any number of times?

    There are a number of places where this problem is unclear. In the first sentence, I'm not sure what

    "can come together" means, but I would guess that the intended meaning is

How many different words of five letters can be formed from seven different consonants and four

    different vowels if no two consonants can occur consecutively and no two vowels can occur

    consecutively and no repetitions are allowed?

    The second sentence is a bit worse. The student misinterpreted that sentence to mean How many different words of five letters can be formed from seven different consonants and four

    different vowels if each letter could be repeated any number of times? But usually, when a math book asks two consecutive questions related in this fashion, the second

    question is intended as a modification of the first question. We are to retain all parts of the first question that are compatible with the new conditions, and to discard all parts of the first question that

    would be contradicted by the new conditions. Thus, the second sentence in our example should be

    interpreted in this rather different fashion, which yields a different answer: How many different words of five letters can be formed from seven different consonants and four

    different vowels if no two consonants can be consecutive, no two vowels can be consecutive, and each

    letter could be repeated any number of times?

    Bad handwriting is an error that the student makes in communicating with

    himself or herself. If you write badly, your teacher will have difficulty

    reading your work, and you may even have difficulty reading your own

    work after some time has passed!

    Usually I do not deduct points for a sloppy handwriting style, provided that

    the student ends up with the right answer at the end -- but some students

    write so badly that they end up with the wrong answer because they have

    misread their own work. For instance,

    455? (5x+2)dx x+7x+C (should be x+2x+C) This student's handwriting was so bad that he misread his own writing; he

    took the "2" for a "7". You'll have to use your imagination here, since this

    electronic typesetting cannot duplicate sloppy handwriting. You do not need

    to make your handwriting as neat as this typeset document, but you need to

    be neat enough so that you or anyone else can distinguish easily between

    characters that are intended to be different. Most students would fare better

    if they would print their mathematics, instead of using cursive writing.

    By the way, write your plus sign (+) and lower-case letter Tee (t) so that they don't look identical! One easy way to do this is to put a little "tail" at

    the bottom of thet, just as it appears in this typeset document. (I assume that

    the fonts you're using on your browser aren't much different from my fonts.)

Not reading directions. Students often do not read the instructions on a test

    carefully, and so in some cases they give the right answer to the wrong


Loss of invisible parentheses. This is not an erroneous belief; rather, it is a

    sloppy technique of writing. During one of your computations, if you think a

    pair of parentheses but neglect to write them (for lack of time, or from sheer

    laziness), and then in the next step of your computation you forget that you

    omitted a parenthesis from the previous step, you may base your subsequent

    computations on the incorrectly written expression. Here is a typical

    computation of this sort:

    453 ? (5x+7)dx 3x+7x+C But that should be 4553 ? (5x+7)dx = 3(x+7x)+C = 3x+21x+C That's an entirely different answer, and it's the correct answer. To see where

    the error creeps in, just try erasing the last pair of parentheses in the line


    A partial loss of parentheses results in unbalanced parentheses. For 4example, the expression "3(5x+2x+7" is meaningless, because there are

    more left parentheses than right parentheses. Moreover, it is ambiguous -- if

    we try to add a right parenthesis, we could get

    44either "3(5x+2x)+7" or "3(5x+2x+7)"; those are two different answers.

    Loss of parentheses is particularly common with minus signs and/or with

    integrals; for instance,

    455–? (5x7)dx x7x+C (should be x+7x+C)

Terms lost inside an ellipsis. An ellipsis is three dots (...), used to denote

    "continue the pattern". This notation can be used to write a long list. For

    instance,"1, 2, 3, ..., 100" represents all the integers from 1 to 100; that's

    much more convenient than actually writing all 100 numbers. And for some

    purposes, an ellipsis is not just a convenience, it's a necessity. For

    instance, "1, 2, 3, ..., n" represents all the integers from 1 to n, where n is

    some unspecified positive integer; there's no way to write that without an


The ellipsis notation conceals some terms in the sequence. But can only be

    used if enough terms are left unconcealed to make the pattern evident. For

    instance,"1, ..., 64" is ambiguous -- it might have any of these


    ? "1, 2, 3, 4, ..., 64" (all the integers from 1 to 64) 2? "1, 4, 9, 16, 25, 36, 49, 64" (that's n as n goes from 1 to 8) n? "1, 2, 4, 8, 16, 32, 64" (that's 2 as n goes from 0 to 6) Of course, in some cases one of these meanings might be clear from the

    context. And just how much information is needed "to make a pattern

    evident" is a subjective matter; it may vary from one audience to another.

    Best to err on the safe side: give at least as much information as would be

    needed by the least imaginative member of your audience. I have seen many errors in using ellipses when I've tried to teach induction

    proofs. For instance, suppose that we'd like to prove

    2222[*n] 1 + 2 + 3 + ... + n = n(n+1)(2n+1)/6 for all positive integers n. The procedure is this: Verify that the equation is

    true when n=1 (that's the "initial step); then assume that [*n] is true for some

    unspecified value of n and use that fact to prove that it's true for the next

    value of n -- i.e., to prove [*(n+1)] (that's the "transition step"). Here is a

    typical error in the transition step: Add 2n+1 to both sides of [*n]. Thus we

    obtain 2222[i] 1 + 2 + 3 + ... + n + 2n+1 = (2n+1) + n(n+1)(2n+1)/6.

    But that says 2222[ii] 1 + 2 + 3 + ... + (n+1) = (2n+1) + n(n+1)(2n+1)/6.

    We've made a mistake already, in the left side of the equation. (Can you find

    it? I'll explain it in a moment.) Now make some algebra error while

    rearranging the right side of the equation, to obtain 2222[*(n+1)] 1 + 2 + 3 + ... + (n+1) = (n+1)(n+2)(2n+3)/6.

