In Florence, Italy

I’ve been away from writing here a long time. Partly it is because I’m in Florence on a research workshop at the Galileo Galilei Institute for Theoretical Physics since three weeks, and before that I did some research as well as wrote on the report from the college trip.

Anyway, the report is finished now, and here it is UPDATED: ReportLibArtMath

Hopefully, I will find time to write here soon. In May I’ll start teaching again …

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Writing the report

This is just a short note to tell you that I haven’t quit thinking about mathematics. I was in the process of writing up the report when I hit upon the subject of calculus in America. There I got stuck and still is. So instead, I took time off to prepare for a research workshop in Florence in Italy (Higher Spins, Strings and Duality) . I’m going there in April. I’ll be away for 4 weeks. But before that the report must be finished and I’ll soon be back writing here!

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Rationale for Only-Answers-Exams

Let me be honest. I hate marking exams. But I enjoy teaching. For me at least, teaching at the university level is the best possible job. I like the freedom of the it, I enjoy explaining things and thinking about well-known things in a new way. If I should complain, I would like to have some more time for research. However, since I have over the years developed a, what I call “Just-in-Time Unpacking” method of teaching that requires almost no time for preparations, I get time for that too (I will describe the method in another post).

But I simply can’t standing marking exams. I remember as a graduate student when I got a pile of 180 exam papers in Mathematical Physics for Electrical Engineers to mark. I split the pile with a fellow student (Ulf B. if you happen to read this) but still it was almost a week of boring job. But that was extreme. Normally you get at most 50 papers or so to mark. Still I can’t stand the drudgery.

Some years ago while teaching a course in introductory thermodynamics I got the idea of simply doing away with much of the marking job and speed it up. The problem was really, as I had discovered by introspection, that marking exams require too much attention, it can’t be automated. It is sometimes alright to do routine work, like painting a fence or hand-wash dishes, because you can think about something interesting during the work. Some types of work, like hanging wallpaper, is really philosophical and very conducive to deep thinking. But not marking exams. There is no way I can think about something else while marking exams.

Why not design an exam with tick-box questions? It took some thinking to reformulate the typical exam problems in this way. Of course, there must be a correct answer to tick for each question, and it took some work to ensure that I did not myself make any stupid errors (and I did occasionally). What was problematic was funnily enough to decide on the wrong answers. All in all it took considerably longer time to put together an exam, but the extra time also made for higher quality in the questions (I think).

The marking was swift!

No student ever complained about this type of exam. But I was once challenged by one parent (!) who thought that guessing made the exam unreliable. Intuitively I did not think so, I had thought about it when setting up the questions (number of alternatives and way of marking), but now I had to work through the statistics. There was no way you could pass the exam by pure guessing.

However, eventually it turned out to be too complicated to design the problems in this way. It took too much effort to make 100% sure that there always was one correct answer to each question. And also coming up with the wrong answers wasn’t that easy. So I switched from tick-box answers to Only-Answers where the students write down their own answers rather than checking against a list and choosing the correct. That worked well. Designing exams was reliable and the marking still swift.

Next step was to do it in mathematics courses.

The standard written exam in say physics or mathematics requires the student to solve a set of problems. They are supposed to write up their solutions in a way that is possible for the teacher to follow. The solution should lead up to the correct answer. The exam requires that solutions should be well motivated with notation and equations explained et cetera.

Over the years I made the following observations:

  • There is a very strong correlation between good solutions and correct answers.
  • There is a very strong correlation between bad solutions and wrong answers.
  • Even the good solutions are seldom, at least not always, up to the standards of “solutions well motivated with notation and equations explained”.

No colleague I’ve talked to have denied this. And of course it is highly expected. Furthermore, any experienced teacher who is marking a problem starts at the end and checks the answer. If it is correct it takes no time to see that the solution is OK, perhaps not up to the requirements, but OK all the same. If the answer is almost correct it is often easy to spot the error, but not always, sometimes it takes a considerable amount of time to find the error. If the answer is plainly wrong but there is a long solution it often takes a long time to evaluate the solution. Then there is a middle ground which is very time-consuming. It is clear that the student don’t really know how to solve the problem but there is a lot of writing to go through. You get into a mode of working where you try to dig out something that is correct. But this wading through badly formulated solutions leading up to wrong answers in order to locate the exact errors and judging how serious they are, or trying to find out how many points to hand out, is largely pointless. After all, it is wrong.

