Bjørn's maths blog

New blog

2009-05-22T20:57:00.000+02:00

This blog has been used for mathematics-related postings. However, I have come to see the scope of this blog as too restrictive, as I also want to post on other subjects concerning teacher education. Instead of starting more blogs on each subject, I will instead start one blog covering all of my work-related activities. This will be called Teacher educator Bjørn. Please come over to my new blog!

Book: Legitimate peripheral participation

2009-04-20T17:41:00.003+02:00

Jean Lave and Etienne Wenger: Situated learning - Legitimate peripheral participation. 1991.

In this interesting book, Lave and Wenger builds on five studies of apprenticeship situations (midwives, tailors, quartermasters, butchers and nondrinking alcoholics, to get to a theory of learning. The concept they create and see as key, is (as the title suggests) "legitimate peripheral participation". This has to do with learning by participating in a social practice. The "peripheral" part has to do with the way a person (for instance an apprentice) can have a small, but still significant, role.

There are at least two ways of reading this book. It could be seen as an attempt to discuss ALL learning. Some would, for instance, argue that what you can learn in a classroom is just how to be a student, and that it can not be transferred into the "real" world. By that kind of thinking, it would of course be hard to admit that I could possibly have learnt something from reading a book. More reasonably, I would like to see the book as an attempt to point out some features of ONE KIND OF learning.

Anyway, it is interesting to view the teacher education I am a part of from the point of view offered by this book. For instance, teacher students probably spend as much effort trying to learn the language neccessary to pass our exams (such as "zone of proximal development" or "cognitive conflict") as they do to learn the actual use of the concepts in everyday teaching.

A quick reading of this book of course did not make me fully grasp their ideas, but at least it is a point of departure for later readings.

Article: Shulman 1987

2009-03-22T09:38:00.003+01:00

Lee S. Shulman: Knowledge and Teaching: Foundations of the New Reform, Harvard Educational Review Vol. 57, No. 1, February 1987

As in his 1985 article, Shulman here is on a crusade to change the discussion of teacher knowledge, which he thought was too little concerned with "comprehension and reasoning, transformation and reflection". Instead of only looking at "the management of students in classrooms" he is concerned with "the management of ideas within classroom discourse".

A powerful example of the importance of content knowledge is Colleen, a teacher student with a master's degree in English. The contrast between her confident and highly interactive teaching of literature and her lack of confidence and her "didactic" style when teaching grammar, is highly illustrative. (The example is actually from Grossman (1985).)

Shulman tries to show what are the sources of the knowledge base for teaching. One of the headings is "Wisdom of practice", and he mentions one of the main problems with teaching as a profession, in my view - I will quote the paragraph in full:

One of the frustrations of teaching as an occupation and profession is its extensive individual and collective amnesia, the consistency with which the best creations of its practitioners are lost to both contemporary and future peers. Unlike fields such as architecture (which preserves its creations in both plans and edifices), law (which builds a case literature of opinions and interpretations), medicine (with its records and case studies), and even unlike chess, bridge or ballet (with their traditions of preserving both memorable games and choreographed performances through inventive forms of notation and recording), teaching is conducted without an audience of peers. It is devoid of a history of practice.

Article: Ball, Thames, Phelps

2009-03-17T09:53:00.003+01:00

Ball, Thames, Phelps: Content Knowledge for Teaching - What Makes It Special?

What I particularly like about this article, is how convincingly it shows that there exists some "special" mathematical knowledge that teachers have (and need) that other professions do not. In discussions about teacher education, people often say that mathematics teachers need to know "the mathematics" and they need to now pedagogy, and often "the mathematics" is thought of as the same mathematics as their pupils are to learn, only more.

The authors gives the example that many professions need to be able to subtract two numbers, and some professions need to be able to check a subtraction to see if the answer is correct. However, only teachers need to be able to see or investigate whether the method used is valid in general or to pinpoint the error in the algorithm.

This is *not* pedagogical content knowledge, because this is not, in itself, concerned with students or learning. It is pure mathematics, but mathematics of a different kind than what is normally taught in universities. In addition to this (which the authors call "specialized content knowledge") comes pedagogical content knowledge, which is concerned with students and learning - for instance knowledge of which typical misconceptions students may have, knowledge of different ways of representing a certain mathematical idea, knowledge of how to choose examples that provide exactly the right progression etc.

