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Inductive Step. Theorem 2. We shall prove the theorem by induction on a. There is nothing to prove here.
Theorem 3. We prove the theorem by induction on b. Basis of Induction: Theorem 2. The reason for my restraint is that writing down conditions 5.
Theorem 4 Don Zagier . This modular exponentiation is widely used in cryptography; we shall return to it later in the book.
To save our theory from collapse, we shall prove the existence of addition after a brief pedagogical digression. This is supported by another childhood story, from BB6 : I would like to point out that the stories are provided by the people who happen to stumble on this blog.
I am sure that the readership of this blog is atypical in terms of mathematical thinking 6 BB was 11 or 12 years old at the time of the story.
He is male, Russian, currently a PhD student in pure mathematics. A cyclic Gray code discussed by BB in his story is a cyclic path on edges of the n-cube passing exactly once through each vertex of the cube.
Most people who have responded have not only gone to deal with unusually abstract concepts in their career, but actually do mathematics. So, the examples here might represent not so much the major difficulties that need to be overcome before an understanding can be reached as in finding the correct way of thinking of division of apples by apples , but the signs that understanding has already been reached, and that the difficulty is purely semantic, i.
I did passably well on the problems, but still I did not understand what induction was really for, until the end-of-year competition.
I failed to solve a single problem: arrange all binary strings of length 10 around the circle so that two adjacent strings differ in precisely one position it is known as a cyclic Gray code of size It was when I was told the solution that I felt that I finally understood the induction.
The missing element was probably the fact that I did not realize that the statement proved by induction is an honest mathematical statement that pertains to concrete numbers like 10, and not only to x, y, n, m and , among which only the latter is a number, but so big and arbitrary that it could as well be denoted by n.
And another story, from RTC:7 I do remember that when I was about 12 or 13 at school we were taught mathematical induction for the first time.
Infinite descent. This poster gave birth to the term Droste effect. And on the label on the bottle there was the same drawing, however smaller and on this picture on the bottle was a smaller bottle etc.
At that age we had only ever proved things directly, and with induction we seemed to be side-stepping the issue and not proving anything.
Golden Rectangle. The quickest proof by infinite descent is perhaps proof of irrationality of the Golden Ratio I found it in a beautiful little book by Tim Gowers [, p.
Theorem 5. The Golden Ratio is irrational. The ratio of lengths of sides of a golden rectangle is called the golden ratio.
Hmm, we did not even care about the numeric value of the golden ratio. A contradiction. Although this is not emphasized by Landau, the proof of consistency of addition does not use Axioms 3 and 4.
Are these axioms of any use at all? We shall return to this question later. Theorem 7. Let M be the set of all x for which this is possible in exactly one way, by A.
Hence 1 belongs to M. Therefore M contains all x. The preface for the student is very short and begins thus: 1. I will ask of you only the ability to read English and to think logically-no high school mathematics, and certainly no higher mathematics.
Our discussion of natural numbers continues in Chapter 7. Exercises Exercise 5. Exercise 5. This is a story from Jo French.
John Baldwin writes: The associative law can only work on two applications of the plus sign. We generalize it to say we can regroup any sequences of additions.
It is quite plausible that this is a distinction one should not make for teachers or at least the question should be at what level you want them to be aware of it.
The second problem is that when minus signs are interspersed this gets more complicated. Since associativity fails for subtraction some further rules are required.
At the time of this episode she was about 8 or 9 years old. This her story continues on Page Subtraction is not associative!
Very frequently, children are placed in a position where they have to figure out rules which have not been made explicit.
Of course, children learn the grammar of their mother tongue exactly that way, by absorption. Striking the right balance of rigor and computational fluency is a hard task.
John Baldwin commented further in my blog: You are well to wonder when I question whether or maybe better how teachers should be taught this.
One of the faults of a preliminary version is that certain facts understood by the writer and immediate audience are not spelled out.