Some students expect to get some points for half-baked attempts at solutions where it is plain that they don’t know how to solve the problem, or don’t even have the basic skills to do it. Encouraging this is bad pedagogy.

What are the good points of Only-Answers-Exams? Here are some:

  • The focus is on getting it correct: To get 10 points you have to do 10 things 100% correct. It is not enough to do 20 things 50% correct. This is really a matter of Quality Management. 90% correct means 9 correct out of 10 and 1 wrong, not all 10 things 90% correct.
  • The marking is fair and objective. Same rules apply to everyone. There is no bias due to bad or good handwriting.
  • Almost no borderline arguing. If the limit for PASS is 20 points and a student gets 19 points, there is no way to find another extra point because there are no more correct answers on the exam. The result, even though sad, has to be accepted.
  • You can still design the exam problems so that students can get points even if they can’t solve the complete problem. You split up the problem in parts and design the questions in steps. I’ve heard complaints that this gives the students too much help in solving the problem. What’s wrong with that? Building the problem step by step helps the student to show how much he or she knows. I’ve seen too many exam questions that are meant to trip the students.
  • The only thing you lose is checking if students can formulate coherent solutions. But that can be checked in other ways. You can have one fairly simple problem where the focus is precisely on writing up a solution.
  • And marking is swift. You get time for the real job: teaching well.
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Applications or Engagement?

Over the years I’ve lost confidence in the idea that applications of mathematics make the subject any more interesting or easier to grasp. It is perhaps not a clever thing to say in public because of the obvious objections. But I’ve gradually swung over to this idea about a humanistic approach focusing the inherent interest and depth of the subject itself with its history and philosophy and weird language. And the applications of mathematics are naturally included under the banner of humanistic mathematics. Applications in science and technology and everyday life are clearly but aspects of human culture. A humanistic approach to mathematics thus naturally subsumes applications and there is no contradiction.

That is a good thing – it puts applications in its proper place – it is but one aspect of mathematics. Thinking about applications in a humanistic way may be an anti-dote against dry, dull, boring and silly standard applications of textbooks. So now I’ve said it.

A happy thing occurred a week ago in one of my calculus classes. Here is the background. I don’t remember if I’ve written on these pages about my experiments, going on for a couple of years now, having mathematics exams where I just require answers to the problems? Yes you read it correctly. I don’t care for the solutions to the problems. I only want to see the answers.

You may be so shocked that you cannot continue reading and perhaps I should explain my rational reasons for doing this. However that would distract me from what I want to say in this post, so bear with me, I will soon write about why I think Only-Answers-Exams is a very good thing in mathematics.

However, there is no denying that there is a point in the students being able to write up a solution to a problem. It’s just that a 4-hour exam is not the proper time to require writing up solutions to perhaps 8 problems. As a compromise, I have now included one problem where the focus is on writing up a solution. It should be an easy problem that almost everyone who has done the course can do so that the focus is indeed on writing up the solution.

Anyway, the calculus class was drawing towards its end and I was trying to come up with a good problem to require a solution to. The students are textile engineers, so I thought about such “applications” and finally came up with the example of calculating how long a thread can be rolled up on a bobbin. For every winding of the thread, the radius increases by the diameter of the thread, and the problem essentially boils down to summing an arithmetic series. I gave the students this problem as an exercise. For the exam I gave them the same problem but now requiring them to solve it by integrating the linear function for the radius as a function of the number of windings. The students became very engaged, they knew that some variant of this problem would appear on the exam, and that always leads to engagement, of course. We had an extra session where the students could ask question as a preparation for the exam. There was much discussion about various approaches to the thread winding problem.

RulleBlogg

I don’t think it was just the fact that the problem was an exam problem that lead to the engagement, these are ambitious and dedicated students. I believe that they found the problem intriguing in itself. It was simply an interesting problem that was engaging.

Could it be said that this is an application of calculus? Yes, probably so, but I prefer not to think of it as an “application”. I prefer to think about it as an “engaging problem”.

