Criminal offences in maths textbooks

[matt grime]

Another vent of frustration:

I hereby nominate the three worst offences committed in maths textbooks as

0. Elementary textbooks asserting a 'fact' that is clearly wrong and going to lead to confusion at a later date. Example: you cannot square root negative numbers. It takes a huge amount of effort to de-program students who have very fixed ideas when they come to university (since complex numbers are even less likely to be taught at high-school anymore).

1. Using an equals sign when you do not mean equals. Example: an A-level textbook I taught from asserted that the square root of 2 _equalled_ 1.41

2. Asking what the domain of, say, $\sqrt{x^2-4}$ is. The domain is part of the definition of a function - that is just an expression at best. See offence 0. Does it suddenly become a function when you learn about complex numbers?

Feel free to add your own.

 

[Kurdt]

A recent gripe of mine has nothig to do with any textbooks I know of but I'm sure there will be books out there because this is part of the primary school syllabus. Instead of teaching standard algorithms for multiplying large numbers and for long division, kids are now being told to 'reason' through these problems by breaking them down into smaller problems they know. This way more work and not very useful in introducing mathematical concepts such as algorithms and long division which comes in handy later on.

 

[mathwonk]

one annoyance i found when writing my algebra book was texts asserting some correct formula, like that for the discriminant of a general (or even reduced) cubic polynomial, that they say are derivable from some particular algorithm, but omitting the derivation. After a finite but huge number of pages, one learns that the stated algorithm is not actually feasible for calculating the given formula. Hence I made it a rule never to say "a lengthy and tedious calculation (or short and easy one) gives...".

If I couldn't do the calculation myelf, reasonably, I showed how to do it on Mathematica, and then also figured out a better way to do it by hand and gave that too, such as the resultant procedure for a discriminant.

 

[mathwonk]

I myself am annoyed by books which give arguments designed for the convenience of the author rather than the student. E.g. in Dummit and Foote, the proof of the decomposition theorem for finitely generated modules over a pid is done in a rather abstract and useless way in the text. Then when a computational version is needed later to actually find a canonical form of a matrix they revert to the standard procedure by diagonalization of matrices using Euclid's algorithm (over Euclidean domains), which they have left to the reader in the exercises.

 

[mathwonk]

Another practice I find objectionable is providing answers for exercises. This effectively deprives the reader of the pleasure and instructiveness of doing them himself, and of deriving checking procedures, and deprives the exercises of usefulness to the instructor who may wish to assign them. There are few things as useful as learning to convince oneself that ones solution is correct.

There is a good reason that here on PF the standard first answer to a question is: "show us what you have tried".

 

[mathwonk]

Errors of course are a fact of life, but authors and editors who decline to acknowledge or correct serious errors in edition after edition puzzle me. E.g. the wonderful book by Shafarevich on Basic Algebraic Geometry has gone through many versions, all containing a false theorem on upper semi continuity of fibres of a morphism. On opage 77, section I.6.3, second edition, vol. 1, the corollary of theorem 7 is false unless one assumes the map is also proper.

Unfortunately this false version is also used later in the discussion of tangent spaces, page 92, section II.1.4, where a modified argument is needed. (One can projectivize the spaces involved to render the appropriate map proper.)

Many other famous books have appeared for decades with the same serious errors uncorrected. Fortunately the errors are usually in the proofs rather tha the statements, but to students learning the reasons the harm is there. Among the many misleading and misstated arguments, the only error in Lang's famous and excellent Algebra book ever corrected to my knowledge was the false theorem to which a counterexample was actually published in a journal, by perhaps James Cannon.

Here is a tiny example of persisting errors. On page 90, second edition of Algebra, after making clear the distinction between products and sums, lang gets it wrong himself. In the second displayed isomorphism, the right side should have a product sign, not a summation sign.

Of course for two summands of modules there is an isomorphism between the two modules, sum and product, so in a twisted sense it is still partly correct, but the whole force of the explanation about their different properties is vitiated by getting it backwards here.

This is the quality of sanctimoniousness of authors: dear reader, YOU should always be careful to understand and notice this important point, (but i won't bother to myself, not even here in my book on the topic!).

 

[neutrino]

Another practice I find objectionable is providing answers for exercises. This effectively deprives the reader of the pleasure and instructiveness of doing them himself, and of deriving checking procedures, and deprives the exercises of usefulness to the instructor who may wish to assign them. There are few things as useful as learning to convince oneself that ones solution is correct.

There is a good reason that here on PF the standard first answer to a question is: "show us what you have tried".

I think it should also be the responsibility of the student to be not tempted by the answers at the back. There are those like me, who like to try a problem many times before looking at the answer, and may not have an instructor for immediate help. Having said that, I don't completely rely on the back-answers to gauge my understanding. It just makes me a tad more confident, that's all.