Thus we were talking in the seminar for whom that was written about future elementary school teachers. And the issue that I was alluding to was at what stage you can make such students self-aware of subtle matters.
Because they have never heard either word. But return to Jo French. I only really felt happy with this after learning what a group is.
Obviously the same is true for division and multiplication, but I remember being unhappy with subtraction more clearly. As we shall see in the next section, she was not alone.
A very clean but excessively detailed calculation. Certainly, it is another question whether this is related to the way teaching was done, or a lack of personal mathematical vocation.
I must say I found this definition very easy to grasp. I have a vague recollection that this was not the case for many of my classmates.
We had very bad teachers in the 6th and 7th grades. My brother pushed me to studying the book myself and finishing in summer the chapters left undone during the school year.
You have already a few examples of new math courses in your book. In hindsight, it is evident she mastered the principles as well as the way to impose it on young pupils.
But many of her colleagues did not. Here is an account of a point when I was ten about negative numbers. I have been interested in mathematics for an early age, at least 7 by my own recollections and papers and if I take my parents word, at least 4 years old, asking questions and reasoning about quantities and counts of things.
This was without any geometric imagery the first introduction I remember of such was in 4e and 3e for introducing coordinate systems and vectors, distinguishing direction and sign, first only in one dimension for weeks then in two dimensions.
For two weeks we made only progressive exercises in this way, reducing in small steps an expression with a lot of parentheses and visually clashing signs to a final result where we were at least authorized to make the final arithmetical operation that we had practiced in primary school.
At first we were told exactly what kind of simplification to do from one line to the next, then we could do it in the order we found the most convenient.
It was a lot of effort and demanded good focus and systematic application of rules before they began to be internalized.
We were at least the few boys and girls I used to chat with so happy to write and treat now the same exercises quickly and efficiently and a little mindlessly or automatically that we felt a little superior to our former selves and began to look at complicated expressions with a sense of familiarity.
When discussing this episode with a former schoolmate, say ten years later while at the University, I was shocked that the only thing he remembered about this is that in math, they were changing rules all the time.
I told him that it was on the contrary a very close simile to what was done in natural languages all the time: you learned quick but not very correct or not very precise ways to say things when speaking orally with your parents, friends and radio, but you learned also in school a full and scholarly way of saying the same thing in much longer form which could be analyzed with grammar and could be used for more varied purposes than the quick and dirty way.
We were studying modern maths, and the lecture of the day was the construction of the relative integers Z from natural positive integers N by taking the cartesian product and factoring it by an equivalence relation; that is, the classical construction of the symmetrization of the commutative monoid.
We could follow, and understand, all the steps in the construction, but the goal of the whole exercise eluded us completely; what were we doing?
I think nobody in the class understood what was going on. After the class, during the 10 minutes break, I went to see the teacher, and asked her what was the point of the construction: after all, we had known negative integers for years, and there was absolutely nothing new in this.
Fortunately, the teacher understood very well the subject she had participated in writing the book , and was able to answer. She explained that we should consider this as a game, and this proved that, to obtain negative integers, we did not need to invent something completely new coming from outside: just playing with what we knew, the positive integers, and using a few tool, we could build negative integers, fractions, real numbers, complex numbers.
The explanation was very convincing, and put a new light on the course; I was very happy to understand that, and very surprised that it had never been mentioned anywhere: we were, in effect, giving complicated answers to tricky, and quite philosophical, questions that had never been asked!
I think that this short conversation had a decisive impact in turning me towards mathematics. What surprised me also is that, a few years later, after my Ph.
Another story from him is on Page The class was having trouble and Mr. I remember being terribly excited about this observation and I went on to explain it to several friends.
A large door in algebra had opened for us, and to this day I think of the subtraction of natural numbers in its equivalent form of adding two signed numbers.
It amazes me that this equivalence is not drilled into children. I am convinced that so many of their problems in beginning algebra would disappear if it were.