I think the concepts "common content knowledge", "specialized content knowledge" and "pedagogical content knowledge" will be useful in further discussions on what we should teach in our teacher education courses.

What happens when you add both the numerator and denominator

2009-03-11T12:46:00.002+01:00

In an article (Rowland et al) is mentioned a kid who conjectures that the fraction you get by adding the numerators and denominators will be between the two original fractions. Example: you get a fraction between 1/2 and 3/4 by calculating (1+3)/(2+4) = 4/6 = 2/3.

This works, of course. One explanation I like is the following: Let's say that 2 persons together own 1 dollar and 4 other persons together own 3 dollars. If they decide to pool their resources, 6 persons will own 4 dollars. It's common sense that not all of them are now richer and not all of them are now poorer.

Another explanation concerns vectors. (Generally it's a terribly bad idea to illustrate fractions with vectors, but in this case it actually works.) If you add the vectors [2,1] and [4,3], you get the vector [2+4,1+3] = [6,4]. Obviously, the slope of the vector [6,4] (which is 2/3) is somewhere between the slopes of the two addend vectors (which are 1/2 and 3/4).

For some reason, some of my colleagues and I started to look at problems associated with this, such as: When is (a+c)/(b+d) exactly halfway between a/b and c/d? (Answer: only when b=d).

In a way, this whole thing illustrates why teachers need skills that other professions don't. While many professions need mathematics to get the right answer, mathematics teachers need to be able to see whether the solution a student proposes is actually correct. That may be with methods within the students' reach or with other methods.

In addition, the teacher of course needs to decide whether he/she should follow up on this - that is, whether the student will learn anything from going deeper into the problem. That is another (but related) kind of skill.

All of this relates to an article by Deborah Ball that I will blog about later...

Article: Rowland, Huckstep and Thwaites

2009-02-20T18:26:00.002+01:00

Tim Rowland, Peter Huckstep and Anne Thwaites: Elementary teachers' mathematics subject knowledge: the knowledge quartet and the case of Naomi

This article is a precursor to the article Turner/Rowland treated a previous post.

I've often discussed with Norwegian mathematics colleagues in teacher education the problem of how students are supervised in their school-based placements. Too often, the discussions with students concern only "administrative" and pedagogical issues, and too little attention is given to the mathematical parts of the lesson. This article is an attempt to help this situation.

The point of the research was to "develop an empirically based conceptual framework for the discussion of mathematics content knowledge, between teacher educators, trainees and teacher-mentors, in the context of school-based placements". The "Knowledge Quartet" is the result:
- Foundations (Propositional knowledge and beliefs)
- Transformations (How the content knowledge is transformed "into forms that are pedagogically powerful" (Shulman))
- Connection (between different parts of a lesson, between lessons or between different parts of the curriculum)
- Contingency (How "to respond to children's ideas")

The authors cite Ma citing Duckworth that intellectual "depth" and "breadth" "is a matter of making connections". This quite deep insight suddenly got very visual for me: isolated points of learning can hardly be have "depth" or "breadth". (My visual image/metaphor soon breaks down, however, so I think I'll stop thinking about it...)

The second part of the lesson, discussing Naomi's lesson, showed how her lesson could be analysed from the four "points of view" given above.

The article is highly recommended.

Just an old sweet song keeps Georgia on my mind

2009-02-19T10:47:00.001+01:00

Did you know that the official song of the state of Georgia, US is "Georgia on my mind"?

Actually, that song is the only thing I associate with Georgia. (Except, when realizing that the state capital is Atlanta, I suddenly also associate it with Coke and CNN, as well as olympics and a bomb.)

Why do I suddenly care about Georgia? Because I'm going there in June - for the InSITE 2009 conference, presenting a paper with a colleague.

Good ideas for things to see and do in Georgia are welcome!

Article: Turner/Rowland

2009-02-17T08:43:00.002+01:00

Fay Turner and Tim Rowland: The knowledge quartet: a means of developing and deepening mathematical knowledge in teaching?

This is a exemplary article. It gives an overview of a theoretical framework and at the same time shows how this framework can be used in practice to provide "discussion points" in teacher education. This is so refreshing, far too many writers leave it to the reader to find practical consequences of the theories.

The Knowledge Quartet is a way of looking at the knowledge that a teacher needs to teach mathematics:
- foundation (what the teacher learned "in the academy")
- transformation ("knowledge-in-action", planning the teaching etc)
- connection (for instance the sequencing of material)
- contingency (the ability to change plans on the go)

I'm about to read another article about The Knowledge Quartet now, which will probably give me more insight into its use.

