What I really hate about some of the older books, such as Knopps' The theory of infinte series and its application is that for a lot of theroems he puts down expressions that have seemed to come out of nowhere. Also a lot of other books of this type do the same. It's pretty annoying and I wonder does anyone else agree with me. I also hate books that give you a taste of a more difficult idea but say "this is beyond this book...", I mean if that's so why mention it? Thank you for your responds.

Yes, that is at times very frustrating. The authors perhaps put things like that in to either cause a person to pursue another topic because of their sparked interest, or they may just talk about a formula because they want to you to have to prove it yourself, although this is very annoying when you want something proven but can't do it (as happens to me all too often!).

Brad

My physics, chemistry, maths, + biology teachers do that all the time. They tell us about something that's a little more complicated (but usually a lot more interesting) than the stuff we're doing, then stop in the middle and say 'but you'll do that in yr 11/yr 12/Uni/you won't ever need to know that.' It's so annoying!!

It's always the way with science. They even tell you things in the first year at University which turn out to be simplifications, ie not even vaguely true - but you only find this out if you study that particular field deeply. Science is full of this but at least in Maths it's difficult for someone to tell you something which is actually wrong...

-Dave

Jack Cohen and Ian Stewart have an expression - "Liar to Children" - which they use for teachers. Basically, when we teach (at any level) we tend to simplify things to the level of the people being taught. Often (for example, teaching the structure of the atom without going into quantum theory) the simplifications are just plain untrue.

For example, you probably first came across trigonometric ratios (sin, cos, tan) being defined as the ratios of sides in a right angled triangle. When you move up along the ladder of mathematical knowledge, you discover that this isn't a good enough definition - it excludes obtuse and larger angles. You then see (probably) a definition based on the coordinates of the end of a unit line at an angle to the positive x axis. Later still, you may see further definitions based on the limits of infinite series. If you were to confront a 14 year old GCSE student with the definition of the sine of an angle as the sum of an infinite series, he or she would stare blankly and learn nothing. Say it's opposite over hypotenuse and you're in with a fighting chance.

Sometimes, however, especially when dealing with people with potential, you want to give a peek at the nice things on the road ahead without having to climb over the obstacles, so you reveal this or that result, whose statement is understandable at this stage, without going into details or proofs (which require more mathematics to be understood). This not only has the benefit of imparting interesting information, it may also enthuse able people to persevere so they can eventually understand that proof which is currently "beyond the scope of the course".

Yes, I agree with Marcus that it really is necessary to teach a simplified model before teaching full details. After all the order facts are taught usually corresponds to the order they were discovered/understood historically. And learning things in historical order is a very good idea in my opinion, as this is the order the ideas naturally develop in the human mind.

I used to remember at the start of secondary school resenting the fact that the physics teachers were wasting our time with all this Newtonian stuff that had long since been shown to be wrong, and why couldn't we move straight on to special relativity (I had read a couple of pop science books on this subject). Of course since then I've realised that you cannot even begin to understand the full picture in physics without looking at incorrect (but simpler) approximations first. It doesn't hurt to read about what's coming later in a superficial manner (for example pop science) but the basics are still extremely valuable even if they are not totally correct!

I think actually Thomas' first point is more important. I have noticed that a lot of mathematical results/theorems are presented in their final polished form, and often the proof provided reveals absolutely no insight into the problem. This means that although you then know the result is true, you may not understand it at all, so may not be able to apply it properly, or even appreciate the full meaning behind it. Of course I'm not saying that all proofs should be presented in a totally intuitive form - for harder results this simply isn't possible. But I think that it would be more useful to outline how the result was originally derived and what made the prover think of using this method. I have a few extreme examples of totally unmotivated proofs from my lecture notes, so I'll see if I can find them (if I even bothered to write them down!)

Ah, but that's the difference, isn't it? In Mathematics you can teach something which is simplified but not actually wrong. For example, the basic definition of trig functions is fine, just a little limited. Compare this with, for example, everyone telling you that nerve cells can't regrow (which is the accepted view even at the start of undergrad science courses) when it is actually the case that they can and indeed do - but it's very difficult to show. That is a blatent lie with no reason behind it; science is full of these - something which you're told as a student is true actually turns out to be nothing like the truth. Actually, the fact that people tell you things in science are "true" in the first place is a lie in itself - there's only theories and models which happen to fit the data better than anything else we've come up with.

-Dave

Negative numbers do not have square roots.

How long did you believe that lie for?

As far as "polished" proofs go, Michael, these are usually the forms in which the proof is easiest to follow, and generally most intuitive. The initial proofs found by researchers are usually quite ugly, as they are often discovered in chunks - proving C implies D, then that A implies B, then that B implies C. After some thought, the published proof that A implies D is developed. Not infrequently, someone else, coming from a fresh perspective, will then find a more elegant proof of the result which has nothing to do with the original steps. This last one is the proof you'll see, and rightly so, since it is much easier to follow. Sometimes you may want to see different proofs of the same result, using different mathematical concepts, depending on the context in which you are studying the result. Sometimes, however, although you are (for example) using a result in probability theory, its proof in group theory, say, might be much easier to follow. Sometimes, then, some proofs are merely of historical interest.

Negative numbers not having square roots isn't a lie if you're only considering the field of real numbers. Not introducing students to a field extension of the reals which is closed under squareroot operations is not really a lie, it's just saving a more advanced concept for later. I think that a better example is the early teaching of calculus which more closely approximates a lie, although even then...

Exactly. I can't think of a single instance where we're actually lied to. When they say "negative numbers have no square root" they're implicitly adding "in the real numbers" because we wouldn't have understood this at the time even if we had been told. However, lies in science tend to be of a form where we could comprehend the truth but it's done more for convenience of the syllabus than anything else.

-Dave

Well, the canonical example is the one I cited above - talking about the structure of the atom without going into quantum theory. The "convenience of the syllabus" is a bit of a red herring - the syllabus is (supposedly!) designed as a sensible way to learn the subject, and the reason certain "lies" are used are because they are good approximations within te desired complexity of the syllabus. For instance, you study Newtonian mechanics at A Level because, in most situations, it's a good enough model without having to go into the complexities of general relativity - astrophysicists may use relativity, but rocket scientists use Newton.

On the other hand, I think many "lies" remain for historical reasons - they have been shown to be false (like nerves regrowing) but have been part of the accepted "truth" for so long that they remain due to ideological inertia.