Let me try to generalize: we should not look for “applications” of mathematics. Since the real “reality” (so to speak)  is so complicated, real “applications” become to difficult. Instead we (textbooks) come up with unrealistic simple “applications” such as: what is the volume of this bottle, the shape of which you get by rotating a function around the y-axis. That problem is silly and boring. In reality: just fill the bottle with water and then …

[I know, however, that the volume of wine bottles was an important problem in the early days of calculus, so there is some humanism in that too …]

Instead “engaging” problems occupy a middle ground between real applications and silly textbook “applications”. They are interesting in themselves, they are not too simple and they require quite some thinking, they can be solved in different ways, and the common sense solution (take the bobbin and wind up a thread, then rewind and measure the length) is not really feasible. The problem should be relevant for the students. And we don’t need lots and lots of such problems (and they are not easy to come up with). Perhaps three or four of them, designed with care, is sufficient for a calculus class.

What I’m trying to say is: thinking “applications” leads the thinking into the desert of boredom, thinking “engaging problems” leads the thinking into more fertile grounds. Thinking “applications” means thinking “lifting mathematics into reality”. Thinking “engaging problems” means “lifting reality into mathematics”.

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The Distributive Law

We have started a book study group at my institute, the idea being to work our way through Finnur Lárusson’s “Lectures on Real Analysis”.  This is a really nice (and thin) Real Analysis book rigorously covering the same stuff as a standard one-variable calculus class. On Wednesday I tried to talk about the first two chapters. I took the opportunity to think more deeply about something that has intrigued me for a long time, namely the role of the Distributive Law.

What makes me think is my observation that many of the problems students have with algebra seems to revolve around the distributive law. I may write on the whiteboard (as part of some calculation):

5(3+x) = 15+5x

“Hey, what did you do!?”

Perhaps I write:

x+3x^2 = x(1+3x)

“What happened here!?”

The law can be used in two ways, of course, and there are names for those operations. In Swedish we say “multiplicera in” and “bryta ut” respectively. I’m not quite sure about the colloquial English expressions? Anyway, the question I’ve been asking myself is: Are there any deeper philosophical or cognitive reasons for these problems?

Some years ago I read in the n-Category Café a post about the fact that the Real Numbers support two binary operations: addition and multiplication and that these operations are closely connected to the exponential functions and the logarithmic functions. Of course, everyone knows that logarithms translate multiplication into addition, something which was indeed one initial motivation behind their invention. Not so well-known, perhaps, is the fact that astronomers in the 16:th century before the advent of logarithms, used trigonometric identities to simplify multiplication. The method went under the name of “Prostaphaeresis”. Now that can be understood through the relationships between trigonometric functions and exponential functions of complex arguments. Another story … but a related pedagogical observation is that logarithms cause even more severe problems for many students. There are certainly many reasons behind this, not the least the abstractness of logarithms and the un-symmetrical notation where we often write exponentials in the suggestive way e^x (based on notation for powers) but logarithms in the more obscure way log x.

My observation is that the connection between addition and multiplication is done via the distributive law at the axiomatic level. Then this connection comes back when we do logarithms and exponentials. Can there be a deep connection explaining the pedagogical problems here? Don’t know – have to think more. So instead back to what I actually tried to say at the seminar.

I started by stating the axioms for addition and multiplication respectively:

Let a, b and c be symbols for real numbers

Addition

A1   a + b = b + a
A2   a + (b + c) = (a + b) + c
A3   There is a real number 0 such that a + 0 = a for all a
A4   There is a real number b such that a + b = 0 for all a

Multiplication

M1   a · b = b · a
M2   · (· c) = (· b) · c
M3   There is a real number 1 such that · 1 = a for all a
M4   There is a real number b such that· b = 1 for all a

Here my colleagues protested that I must exclude a = 0 in the last axiom.

No you dont’, if these are the only axioms you have. What we have are two Abelian groups with no connection whatsoever between them. Because what excludes a = 0 in the last axiom for multiplication is precisely the distributive law.

The Distributive Axiom

  a · (b + c) = a · b + a · c

In a way it is remarkable (as I read somewhere) that there is just one axiom connecting the operations of addition and multiplication. Note also that it is multiplication that distributes over addition and not the other way around. Are we pointing this out to the students?

Anyway, if you instead start your mathematics with the natural numbers Ν and the Peano axioms, then addition and multiplication must first be defined recursively from the successor operation and then step by step be extended to Z, then to Q and then to R.