Article: Lee S. Shulman

2009-02-15T15:49:00.003+01:00

For some reason, I haven't read anything by Lee Shulman before reading his short 1985 article "Those Who Understand: Knowledge Growth in Teaching" today. This does not mean that his thoughts haven't influenced me, however - last year I even wrote a short paper (for a course I'm taking) partly based on some of the ideas.

The main point in this article is perhaps distinguishing "among three categories of content knowledge":
(a) subject matter content knowledge
(b) pedagogical content knowledge
(c) curricular knowledge

There is a tension even in the Norwegian mathematics education field between people focussing on the mathematics and other focussing on the pedagogy. Shulman's categories introduced more nuances into that discussion.

An example of "pedagogical content knowledge" that I teach in my courses in mathematics in teacher education is students' typical misconceptions and how to overcome them. Another example is which bits of history of mathematics are useful to motivate students in particular parts of the curriculum. Examples of particular methods of teaching should also be included here.

"Curricular knowledge" includes both knowledge of different materials available (including digitally) and when they can usefully be used. Knowledge of what goes on in other subjects in the same grade or in mathematics in other grades are also included.

These three categories will probably prove to be useful for forming the discussions we will have now that the curriculum (in teacher education) will be changed again.

Shulman's article is written as an answer to George Bernard Shaw's "He who can, does. He who cannot, teaches." Shulman's version is: "Those who can, do. Those who understand, teach."

HPM 2008 Day 4

2008-08-08T01:39:00.000+02:00

Day 4 of the conference was actually just half a day. Glen Van Brummelen talked on “Crossing Cultures, Seas and the Cosmos – In Search of the Origins of Trigonometry”. Fascinating though the history of trigonometry may be, the most interesting part of this talk for me, was the discussion he had on how to look at history. He warned against reading history as “the royal road to us”, for instance by reading original sources, but only noticing the parts that are part of the history leading to our current knowledge. This seems to put quite high demands on the teachers – for me, it will not be a question of avoiding anachronisms, but rather how much anachronisms to avoid. Too high demands on teachers in this regard will effectively mean removing all history from schools.

Kristin Bjarnadóttir held a fascinating talk on “A Puzzle Rhyme from 1782”. It is interesting to see how the “same” problem crosses borders and get immersed in new cultures.

Finally, Cecilia Costa talked about “The Alto Duoro ‘wine coopers’ mathematics”. The point was to look at what kind of mathematics is involved in making the barrels used for wine production, and this was done by interviewing the craftsmen themselves. The talk ended up in a discussion on why the barrels are not sylindrical, in which David Pengelley argued that it could make it easier to avoid leaks, while Peter Ransom argued that non-sylindrical barrels could more easily be rolled (not only in straight lines, but also in curves). I’m sure both of them have a point.

Thereby ended the HPM 2008 conference. The next HPM conference will be held in 2012 somewhere near Korea (where ICME12 will be held). But before that, there is the CERME in Lyon in January-February 2009 and the ESU in the Netherlands or in Greece in July 2010. Moreover, there will probably be other opportunities to meet the HPM family as well. But at the moment of writing, it feels good that the conference is over – my brain can’t take more input at the moment, and it will be nice to have a few weeks of vacation before trying to work on some of the issues here.

HPM 2008 Day 3

2008-08-08T01:08:00.001+02:00

Karen Parshall held the opening lecture of the third day of the HPM 2008. Her talk was on “The Evolution of a Community of Mathematical Researchers in North America 1636-1950”. For me, particularly interesting was the fact that North America for quite some time remained “loyal” to the English way of doing thing, which made them, for a time, slow to pick up on the progress in continental Europe. (My Master thesis concerned the 1600s and early 1700s, so it’s particularly interesting for me.)

Then recent Felix Klein award winner Ubiratan d’Ambrosio held a talk with the long title “The transmission and acquisition of Mathematics in Colonial and Early Independent Countries in the Americas, and a brief Reference to the 20th century.” Who else than Ubi has the knowledge to give a one-hour talk on the mathematics of a whole continent over a period of 500 years? It is always a pleasure to hear Ubi talk, and I hope I will get many more chances in conferences in years to come.