 

[mathwonk]

i knew that would get an opposing view. I believe its still true though.

heres another reason i dislike them:

Once in college i handed in a particularly short answer to an easy problem, and the grader gave me a zero and insulted me for "copying the answer from the back"!

Of course I was completely unaware there were any answers in the back since i had never looked there. since he did not seem to respect me, i lost all respect for that grader from then on.

Of course I have since learned he was merely speaking from experience.

 

[mathwonk]

Crimes of the lecturer or author:

I always thought it inappropriate for the instructor to decline to read the homework, and assign the grading to an assistant less competent than those students actually taking the class for the first time.

E.g. an offense commited by some teachers and authors is to write in an authoritarian vein, with words like "obviously" and then expect the student to write homework in an entirely different way, with full clear explanations.

Don't they know the student is learning to write mathematics from what he is reading and hearing?

this happened to me in graduate school.

I worked hard on one homework until i understood it thoroughly, then wrote it up in two versions, one short and trivial, followed by a complete explanation, (in the same paper).

But the grader who did not know the topic himself, stopped reading in despair after the short version, never realizing i had given the details he wanted immediately afterward.

He then explicitly criticized me for writing (initially) like the author of our text! Well who else was one supposed to imitate? And in fact I had gone much further than the author's example, if the grader ahd bothered to read on.

But one crime here is expecting the student to behave differently from ones own explicit example.

Thats why I say not to look at, nor provide, the answers. It is pleasing and time saving, but harmful. I admit there are times when I myself like to check them (especially on uninteresting and tedious problems), but it is still a bad habit, even if i may be tempted to yield to it when available.

Another crime is letting feedback come from incompetent persons of ones own choosing, without giving adequate supervision.

 

[mathwonk]

of course i will cut an author more slack if he owns up to his intentions in a book. i especially enjoyed the introduction to sternbergs differential geometry. he explained that he gave proofs of hard theorems right away in order to help tie down the definitions.

He also said he hoped he had avoided serious errors, but could make no such pretension in regard to sign errors and factors of pi etc, as he could never keep those straight himself. Anybody that honest, and that smart, can be forgiven much, (and it is probably unlikely much will be found to object to).

 

[DeadWolfe]

I have to say that asking about the domain of a function is the worst. Students often have the idea that a function is a computable expression, which leads to a lot of trouble later, and questions like that seem to intentionally encourage that confusion.

 

[CRGreathouse]

1. Using an equals sign when you do not mean equals. Example: an A-level textbook I taught from asserted that the square root of 2 _equalled_ 1.41

I'm curious -- do you extend this to big O notation? I mean, do you object to statements like
$$\pi(x)=\frac{x}{\ln x}+\mathcal{O}(\sqrt x\ln x)$$

 

[matt grime]

Why would I, or anyone else, have a problem with that if it is true? (Or as a statement of a conjecture, as well.)

 

[roger]

Another vent of frustration:

I hereby nominate the three worst offences committed in maths textbooks as

0. Elementary textbooks asserting a 'fact' that is clearly wrong and going to lead to confusion at a later date. Example: you cannot square root negative numbers. It takes a huge amount of effort to de-program students who have very fixed ideas when they come to university (since complex numbers are even less likely to be taught at high-school anymore).

1. Using an equals sign when you do not mean equals. Example: an A-level textbook I taught from asserted that the square root of 2 _equalled_ 1.41

2. Asking what the domain of, say, $\sqrt{x^2-4}$ is. The domain is part of the definition of a function - that is just an expression at best. See offence 0. Does it suddenly become a function when you learn about complex numbers?

Feel free to add your own.

Why were you teaching from an A-level textbook? were you giving private tuition?

 

[CRGreathouse]

Why would I, or anyone else, have a problem with that if it is true? (Or as a statement of a conjecture, as well.)

You said you had a problem with misuse of the equals sign, and this is such a misuse -- but one that is commonly accepted and used. If the equal sign was literal, we could conclude from f(x) = O(x log(x)) and g(x) = O(x log(x)) that f(x) = g(x).

That is, are you complaining about approximations being shown as equalities, or all abuses of notation (even relatively benign ones) involving equalities?

 

[matt grime]

That f(x)=O(x log(x)) has a specific meaning. That equality is being used in a different, and explicit sense, from numerical equality. Personally, I'd prefer that they use ~ to indicate that they are using an equivalence relation, but that is just me.

Asserting that sqrt(2)=1.41 is just plain wrong. If they were to have written sqrt(2)=1.41 (2 d.p.), then that would be fine. They didn't.

 

[NateTG]

You said you had a problem with misuse of the equals sign, and this is such a misuse -- but one that is commonly accepted and used. If the equal sign was literal, we could conclude from f(x) = O(x log(x)) and g(x) = O(x log(x)) that f(x) = g(x).