As long as one of the numbers in the product a · b is a natural number, we can think of multiplication as repeated addition, and that is the basic intuition behind defining multiplication in Ν from addition. Now if we start with R directly, then we need something else that provides a connection between the two binary operations. That is precisely what the Distributive Law does.

If we don´t have it, then we cannot write a + a = 2 · a. Let’s prove it using D.
Th1   a + a = 2 · a
Proof:
a + a = [M3] = a · 1 + a · 1 = [D] = a · (1 + 1) = [Def:  2 ≡ 1 + 1] = a · 2 = [M1] = 2 · a

The distributive axiom is crucial here. In this sense, it “captures the idea of multiplication as repeated addition” and generalizes it to both numbers in a · b being arbitrary.

Then for division by zero. Suppose there is a real number b such that
b · 0 = 1

Then 1 = b · 0 = b · (0 + 0) = b · 0 + b · 0 = 2 · b · 0 = 2 · 1 = 2
where I have only used axioms and Th1. A contradiction arises. Perhaps one should add that we must suppose that 1 ≠ 0.

Update: As a colleague was kind to  point out I can arrive at the contradiction easier by just adding the additive inverse to b · 0 (axiom A4) to both sides of
b · 0 = b · (0 + 0) = b · 0 + b · 0
Then we get at once 0 = 1 and we see that we indeed need to assume 1 ≠ 0. Thanks Bengt!

Let me end with the pedagogical reflection. No one has any problems with addition. Multiplication is a little bit more difficult, but no students have any serious problems with it. The problems start precisely where the operations starts to interact through the distributive law. We as teachers takes it for granted being an absolute triviality. I don’t think it is. I think it is deep.

One operation gives us group theory. Two connected operations give us algebra. Is it a wonder that it is difficult, especially since we don’t really do any group theory as a preliminary?

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From the World of Mathematics II

When I came home with the World of Mathematics, volume 3 had a bookmark (an old yellowed postal giro form) at the article Mathematics and the Metaphysicians by Bertrand Russel. The text is from 1901 I think, and it shows. This is the article from where the famous quip comes:

“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

Not a very humanistic point if view!

Russel discusses “pure mathematics” which he thinks of as consisting

“entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true.”

This is the view put forward in his book The Principles of Mathematics published in 1903 (most of it written in 1900). Mathematics consists of implications P → Q, one proposition P implying another proposition Q.

The article is really fun reading, written as it is by an obviously very enthusiastic Russel (and young – he was 28). The paradoxes of calculus had been solved by Cauchy and Weierstrass through the limit concept and the epsilon-delta approach. The infinitesimals of the pioneers had been abolished (once – but not for all – as it turned out). The paradoxes of Zeno had been solved and the concept of the infinite reined in by Peano, Dedekind and Cantor. The story is told by someone who was there close in time. Russel pokes fun at the philosophers, who haven’t been able to make any progress with the questions of the infinitely small and the infinitely big, since the Greeks. Now the problems are solved by the mathematicians – but have the philosophers noticed it?

But this was before Frege, indeed before Russel’s own discovery of the Russel paradox in the theory of infinite sets and the flaw in Frege’s logic. The Sisyphus work of Principia Mathematica was due in ten years time. The work where Russel and Whitehead tried to reduce all of mathematics to logic. This idea, that mathematics is nothing but logic, is spelled out in the article. This is what became the foundational school of Logicism. But also Formalism is implicit in the article, the idea that mathematics is a pure play with symbols using the laws of logic.

And then came Gödel … and the rest is history, as they say.

Neither logicism, nor formalism are very humanistic in their approaches to mathematics, but the history of why these approaches arose and their successes and failures, that history is truly humanistic. Basically it is the history of understanding the concept of number.

Reading and Writing and Calculating are core subjects in all schools all over the world and has been so for a long time. Understanding numbers ought to be a humanistic endeavour, just as understanding how to read and write.

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From the World of Mathematics I

Long time ago I read all six volumes of the Swedish edition (called Sigma) of The World of Mathematics. Last spring, when I dug out volume 6 from the bookshelf it fell open on Oswald Spengler’s Meaning of Numbers. A few weeks later I found the American edition in a nearby antiquarian bookshop (Röde Orm on Husargatan 18D in Haga, Goteborg, for locals or if you happen to pass by. At the time of writing the present text, there is one more copy to be found there.).