BS, aged 6 Each morning in class we spent about 90 minutes on arithmetic, and were usually issued with postcards containing three additions of 3 two or three digit numbers and three subtractions of three digit numbers.
I never got to the subtractions, being too slow: but finally worked out that they must be easier, as they involved only two numbers.
So I started doing the three subtractions first; of 8 TE is male, American, a professor of mathematics education.
Does anyone know? We had a blackboard-full of subtractions using decomposition to do. Therefore I remember still subtracting the smallest digit from the largest, which was incorrect.
The next day I think we moved on to the next aspects of mathematics. We had not covered them at school, and she is not very mathematical.
I think she even expressed some scepticism 9 10 BS is male, English, has a doctorate in mathematics and a lifetime teaching mathematics in university setting.
AH is female, a New Zealander living in England. She has a PhD in Mathematics Education. She is a mathematics teacher educator. This episode occurred during her schooling in New Zealand.
All the same, I got the basic idea, and was very excited about these strange new numbers. Next time we were studying subtraction in school, I got all the questions wrong!
Looking back, I had not understood that while addition is commutative, subtraction is not! The rest had all had positive answers so this stumped me.
She did so by saying you do not get a negative number of apples do you there are only 0 or more than 0.
I took this to heat in such a manner that later on in my schooling when negative numbers were introduced, I was convinced they were always the result of mistakes.
It took me a long time to realize that they were valid mathematical concepts, that could be useful, they just do not relate very well to apples.
This turns out to be one of the basic examples of the so-called nvalued groups as introduced by Sergei Novikov and Victor Buchstaber; see  and Exercise 6.
To express the confusion which operation is which, [ ] denotes 11 12 RE is male, English, holds a PhD in mathematics.
KP is female, British, holds a PhD, is researcher is computer science. The origins of the theory of n-valued groups lie in algebraic topology.
Exercises Exercise 6. It goes like this. Exercise 6. But Landau was not in hurry to introduce subtraction; he first developed the theory of fractions, real and complex numbers, and only then proved the properties of subtraction, simultaneously for all these number systems.
We shall use square brackets [ ] to write unordered n-tuples. For example, [1, 2, 1] and [1, 1, 2] represent the same element of N 3 , while [1, 1, 2] and [1, 2, 2] are different elements of N 3.
We shall call X n the n-multiset of X. The following three axioms define n-valued group structure.
Unfortunately, most people who are not computer scientists believe these two modes of thinking to be the same.
The purposes, nature, frequency and levels of abstraction commonly used in programming are very different from those in mathematics. This statement may appear to be extreme, but let us not to jump to conclusions and look first at a very simple example.
I suggest to have a look at M ATLAB, an industry standard software package for mathematical mostly numerical computations.
I apologize to my computer scientist colleagues who on a number of occasions explained to me that M ATLAB is nothing more but a glorified calculator.
I choose M ATLAB because, for a lay person, it provides an easy, even if limited, insight into what is going on in computer realizations of natural numbers.
We shall look in the next chapters at how they keep control of this bestiary. For the time being, we have only to take note that we have to be prepared to look at many different number systems satisfying the Dedekind-Peano axioms.
In England, a popular slander about Yorkshiremen is that they use special numerals for counting sheep. In Wensleydale, for example, the first ten sheep numerals are said to be yan, tean, tither, mither, pip, teaser, leaser, catra, horna, dick.
One lesson is that we have to distinguish between ordinal numerals, which express relative order of objects, first, second, third,.
In languages around the world, there is a remarkable diversity of systems of numerals, both ordinal and cardinal.
We have already discussed in Chapter 1 a special class of distributive numerals in the Turkish language. The Japanese language provides one of more striking examples of diversity of numerals.
Here, different numerals are used for counting, for example, flat objects like sheets of paper and long slender objects like pencils.