Jodelle Magner had a talk on “Napier’s Rods In Today’s Classrooms” which showed one way of working with different multiplication algorithms. Then Peter Ransom had a talk on sundials – decided on at short notice to fill the gap left by people who didn’t come to the conference. It is always nice to hear Peter talk – even though this was (of course) closely resembling what he did at ICME the week before.

David Pengelley and Janet Heine Barnett’s workshop had the title “Learning Discrete Mathematics and Computer Science via Primary Historical Sources: Student projects for the classroom.” It consisted of two sets of worksheets, which the participants worked on: one on the bridges of Koenigsberg, another on Pascal’s triangle (according to Pascal). This was quite interesting.

Ewa Lakoma talked on “On the role of the history of mathematics in mathematics education for the knowledge-based society”. She only mentioned in passing a survey that has been done in Poland, showing how history of mathematics is perceived in different groups, and I would like to know more about this aspect of her talk.

HPM 2008 Day 2

2008-08-07T03:28:00.001+02:00

The second day of HPM 2008 started with Evelyne Barbin (the new chair of HPM) and a talk titled “Dialogism in mathematical writing: historical, philosophical and pedagogical issues”. She gave a quick introduction to some aspects of Bakhtine’s work, and in particular the three “bullet points” active responsive attitude of the listener, addressivity and speech genres. I see that Bakhtine can indeed be interesting in the context of HPM, and need to make sure to learn more about him.

I skipped the next plenary lecture (as I had to check my email) but then went to Edel M. Reilly’s talk “Mathematics Apart: Examining the History of Subject Isolation and Its Implications for Mathematics Education”. According to her abstract (and title) this should be a historical talk, but her actual talk focussed on the present and future. I will have to read the paper to see if the historical element is there… Anyway, it was interesting to hear to which degree the US school system has chosen to compartmentalize mathematics (in most of the rest of the world, students take “mathematics courses”, not “algebra” and “geometry” without connections between them…)

Mala Saraswathy Nataraj talked about “Using history of mathematics to develop student understanding of number system structure. She described ways in working with pupils to get them aquainted with our numeral system through work with “toothpicks”. It also had a very interesting idea: that the work on writing large numbers (using powers of 10) is a useful prerequisite for later understanding the notation in algebra. It seems very reasonable, I just hadn’t thought about it like that before.

Chorlay Renaud and Anne Michel-Pajus presented the paper “The multiplicity of viewpoints in elementary function theory: historical and didactical perspectives.” This was quite fascinating. One major point was that functions could be presented in “two different worlds”: the world of quantity and the world of sets. Each of these worlds have semiotic and conceptual coherence – however, mixing them may confuse the students. In France, it is obviously the “world of sets” which is the “correct” one – functions are seen as a relation between sets. In Norway, I guess we cling to the “world of quantity” almost until university, although some definitions or ways of formulating things from “the world of sets” may creep in here and there. It would have been interesting to look at this…

Staffan Rodhe talked about “Emanuel Swedenborg’s work on differential calculus” with a few more details than last time I heard his talk (four years ago).

In the lunch break, there was a meeting of the “advisory board” of HPM – there’s never any rest… 

After lunch, George W. Heine talked about “Euler’s Contributions to Mathematical Cartography”. The first part of the talk was particularly fascinating, in which Heine discussed Nicolas Delisle’s attempts (?) to make maps of Sibiria. Although given huge resources, Delisle never succeeded, and he stubbornly denied seeking help from Euler – which may be a sign that he was actually a spy in Moscow, just trying to keep the resources flowing in while at the same time shipping maps out of the country (which he apparently did).

Then today’s highlight: Peter Ransom’s act as a seafarer in the workshop “Yo Ho Ho-ratio: some mathematics of Trafalgar (How Lord Nelson inspired curriculum development in mathematics)”. Peter’s workshops are always good, so obviously we had a good time – while making things, doing calculations, learning about history… And not only had Peter put all the resources onto a CD which we got for free – he had also made pencils with the name of the workshop on it… That’s what I call dedication…

In the evening there was a very nice private social gathering, and then – on Wednesday – we had a “free day”, on which I joined many of the others to Teotihuacán – but I should write about that in another blog…

HPM 2008 Day 1

2008-08-05T18:01:00.001+02:00

The first day of the HPM 2008 was very good. It takes place in a very nice building in the very centre of Mexico City. The first talk was David Pengelley's "The use of original sources in the teaching of mathematics pupils at all levels". He discussed his dream: that pupils at all levels should learn all of their mathematics from primary sources. I surely do not share that dream, but I certainly think that primary sources has their place in the mathematics classrooms, just as pupils in Norway will never be allowed to go through 10 years of education without reading Ibsen (instead of just short summaries of his works). David also came up with the idea of having the HPM maintain a bibliography of the HPM field. That is certainly both an incredibly big task, but it would also be very useful. Just imagine an online bibliography where you could search for particular mathematicians, levels (in school), type of paper (empirical research, ideas from the classroom...), language. It could certainly not be the work of one person, but still some person must start and dedicate a significant amount of time to it...