That is, are you complaining about approximations being shown as equalities, or all abuses of notation (even relatively benign ones) involving equalities?

True, it seems like those should be written as:
$$f(x) \in \mathcal{O}(x \ln x)$$
since big-O is really equivalence classes.

And:
$$f(x)=x + \mathcal{O}(\ln x)$$
might be re-written something like:
$$f(x) \equiv x \left(\mathcal{O}(\ln x) \right)$$

But there's a similar thing with constants of integration:
$$\int 2x \mathrm{d}x = x^2 + C$$
so
$$\int 2x-2x \mathrm{d}x = \int 2x \mathrm{d}x - \int 2x \mathrm{d}x =(x^2+C)-(x^2+C)=0$$

 

[roger]

That f(x)=O(x log(x)) has a specific meaning. That equality is being used in a different, and explicit sense, from numerical equality. Personally, I'd prefer that they use ~ to indicate that they are using an equivalence relation, but that is just me.

Asserting that sqrt(2)=1.41 is just plain wrong. If they were to have written sqrt(2)=1.41 (2 d.p.), then that would be fine. They didn't.

What would be wrong with defining pi=3.14 to make it easier for the student to solve that particular problem?

 

[neutrino]

What would be wrong with defining pi=3.14 to make it easier for the student to solve that particular problem?

Defining pi as 3.14? I think I just heard one more of Matt's veins pop.

First, the definition of pi is that it is the ratio of the circumference of a circle to its diameter. Moreover, as Matt has already stated, you can qualify the equality by stating that it is true to certain number of decimal places. When I was first introduced to pi, I was told that it was equal to 22/7. :grumpy:

The teacher can always say something to the effect of, "Kids, use the approximate value of 3.14, but remember that pi is NOT equal to 3.14."

 

[NateTG]

Defining pi as 3.14? I think I just heard one more of Matt's veins pop.

I'm thinking the post might have been facetious - a reference to the infamous legislation.

That said, a good exercise might be to determine the 'diameter' of a circle inscribed on the surface of a sphere of radius 'R' so that the ratio of circumference to 'diameter' is 3.

 

[neutrino]

I'm thinking the post might have been facetious - a reference to the infamous legislation.

Me not being American, and the lack of emoticons to suggest that it might be a joke, made me take it seriously.

That said, a good exercise might be to determine the 'diameter' of a circle inscribed on the surface of a sphere of radius 'R' so that the ratio of circumference to 'diameter' is 3.

You can always add that it's a Euclidean circle that you're talking about when defining pi that way. :tongue2:

 

[Gib Z]

I am a victim of matt's 3rd hate, the domain of a function :(

Only recently have I learned that the domain of a function is incorporated within the definition of the functions, rather than something that must be worked out. I used to think, ok a functions a function if I don't get 2 y values for any x value. Even though now I know I was wrong, I am extremely confused >.< Can someone explain to me matt's comment on how the sqrt operation only becomes a function when complex numbers are introduced? I thought it was a function as long as only the positive ( or negative) root was taken, and that would be fine without complex numbers :(

 

[Gib Z]

In fact some notation with functions confuse me, so I am going to guess what it means:

f: R -> R means that the function, f, accepts any Real values and the outputs them onto the reals? and The first R, after the F, is the domain, while the 2nd R is the y values? Something Like that? In the terrible textbooks matt talks about, and the ones I've learned from :( f(x) = some expression is the function, and some exercises might be find the domain..

 

[Parthalan]

In the notation $f:A \rightarrow B$, $A$ is the domain, and $B$ is the codomain, of which the range (which is $\{y \in B : y = f(x), x \in A\}$) is a subset.

 

[matt grime]

Ok, Gib. When you were doing those exercises 'what is the domain of the following function' you were doing very badly worded exercises.

It is *presumed* that the 'function' must only map to R, thus given sqrt(x), then because you only know about real numbers, you write 'the domain is the positive real numbers and zero'.

Why? Why did you restrict yourself to thinking only about real output? Why not complex numbers? Why not an answer in the extended complex plane? If you're allowed to consider complex output, then the domain is all of R. What changed?

And why did you write down the maximal domain. The square root is a prefectly good function from [1,infinity) to [1,infinity), so why didn't you write that as the domain?

The questions are nonsense, and give the wrong definition of function. A function can be defined in many ways. The idea of it as a relation, a subset of XxY is not the most helpful, but the idea that it is just an expression without any domain is even more unhelpful.

Given an expression, a symbol, you cannot simply talk of 'its domain' as if there were a unique and god given answer. Even if we assume that it must take real values, sqrt can have as a domain the interval [1,infinity).