Since I now have started to write a report on the project I’ve also decided to do some reading. I’ll start with Leslie A. White’s article The Locus of Mathematical Reality: An Anthropological Footnote vol. 4 pages 2348-2364 (vol. 6 pages 2441-2458 in the Swedish edition).

White (1900-1975) who was an American anthropologist, addresses the question of where the mathematical concepts and truths reside. Do they belong to the external physical world or are they human mental constructions? His text is interesting in many ways. He refers back to earlier discussions about this issue and he is very eloquent about his own point of view: mathematics is purely a cultural phenomenon.

Another way to phrase the question is to ask whether mathematics is discovered or invented. White reviews how mathematicians have the feeling that they are discovering something which is external to themselves, citing G.H. Hardy as an example (among others). On the other hand, it seems just as clear that mathematical concepts are human inventions. So the answer would be that mathematics is both discovered and invented. (Perhaps one could say that the concepts are invented whereas the truths are discovered?) He clearly renounces any notion of a Platonic abstract realm where mathematics reside.

White’s answer to the dichotomy of discovery vs. invention is to claim that mathematics is a cultural phenomena. This sounds reasonable, but he does it in an anthropological framework which to me takes it a bit too far. When thinking of mathematics as a mental phenomenon there are of course two senses to the concept of a mental: mental concepts in the individual human being and shared mental concepts of the species. Here are some quotes from White (where the italics are mine in the following quotes):

“What we propose to do is to present the phenomenon of mathematical behavior in such a way as to make clear,  on the one hand, why the belief in the independent existence of mathematical truths has seemed so plausible and convincing for so many centuries, and, on the other, to show that all of mathematics is nothing more than a particular kind of primate behavior.”

Clearly it is too simple-minded to consider “external physical reality” and “internal mental reality” as the only possibilities for where mathematics could reside. The human culture is another possibility. White writes:

“Mathematical truths exist in the cultural tradition into which the individual is born, and so enter his mind from the outside. But apart from cultural tradition, mathematical concepts have neither existence nor meaning, and of course, cultural tradition has no existence apart from the human species. Mathematical realities thus have an existence independent of the individual mind, but are wholly dependent upon the mind of the species.”

This makes sense to me, but then he continues with:

“Or, to put the matter in anthropological terminology: mathematics in its entirety, its “truths” and its “realities,” is a part of human culture, nothing more. ”

It’s the nothing more part I don’t agree with. These quotes are from the beginning of the text. White then goes on to argue his case and much of it makes perfect sense, but he takes it too far. Even physical theories like Maxwell Electrodynamics and Einstein General Relativity becomes purely cultural in White’s view. Certainly, the detailed formulation of the theories of fundamental physics are dependent upon culture, but surely there is a basis in physical reality here – independent of human culture – or any culture anywhere. But to White, culture was a “super-human” entity that could only be explained in terms of itself. The role of the individual is reduced to null: discoveries are not made by individuals – it is something that happens to them. In this way, culture becomes almost meta-physical.

In more recent times, Reuben Hersh has argued for mathematics being a cultural phenomena. The argument is popularized in his book What is Mathematics, really? Another reference is his article Some Proposals for Reviving the Philosophy of Mathematics in New Directions in the philosophy of Mathematics (ed. T. Tymoczko). Hersh does not take the argument as far as White.

The arguments for mathematical concepts residing in the common human culture are very convincing, but does it explain everything? Still we have the Wigner problem of the “The unreasonable effectiveness of mathematics in the natural sciences” (Comm. Pure Appl. Math, 13:1, 1960). I cannot escape the feeling that there must be something in physical reality that serves as a basis for mathematics. A point of view very different from the “cultural basis view” is put forward by Roger Penrose in his The Road to Reality. Penrose seems to be a Mathematical Platonist and his discussion on the interactions between three worlds: the Platonic mathematical world, the Physical world and the Mental world in the first chapter is very intriguing. Where is culture in that picture?

Come to think of it, these questions also reminds me of Gottlob Frege’s struggle (The Foundations of Arithmetic) with defining numbers independently of individual mental states in order that mathematics not become a part of psychology.

Much more here to think about, obviously, and to write about in the report!

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