The next plenary lecture was Rosa María Fanfár, who talked on "Matemática educativa. La convergencia de series infinitas". I must admit that this area of mathematics is one I do not usually work on, so that it was a bit of an overload on my brain to try to understand all of the mathematical examples based on the simultanous translation into English - it didn't help that I hadn't made sure to sit close enough to the screens to actually see them. Luckily, the paper is in the proceedings.

Then it was time for parallell sessions, and I was chairing one of them. The talks there was good: Maria Christina Araújo de Oliveira and Ruy Pietropaoplo talked about a magazine for uneducated school teachers in Brazil in the 1950s and 1960s. Ildikó Pelczer gave "A historical overview of analysis exams in Rumania", which showed patterns that I suspect would also be found in Norwegian teacher education exams of today. Hans-Stefan Siller discussed the emergence of Informatics as a subject in his talk "Informatics - A subject developing out of mathematics", while Flávia Soares discussed examinations for teachers in the 1800s in Brazil.

In the second parallell session, Funda Gonulates discussed a project which she has done, where she tried to measure whether working on history of mathematics did change the students' attitudes to history of mathematics and their knowledge of how to include it in teaching. However, she did not find significant increases. This does point to one of the problems of doing empirical research in the HPM field - the number of students involved needs to be high, and even if there were significant results, critical voices would certainly demand a control group. And given a control group, some would also question if the teaching given to the control group was "good enough". While this kind of research is certainly useful, for the individual teacher who wants to try to include history of mathematics, I think it is even more useful to get examples which can motivate him. "The proof is in the pudding": if the teacher thinks that the students are benefitting from the history, he will keep including it. On the other hand, no teacher will include anything based only on research papers telling him that something works for some faraway students.

Bjørn Smestad (who - again - happens to be me) talked about an interview study he has done trying to find out things about teachers' conceptions on history of mathematics. This talk went much better than the talk in Monterrey - maybe because it was much better prepared. There were interesting questions afterwards as well.

Robert Peard gave interesting ideas of courses he has had for teacher students in which he did not try to teach them the "basics" which they had failed to learn through 12 years of schooling, but instead focused on a few examples in which mathematics is important for understanding the world around us - for instance the calendar.

Beverly M. Reed talked about "The effects of studying the history of the concept of function on student understanding of the concept". Her theoretical basis was APOS theory, based on Piaget's theories. Her approach was qualitative, and she had promising conclusions, although I guess critics would still argue that her results could also have been obtained without history of mathematics.

That concluded the formal part of day 1. A very good day indeed. And now I'm free to do "whatever I want" for the rest of my stay in Mexico, as my talks are finished...

ICME Day 5 Part 2

2008-08-05T17:56:00.000+02:00

Day 5 became the last day of the conference for me, as I preferred to stay in my hotel to prepare Monday's talk at HPM instead of going to the conference on Sunday.

Thus, the last I got from ICME was the presentation of three new ICMI Studies, at least two of which I will certainly try to get time to read as they become available. ICMI Study 15 concerns "The Professional Education and Development of Teachers of Mathematics", while ICMI Study 17 is titled "Technology Revisited". The presentations actually concerned the making of the studies more than the actual ideas inside them.

So, that's the end of ICME11 - I hope I'll get to ICME12 in Seoul in four years' time. But now: HPM 2008...

ICME Day 5 (Part 1)

2008-08-01T22:27:00.001+02:00

On Day 5 of ICME11, I've so far only heard one talk. That was Jan van Maanen's excellent talk "Professional development of teacher educators, the ELWIeR initiative". Why excellent? Because it was funny, based on an important work and gave me useful information. It was not surprising that it was funny - Jan's talks are almost always entertaining. The project was a large one, but Jan focussed on two aspects - the development of new "handbooks" and on some of the research in the initiative.