    And now it appears that we're done. But there was an algebra error on the

    right side: (2n+1) + n(n+1)(2n+1)/6 actually is not equal to (n+1)(n+2)(2n+3)/6. (You can check that easily.) The error on the left side was more subtle. It is based on the fact that too

    many terms were concealed in the ellipsis, and so the pattern was not

    revealed accurately. To see what is really going on, let's rewrite equations [i]

    and [ii], putting more terms in:

    222222[i] 1 + 2 + 3 + ... + (n-2) + (n-1) + n + 2n+1 = (2n+1) + n(n+1)(2n+1)/6.

    222222 + 2 + 3 + ... + (n-2) + (n-1) + (n+1) = (2n+1) + n(n+1)(2n+1)/6.

    [ii] 12And now you can see that the left side is missing its n term, so the left side

    of [ii] is not equal to the left side of [*(n+1)].

Algebra Errors

    Sign errors are surely the most common errors of all. I generally deduct

    only one point for these errors, not because they are unimportant, but

    because deducting more would involve swimming against a tide that is just

    too strong for me. The great number of sign errors suggests that students are

    careless and unconcerned -- that students think sign errors do not matter. But

    sign errors certainly do matter, a great deal. Your trains will not run, your

    rockets will not fly, your bridges will fall down, if they are constructed with

    calculations that have sign errors.

    Sign errors are just the symptom; there can be several different underlying

    causes. One cause is the "loss of invisible parentheses," discussed in a later

    section of this web page. Another cause is the belief that a minus sign means a negative number. I think that most students who harbor this belief

    do so only on an unconscious level; they would give it up if it were brought

    to their attention. [My thanks to Jon Jacobsen for identifying this error.] Is x a negative number? That depends on what x is.

    ? Yes, if x is a positive number.

    ? No, if x itself is a negative number. For instance, when x = 6, then

    x = 6 (or, for emphasis, x = +6).

    That's something like a "double negative". We sometimes need double

    negatives in math, but they are unfamiliar to students because we generally

    try to avoid them in English; they are conceptually complicated. For

    instance, instead of saying "I do not have a lack of funds" (two negatives), it

    is simpler to say "I have sufficient funds" (one positive).

    Another reason that some students get confused on this point is that we

    read "x" aloud as "minus x" or as "negative x". The latter reading suggests

    to some students that the answer should be a negative number, but that's not

    right. [Suggested by Chris Phillips.]

Misunderstanding this point also causes some students to have difficulty

    understanding the definition of the absolute value function. Geometrically,

    we think of |x| as the distance between x and 0. Thus |3| = 3 and |27.3| = 27.3, etc. A distance is always a positive quantity (or more precisely,

    a nonnegative quantity, since it could be zero). Informally and imprecisely,

    we might say that the absolute value function is the "make it positive"


    Those definitions of absolute value are all geometric or verbal or

    algorithmic. It is useful to also have a formula that defines |x|, but to do that we must make use of the double negative, discussed a few sentences ago.

    Thus we obtain this formula:

which is a bit complicated and confuses many beginners. Perhaps it's better

    to start with the distance concept.

    Many college students don't know how to add fractions. They don't know

    how to add (x/y)+(u/v), and some of them don't even know how to

    add (2/3)+(7/9). It is hard to classify the different kinds of mistakes they

    make, but in many cases their mistakes are related to this one:

    Everything is additive. In advanced mathematics, a function or

    operation f is called additive if it satisfies f(x+y)=f(x)+f(y) for all numbers x and y. This is true for certain familiar operations -- for instance,

    ? the limit of a sum is the sum of the limits,

    ? the derivative of a sum is the sum of the derivatives,

    ? the integral of a sum is the sum of the integrals.

    But it is not true for certain other kinds of operations. Nevertheless, students

    often apply this addition rule indiscriminately. For instance, contrary to the

    belief of many students,

We do get equality holding for a few unusual and coincidental choices

    of x and y, but we have inequality for most choices of x and y. (For instance,

all four of those lines are inequalities when x = y = π/2. The student who is

    not sure about all this should work out that example in detail; he or she will

    see that that example is typical.)

    One explanation for the error with sines is that some students, seeing the

    parentheses, feel that the sine operator is a multiplication operator -- i.e., just

    as6(x+y)=6x+6y is correct, they think that sin(x+y)=sin(x)+sin(y) is correct.

    The "everything is additive" error is actually the most common occurrence

    of a more general class of errors:

    Everything is commutative. In higher mathematics, we say that two operations commute if we can perform them in either order and get the same

    result. We've already looked at some examples with addition; here are some

    examples with other operations. Contrary to some students' beliefs,

etc. Another common error is to assume that multiplication commutes with

    differentiation or integration. But actually, in

    general (uv)′ does not equal (u′)(v′) and ? (uv)does not equal (? u)(? v).

    However, to be completely honest about this, I must admit that there is one

    very special case where such a multiplication formula for integrals is correct.

    It is applicable only when the region of integration is a rectangle with sides

    parallel to the coordinate axes, and

    u(x) is a function that depends only on x (not on y), and

    v(y) is a function that depends only on y (not on x). Under those conditions,

(I hope that I am doing more good than harm by mentioning this formula,

    but I'm not sure that that is so. I am afraid that a few students will write

    down an abbreviated form of this formula without the accompanying

    restrictive conditions, and will end up believing that I told them to

    equate ? (uv) and (? u)(? v) in general. Please don't do that.)

    Undistributed cancellations. Here is an error that I have seen fairly often,

    but I don't have a very clear idea why students make it.

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