I give a table of some of them: regular numbers 1 2 3 4 5 6 7 8 9 10 simple objects flat things long slender things ichi hitotsu ichimai ippo ni futatsu nimai nihon san mittsu sanmai sanbon shi or yon yottsu yonmai yohon go itsutsu gomai gohon roku muttsu rokumai roppon shichi or nana nanatsu nanamai nanahon hachi yatsu hachimai happon ku or kyu kokonotsu kyumai kyuhon ju or jyu tou jumai jyuppon S HADOWS OF THE T RUTH V ER.
We shall soon see that this is because, as frequently happens in mathematics, existence of nice additional structures on a mathematical object is closely related to the uniqueness of this object.
Indeed, we had seen in Sections 7. In the same time, we for some reason believe that they all are the same object: a unique canonical object which we call the system of natural numbers and denote by letter N.
In his paper  David Pierce summarised difficulties demonstrated in Section 5. Historically, this observation was made explicit by Dedekind [17, II.
In a more algebraic language, the issue is clarified by Henkin in . Notice that every unary algebra automatically satisfies Axioms 1 and 2.
Theorem 8. But, as we have just seen, the argument takes some work. A unary algebra with commutative and associative addition with thanks to David Pierce.
And would not the reader agree that little Elizabeth Kimber and little AB had good reasons to be confused? If, in addition, an induction algebra satisfies Axioms 3 and 4, we shall call it a Peano algebra.
Theorem 9. Let p be a prime. The following proof belongs to Leonard Euler . It had been presented by Euler to the St.
Petersburg Academy of Sciences on 2 August [, p. Gauss reproduced it in his Disquisitiones Arithmeticae [, art.
Each morphism f has a unique source object a and target object b where a and b are in ob C. We write hom a, b to denote the class of all morphisms from a to b.
Now we can reformulate Theorem 9 in the language of categories. Theorem Any Peano algebra is an initial object in the category of unary algebras.
In particular, any two Peano algebras are isomorphic. It is, then, logistic which treats on the one hand the problem called by Archimedes the cattle-problem, and on the other hand melite and phialite numbers, the latter appertaining to bowls, the former to flocks; in other types of problem too it has regard to the number of sensible bodies, treating them as absolute.
Its subject-matter is everything that is numbered; its branches include the so-called Greek and Egyptian methods in multiplications and divisions, as well as the addition and splitting up of fractions, whereby it explores the secrets lurking in the subject-matter of the problems by means of the theory of triangular and polygonal numbers.
Its aim is to provide a common ground in the relations of life and to be useful in making contracts, but it appears to regard sensible objects as though they were absolute.
Quoted from [, pp. Exercises Exercise 7. If the definitions are consistent, are the corresponding binary operations commutative?
Will the distributive law of multiplication with respect to addition hold? Exercise 7. Indeed, how can we claim that the set of all flying pigs equals the set of all mermaids?
There is nothing in common between pigs and mermaids. Prove that any two empty sets are equal by considering a category with sets as objects and appropriately defined morphisms, and such that empty sets are initial objects in that category.
Does it come with legs? Waiter: It comes with one leg. A conversation in a diner. Adding fractions is a notorious issue in mathematics education, and appears to grow in its notoriety.
We start our discussion by quoting a testimony from Simon J. Shepherd1: My main problem while quite young was adding fractions.
I was quite happy multiplying them, since you simply multiplied the two top numbers to get the answer top number, then multiplied the two bottom numbers to get the answer bottom number.
But with adding you had to find a common denominator by cross-multiplying and I remember it took me years to crack this technique!
Funnily enough, many of our foundation year students have exactly the same problem. Perhaps learning the multiplication before the addition actually inhibits learning the addition rule?
My personal opinion is that, indeed, addition should come before the multiplication. I will try to explain that in the next section.
His other stories are on Pages 10 and I am surprised how frequently such memories are related to the subtle interplay of hidden mathematical structures, like the dance of shadows in a moonlit garden; these shadows can both fascinate and scare an imaginative child.