Apparently, the textbook situation in teacher education has been quite different from that in Norway. In Norway, there are new textbooks "all the time", while in the Netherlands, Van Dormolen's 1973 textbook "Didactiek van de wiskunde" is still in use. The creation of the new handbooks has involved most of the teacher educators in the Netherlands, and has led to considerable professional development within the community. This of course makes me think of whether there may be a similar project that Norwegian teacher educators could collaborate on, to get similar positive results. At the same time, of course, I'm eager to see the results of the Dutch efforts - that is, I want to read the finished handbooks...

He also discussed some research findings, but I will not try to give an impression of them here. However, it is interesting to see that this research also focussed on teacher students' ability to understand the pupils' ways of thinking (just as at least two of yesterday's speakers). I'm happy to say that we have had at least some work on that with the students at my institution, although we could certainly do more.

It the final session of the TSG23, Bjørn Smestad (who happens to be me) held a talk on three years of student projects on history of mathematics. I was not too happy with the outcome of the talk - it went just as badly as I feared in advance. My idea was to look at some student projects that I've done with my students. In these student projects, the students were given little input from me (even though they could certainly have asked for more), and I therefore thought that the products of these projects could give an idea of the sorts of problems that also ordinary teachers in school might have faced if they had taken the curriculum requirements to include history of mathematics in their teaching, seriously. The discussion afterwards focussed more on how such students projects could have been done, however, rather than how teachers in schools might better be helped. However, as I'm having a talk on Monday with a similar question in the end, I may hope that that will work better...

Afterwards, there was more discussion on the way to go on. For me, it is clearer than before that we have to treat two issues seperately: on the one hand how we - as educators who are very interested in the history of mathematics - may include history of mathematics in our teaching, and on the other hand how we can help other teachers include history of mathematics in their teaching, preferably on a large scale (that is, not only five or six teachers supported by an expert, but thousands of teachers...) Both issues are very interesting, but it is not at all obvious that what a special person such as Jan van Maanen can do in his classroom, can be replicated by less knowledgeable teachers. On the other hand, neither is it clear that history of mathematics has a place in the teaching of uninterested teachers.

After this discussion, I had lunch and went to the computer room to write this and check my email...

ICME Day 4

2008-08-01T21:37:00.001+02:00

Day 4 of the scientific activities of ICME (which was the 5th day of the conference, as the fourth day was excursion day), started (for me) with a talk by Caroly Kieran; "Conceptualizing the learning of algebraic technique: Role of tasks and technology". She had convincing examples of how CAS (computer algebra systems) can be used to improve both the students' technical aptitude and conceptual understanding. This is not self-evident, indeed, I have myself been suspicious about CAS, thinking that they will only help students avoid doing the computations themselves. But of course, just as with calculators, CAS can also be used in an investigative manner - temporarily removing the students' need to do the algebraic manipulations themselves make them better able to make conjectures and check their conjectures on new examples. One of her examples was to let the students do (x-1)(x+1)= and (x-1)(x*x+x+1)= and then conjecture what (x-1)(x*x*x+x*x+x+1) and so on is (sorry for the terrible ASCII notation here). The use of CAS made the students confident that they had the right answer, and therefore the confidence needed to make hypotheses. I will certainly consider using this at some point in the future when teaching algebra to my students. (Kieran stressed, however, the importance of task design - she did not hide the fact that CAS can also be used terribly...)

The next part of my program was the TSG23. Uffe Thomas Jankvist held a talk titled "On empirical research in the field of using history in maths education". He argues that the HPM group has had too little "empirical research". Moreover, he has had a very interesting example of such research, which seems very good. His paper had a combination of being practical and theoretical at the same time, which was very good. My only "protest" was that he used the phrase "armchair research" as the opposite of "empirical research". I think this gives the wrong impression. In the history of HPM, it is true that there has not been much empirical research, but the papers have been divided into (at least) TWO other cathegories - the papers that have indeed been empirical (but not research) and the once that may have been research, but not empirical. Thave been lots of papers discussing individual experiences from the classrooms, as well as lots of papers discussing theoretical issues without the important empirical components. A discussion of the virtues of empirical research should at least take both of these other types of work into account.

There was also a paper by Lenni Haapasalo from Finland.