 

[cliowa]

One annoying feature - particularly for beginners - of most math books is "overloading the notation". Often the authors tell you at some point that they have designated two different things using the same symbol, but "it will be clear from the context what is meant". This is sometimes true, but unfortunately many times it is a great pain, don't you feel?

 

[ObsessiveMathsFreak]

Elementary level:
$$\pi = \frac{22}{7}$$
or variants thereof.
My primary school maths book used this approximation and it was terribly confusing. I remained muddled on the subject of pi for quite some time until I eventually learned that pi could never be written exactly as a fraction or a decimal. I believe texts do things like this in order to avoid potentially "uncomfortable" facts, which are nonetheless very enlightening.

Commutative:
For shame. This word should not be uttered at the elementary level. The same goes for associative, etc.

Intermediate Level:
Trigonometric Functions:
Of course, you must begin with triangles. But eventually you will have to introduce the unit circle and explain why cos(1000000^o) is defined. Also, a proper discussion of radians would help.

Logs:
Introducing logs with the Pavlovian, "The log of a number to any base..." mantra is a cardinal sin, serving only to force the student to "divide by log" in later life. Don't be afraid to show some old yellowed log tables to get the real point across.

Determinants:
Just ... stop ... talking ... about ... them. Most authors harp on these for far too long and never get anything across. Any text that doesn't draw a parallelepiped and spends more than one chapter on determinants is wasting your time.

Higher level:
I'll Posed Questions:
When it takes longer to figure out what the question is actually asking for than it does to answer it, the question is really a waste of time.

Symbol Pushing:
It's all very well to introduce an operation of some description, but if you don't explain what it actually does or why it is useful, you may as well have not introduced it at all. It becomes a symbol to push and is devoid of meaning.

Axiomatisation:
If you're going to introduce a subject through axioms, you have already failed, and may as well give up. There is a time and a place for the axiomatic method. Ninety nine times out of one hundred, your book isn't going to be it.

Pictures:
There is a decided lack of these at the higher level for some reason. Entire books on differential geometry have been written without a single diagram. This is likely reflective of some underlying pathology within the mathematical community.

 

[morphism]

Axiomatisation:
If you're going to introduce a subject through axioms, you have already failed, and may as well give up. There is a time and a place for the axiomatic method. Ninety nine times out of one hundred, your book isn't going to be it.

What?!

min post length

 

[mathwonk]

at some point in our history someone convinced many people that pictures are for kindergarten students. this idiocy has persisted to this day in many otherwise intelligent people. i have met many people who think they are more intelligent because they decline to read books with pictures in them. these faux intelligentsia do not seem to realize that almost all the great classics of literature, especially nineteenth century novels, were originally published with gorgeous illustrations, which were only removed to save the cost of reproducing them.

math books are probably suffering from the same phenomenon. the more pictures the better.:!):tongue2:

 

[morphism]

I think a professor of mine once called it the Bourbaki syndrome. :tongue2:

 

[matt grime]

The Boubaki syndrome really means the writing of mathematics in its 'purest' form, i.e. statements with minimal hypotheses and a very dry style without motivation and in principle from the ground up. The lay reader at this point should not think that by motivation I mean solving a problem in the real world. I mean explaining the reasons why one might wish to prove such a theorem.

The style has its benefits, and its drawbacks, naturally. The reader can decide for themselves if they want articles with lots of statements like 'let M be a monoid, now...'

 

[jostpuur]

Theorem n.m

Blablabla (insert an arbitrary theorem here) blabla.

Proof: See exercise p.q $\square$

...
...
...

Exercise p.q.

Prove theorem n.m.Okey I understand that sometimes it is reasonable to leave proofs as exercises, but on the other hand sometimes it is not even reasonable. It can be impossible for a student to do it without asking somebody how knows it. When I saw this for some couple first times, I actually eagerly searched the exercise in hope of finding hints for the proof. But there's no hints ever, just the dry "Prove theorem n.m." :grumpy: Besides, if there's no hints given in the exercise, then why should the reader see it for the proof?

 

[mathwonk]

this exercise should be followed by the words "gotcha!"

 

[Kurdt]

It would be reasonable if they provided you with enough information to do the proof yourself but in some cases I've known they use something that isn't even covered in the text.

 

[jhat21]

Criminal offenses in math textbooks:

#2 Useless diagrams.
Refer to fig 1.a on page#<somepagefaraway>,
fig 1.a (a right triangle with the right angle <ABC labeled 90 degrees)
what is the sin of angle <ABC?
Just say what is sin(90) !

When they make you do problems according to the diagram *they* assign, using the variables *they* pick, so you have to flip back and forth to their stupid pictures and premises. Just so you have to use *their* book! I know what sin90 is but what the hell is sin (<ABC) is unless I buy their textbook?!