AG was less fortunate: The word for fraction in Romanian is fractie, and that was the terminology used by my teacher and in textbooks.
In effect, when dealing with quarter apples, we are working in the additive semigroup 14 N generated by What happens next is much more interesting and sophisticated: we have to learn how to add half apples with quarter apples.
In primary school, of course, taking the inductive limit is called bringing fractions to a common denominator. Fraction-of-an-inch Adding Machine, , Patent no.
Photo by Windell H. B OROVIK 90 8 Fractions A directed set is a nonempty set A together with a reflexive and transitive binary relation that is, a preorder , with the additional property that every pair of elements has an upper bound.
Notice that the directed set in our definition is the set N ordered by the divisibility relation. It is not linearly ordered and has a pretty sophisticated structure by itself!
Let I, be a directed set. It is not a simple and straightforward operation. In words of SJS4 : My main problem while quite young was adding fractions.
But with adding you had to find a common denominator by cross-multiplying and I remember it took me years to crack this technique!!!
But let us return to construction of an inductive limit. For a child, this may look like a magic, as a story from Tony D.
Gilbert5 confirms: I greatly liked arithmetic at primary school. Among other things she taught us fractions it seems amazing now that she taught us this, very successfully as far as I was concerned, whereas I regularly see first year students of Science who are quite unable to cope with these.
My recollection is that she used dividing up a cake as her model for fractions just as I have done with many a student since!
Having established fractions in lowest terms, she then went on to deal with canceling down, multiplication and division, and also addition and subtraction of fractions with the same denominator.
The point of this story was my reaction to that final explanation. I can still remember my puzzlement before the explanation as to how to one would deal with the problem of distinct denominators, and my really wanting to have the problem resolved.
Then once we had patiently and in my case enjoyably gone through the detour of HCFs and LCMs, my pleasure and appreciation of the cleverness or so it seemed to me of the resolution of the problem, once the explanation was given.
My teachers could not explain, but I was used to that. Finally I asked my father, who was an accountant.
He said: if you divide everything into halves, you have twice as many things. Suddenly not just fractions but the whole of algebra made sense for the first time.
Some my correspondents found this, at a first glance more formal and purely symbolic approach to fractions and rational functions easier to grasp.
Listen to a testimony from RW7 : I recall very few difficulties with mathematics in my formative years: on the whole it was a case of learning a simple rule and doing exactly what you were told.
I do recall being introduced to fractions at age about 7, and wanting to know how to add them together. At least I think it was an algebraic formula, but this might be a false memory implanted by my later understanding: it might really have been an algorithm, or even an example.
Presumably my difficulties went away once I was taught it properly. Victor Maltcev8 preferred a purely axiomatic approach: When I was 8 we learned at school 2nd grade rational numbers.
I had very big problems in grasping them the whole 2nd and 3rd. Eventually I started feeling I will never get along with them and every time before classes in Maths I felt depressed that again we 6 7 8 BC was 10 years old.
He is male, a non-English Western European. RW is male, British, a professor of mathematics. When I was 10, 5th grade omitting the 4th , in one of the classes I realized that in order to understand rational numbers we should not understand them!
This gave really a big push to studying Maths after school classes. Alexey Muranov: I also remember that probably in the 5th grade I added up fractions with different denominators without being taught or asked :.
And this is also tricky. This should demonstrate the commutativity. I had a vague feeling, that this demonstration is not really a proof, but did not dare to say aloud such a nasty thinking to the teacher.
Today I would subsume this epistemological problem under The Unreasonable Effectiveness of Mathematics.
Eugene P. Wigner J is an intriguing case as she has a severe aversion to all things mathematical and, in particular, division.
In fact, from what I have seen, it is beyond me as to how 9 10 BB is male, Austrian, a professor of mathematical physics.
Ok, how many 5s are there in 20? Ok, how many 4s are there in 20? I found this fascinating. I decided to try again.