I had actually planned to skip the next plenary session, but ended up going anyway. I'm glad I did. The two professors Fujii and Even (Fujii from Japan and Even from Israel) had a talk each on the topic "Knowledge for teaching mathematics". Besides being genuinely funny, Fujii's talk was also a very interesting glimpse into the world of Japanese Lesson Study. He gave many interesting examples of mathematical discussions this led to. The main realization for me was that even though we do ask our teacher students (in Norway) to write detailed plan of every lesson, we never ask them to write in these plans what they expect their pupil's (mathematical) reaction to be. They write a lot on the pupils' actions (what they are supposed to DO), but not on which strategies they will probably use or the errors they will probably do. This is an important weakness. In Japan, the anticipation of such difficulties is an important part of lesson study, including (of course) the planning of how to approach (or even make use of) them, should they occur.

Prof. Even had a similar point of view, although from another view point. Her work with teachers has (among other things) included teachers reading research papers and replicating the work in the papers (for instance giving their pupils the same tasks as discussed in the research papers). The teachers are often astounded that the misconceptions they read about in the research papers, are often present in their own classrooms as well. This made the teachers more capable of understanding students' thinking, but Even also stressed the importance on working on how to translate that understanding into teacher practice.

The last post of the programme was the ASG for the HPM (the 2nd part). First, Ubiratan d'Ambrosio held a very interesting talk on Julio Gonzalez Cabillon. (I'm sorry I can't get the name right on this keyboard.) Cabillon was the moderator - for years - of the Historia Mathematica email list, which was an amazing resource for researchers worldwide. As a moderator, Cabillon both provided a lot of the answers, but also intervened in a diplomatic manner when discussions got heated. Ubi has researched who Cabillon is, and has at least found him and talked to him. Today, Cabillon is, luckily, still very much alive, but has just decided to persue other interests than this list, which took so much effort for such a long time. It is a pity, but understandable, that noone has managed to create a list to replace the HM list.

The rest of the ASG was spent on Costas Tzanakis giving a report on the work of the HPM for the previous four years and on discussions on the future. These discussions will continue in Mexico City next week.

Thus ended the evening at ICME - and I went out with some colleagues for dinner. Another nice day at ICME11.

ICME Day 3

2008-07-31T09:12:00.000+02:00

The third day of the ICME started with José Antonio de la Peña’s talk on “Current trends in mathematics”. This seems as an almost impossible topic to cover in one hour, but la Peña had chosen to focus on just a few highlights, and to take a “light” approach. I think this was wise. He also pointed to the Mathematics in the movies site, which features video clips from movies. I’ll check it out…

Next, I heard an excellent talk by Jeremy Kilpatrick titled “A Higher Standpoint”. The point of departure was Felix Klein’s “Elementary Mathematics from a Higher Standpoint”, published in the first decade of the 20th century. Klein’s object was to “bring to the attention of secondary school teachers the significance for their professional work of their academic studies”. He pointed to a “double discontinuity”, whereas the student who goes on from school to higher education, feels that the mathematics is completely different, and at the same time the teacher who has finished his higher education and goes back to school to teach, does not see the connection either. Klein wanted to remedy this by updating the school curriculum and by revising the university instruction to take into account the needs of the school teacher. The talk went on to discuss the books in more detail, and also discussed Pólya and Freudenthal in the connection with Klein’s ideas.

Kilpatrick’s talk made me want to read Klein’s textbooks, which was certainly one of his objects. It also gave lots of food for thought.

At about this time in the conference, there was distributed a leaflet giving information on the 12th ICME, taking place in Seoul July 8th-15th, 2012. I’m already looking forward to it! The website is: http://icme12.org.

The next thing I attended was the TSG23. Louis Charbonneau talked about “Astronomical and mathematical instruments as pedagogical tools”, putting emphasis on the emotional aspects of being able to touch instruments that have been used to measure heaven and Earth… It was a very interesting and enthusiastic talk. Snezana Lawrence gave the answer to my concluding question in Saturday’s talk (which means I have to update my talk a little…) My question is what we can do to make teachers able to include history of mathematics in their teaching. Lawrence has used history of mathematics as the focus of a teacher development program that seemed very good. I really need to get to know more about this project (and I guess I can find out more on her website, mathisgoodforyou.com).

Liliana Milericich read the paper “The teaching and learning of integral calculus from a historical perspective”, which pointed to some pitfalls in the teaching of integral calculus. As this is not part of what I teach, I didn’t note down particular things that I need to remember from that talk.