Yet again, the multiplication was no real problem, the first division only a minor problem, but the second division caused major problems.
By contrast, a story from Jonathan Kirby: Here is my favourite [true! What is five times three? Are you sure?
To me, five times three meant adding five copies of three, which was clearly different from adding three copies of five.
I was puzzling over why they should be the same for the next few days, then suddenly I realised that I could see why they were the same.
On squared paper, you could colour in five rows of three on top of each other, and the total number of coloured squares was the product, But then you could rotate the paper and have three rows of five, and the same number of coloured squares.
I have loved mathematics [almost] ever since. And Alexey Muranov reminds us about the role of visualization: I also vaguely remember that some of the algebraic laws studied in the second grade, such as associativity and distributivity, were posing problems.
I do not remember now which exactly, but it could have been the associative laws for multiplication. I am not sure, but this could be because the other laws could be explained on 1or 2-dimensional pictures, and in class we only used 2-dimensional pictures, but the best way to explain the associativity for multiplication would be to arrange objects in a 3-dimensional parallelepiped.
I did not think in terms of pictures, I somehow was able to imagine those laws, but probably I used geometry subconsciously.
Exercises Exercise 8. Exercise 8. Let us take a short break, to pause and reflect. I wish to devote a few pages to a discussion of the structure of this book.
Have you noticed that I started with multiplication of natural numbers and only then moved to addition? This is not the natural logical order for the two topics.
I deviated from the natural logic of development of mathematical theory for a simple reason: for me, it made pedagogical sense to postpone the application of severe rigor and the axiomatic method to later chapters.
I would be delighted to have comments from my readers: do they agree that it was right choice? Notice that in this chapter I will discuss mostly the issue of university mathematics education.
Even if the topic of the book is elementary mathematics, it is discussed at a university mathematics level. A fragment of The Rape of Helena, The concept of didactic transformation is fairly old and can be traced back to Auguste Comte [, preface]: A discourse, then, which is in the full sense didactic, ought to differ essentially from one [that is] simply logical, in which the thinker freely follows his own course, paying no attention to the natural conditions of all communication.
In French literature, there is a concept of transposition didactique; the term has been devised by Yves Chevallard  and has become a mainstream element of the French mathematics education theory.
We have to remember that even a simple change in order of exposition could have dramatic effects and usually requires a serious rethinking of the underlying logical development of the material.
This depends on having a theory of integration that enables the integration of continuous functions, but we would want this anyway.
To develop the theory based on the extension of ax from x rational to x real would need some subtle analysis, possibly including uniform continuity.
I therefore favour the definition of ln as an integral. We have to accept that, in mathematics, didactic transformation is indeed a form of mathematical practice.
Moreover, it is in a sense applied research since it is aimed at a specific application of mathematics: teaching. It remains mostly unpublished, underrated and ignored because it is frequently confined to early stages of course development or to ephemera of classroom practice.
Unfortunately, the principles of didactic transformation are frequently neglected in the mainstream mass mathematics instruction. No mathematical idea has ever been published in the way it was discovered.
Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty.
This then if if has affected teaching matter, is the didactical inversion. Not in the trivial abridged version, but equally we cannot require the new generation to start just at the point where their predecessors left off.
Instead, I wish to turn to a case study. Its choice is motivated mostly by references to the concepts of convergence and limit in the later chapters of this book.
Moreover, definitions and axioms often neither formalize nor generalize human everyday concepts. A clear example is provided by the modern definitions of limits and continuity, which were coined after the work by Cauchy, Weierstrass, Dedekind, and others in the 19th century.
Anyone who has taught calculus to new students can tell how counter-intuitive and hard to understand the epsilon-delta definitions of limits and continuity are and this is an extremely well-documented fact in the mathematics education literature.
The reason is cognitively simple. Here is a story from Azadeh Neman: As for the limits, we were told to imagine an animal getting closer to point on the plane but not really arriving at it.
This is a very poor description and gave me many years of unnecessary pain while dealing with limits. The thing is this animal might actually decompose itself into many different animals who have nothing to do with the first one and whoever acts more dramatically close to a certain point will guide the limit close to that point.
There is another approach, due to the famous logician Abraham Robinson , 2 AN is female, Iranian, learned mathematics in Persian and later in English; at the time of this episode she was 15 and taught in Persian.
AN holds a PhD and is a postdoctoral researcher in mathematics. B OROVIK 9 Pedagogical Intermission:Didactic Transformation which places infinitesimals quantities and variables which are bigger than zero but smaller than any positive real number , purged from calculus in 19th century, back at the core of the subject.
So far, this approach has made only relatively modest inroads into mainstream teaching; see Keisler [43, 44, 46, 47]. Internal set theory proposed by Nelson [73, 74] blurs the difference between finite and infinite in a very simple, effective and controlled way.
A brilliant little book Nonstandard Analysis by Alain Robert  demonstrates the didactic power of this approach. The Introduction to the book starts with a quote from Euler: Since L.
Euler was among the most inspired users of infinitesimals, let him have the first word. Jerome Keisler . My correspondent Frederick Ross has brought my attention to yet another [.
This is [. The interested reader can find a comprehensive exposition of synthetic differential geometry in Anders Kock .
Next, there is a lecture course by Donald Knuth on calculus in O-notation, taught by Knuth for may years and exceptionally well polished pedagogically.
I recall being comfortable with the definition of derivative as a limit. As you correctly point out, it takes a considerable amount of mathematical training to formulate precisely what the problem was.
In retrospect, what I must have been bothered by is the non-constructive nature of this definition.
Actually I am currently writing a text on constructivism, and it could be that even after all these years I would still be unable to identify the source of the anxiety were it not for the fact of having understood constructivism better recently.
It is a timely reminder: didactic transformation is necessarily a very subtle balancing act. There is also a very promising approach to calculus based on eliminating the concept of a limit and replacing it by uniform Lipschitz bounds [60, 64].
This definition can be applied only to a narrower class of functions than the one usually called differentiable. However, the class of uniformly Lipschitz differentiable functions suffices for most engineering applications and is open to a rigorous and didactically efficient treatment in teaching.
The uniform bounds approach allows to introduce yet another simplification: at an entry level, differentiation can be introduced as a mere factoring.
It is only natural to suggest that didactic transformation should form a part of a professional toolbox of a mathematics lecturer. Mathematics provides a bewildering variety of apparently incomparable approaches to the same topic.
We also have to remember that we cannot freely bend them into the desired shape or pick and mix elements of different approaches: each of them has its own internal logic which cannot be interfered with.
In that case, how do we predict and assess the relative advantages and disadvantages of a particular approach in teaching to a given group of students in a given course?
Every mathematician is aware of the existence of so-called mathematical folklore, the corpus of small problems, examples, brainteasers, jokes, etc.
Occasionally, these observations find their way to print. But in general the collective pedagogical experience of university mathematicians remains uncharted territory.
One also has to take into consideration cultural differences between various countries and various university systems.
In Britain, where I live and work, almost every course in university mathematics departments and most mathematics courses in service teaching are tailor made.
I argue that this hidden work of teachers is essentially a form of mathematical research; it uses the same methods and is based on the same value system.
The difference is the form of output; instead of a peer reviewed academic publication or a technical report for the customer, as it is frequently the case in applied and industrial mathematics the output may take the form, say, of a detailed syllabus for a course which exposes classical theorems in an unusual order, or just a page of lecture notes with a new treatment of a particular mathematical topic.
The mathematical problems solved by a lecturer in the process of course development and conversion of mathematical material into a form suitable for teaching are far from glamorous.
As mathematical research stands, this kind of work is perhaps unambitious, but it is nevertheless mathematical problem solving made very challenging by severe restrictions on the mathematical tools allowed.
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