Later in the day, I attended the SEG (Sharing Experiences Group) on interactive whiteboards (IWBs). This was incredibly interesting to me, as I’m just starting out in this area, and having written a paper for a conference just before leaving for Mexico. There were lots of interesting thoughts there, and seeing the webpage of these people was also very interesting. Moreover, I was pointed to an interesting conference in Cambridge in June 2009, which I will consider going to.

That ended day 3 of the conference. As day 4 was an excursion day, I will go on with this blog with day 5 later…

ICME Day 2 (part 2)

2008-07-31T08:48:00.000+02:00

The first ASG meeting of the HPM group consisted of two interesting talks. Gert Schubring’s talk was titled “Researching into the History of Mathematics Education – an HPM perspective”. He discussed how teachers were educated in different states from the middle of the 18th century. (Before that, states took little interest in this.) The talk was packed with information, and I can obviously not repeat it here. However, it was interesting to hear how there was a huge effort after the French revolution to teach teacher educators, while in France in 1763, it was argued that it was needless to educate teachers, as good textbooks would make sure that the teachers could educate themselves.

Fulvia Furinghetti talked on “The emergence of women in the international arena of mathematics education – Just so stories”. Although it was a talk about the women who has played a role, the main picture is of course that there have been very few women who has been prominent in the field until recently. Lately, that has luckily changed.

ICME Day 2 (part 1)

2008-07-28T22:00:00.004+02:00

Today's plenary speaker was Celia Hoyles, and the topic was "Technology and mathematics education: Transforming the mathematical practices of learners and teachers through digital technology". To me, it was a great talk, as it functioned as an introduction into a field in which I will work more in the coming years. She described six areas in which ICT is transforming mathematics education:

dynamic and visual tools to explore in shared space
tools to outsource processing power
new representational infrastructures
connections between school and learner's culture
connectivity
intelligent support for the teacher.

She had examples in all of these areas, and her talk certainly made me eager to get hold of the forthcoming ICMI Study on these topics.

She also referred to the National Centre for Excellence in the Teaching of Mathematics website, and in particular the Mathemapedia - a Wikipedia for mathematics, apparently. I haven't had the time to look at it yet, but should certainly check how this site compares to my plans for a teacher education wiki locally in Norway. Maybe there will be common areas of interest.

The next lecturer was Anna Sfard. Her talk was titled "Learning mathematics as developing a discourse". Her talk was - unsurprisingly - well attended, so the room was far too small, as was her time. Anyway, I found it most interesting. She discussed what she terms as a transition from the metaphor of "aquisition" to a metaphor of "participation" as far as learning is concerned - moving from a Piagetian world to a Vygotskyan. This has far-reaching consequences. She looks at mathematics as a discourse, and a number as a "discursive construct". This has far-reaching implications when it comes to interpreting children's approach to mathematical problems.

The third item on my agenda today was the TSG23 - the topic study group on history of mathematics in education (not to be confused with the topic study group on the history of mathematics education, of course). Due to some technical difficulties, one of the talks were postponed to tomorrow's session, and only Costas Tzanakis had his talk. This was a very interesting talk on how history of mathematics may throw light on the problems pupils face when trying to learn the variance concept. One particularly striking point (to me) was how educators have tried to make the topic "soft" by including examples from social sciences instead of from physics and geometry, but have ended up making it harder. This is because the terms mean - and even variance - have clear, concrete interpretations in certain examples from physics and geometry, and because they offer the possibility of experiments.

Now, I'm having lunch break, but will go to the ASG meeting for the HPM group later this evening. But it has already been a very interesting day. (And I've also had the time to look at my own talk, which will be on Saturday. I think it will be ok.)

(As mentioned before, these posts are delayed by 20 days.)

ICME Day 1

2008-07-28T04:45:00.002+02:00

The ICME 11 has started today - the main conference for mathematics educators worldwide. As always, the first day is mainly spent on formalities and on setting the stage for things to come.

The opening consisted of a whole series of speeches - inevitable, I guess - and the distribution of the Felix Klein Awards and the Hans Freudenthal Awards for 2005 and 2007. This was a nice thing to see - both that everybody took the time to listen to praise for the lifetime achievements of some of our most wonderful colleagues, and that the prize winners showed that the prizes were actually quite important to them as a recognition of a lot of heavy work.

The winners were:
Felix Klein Award 2005: Ubitaran d'Ambrosio
Hans Freudenthal Award 2005: Paul Cobb
Felix Klein Award 2007: Jeremy Kilpatrick
Hans Freudenthal 2007: Anna Sfard

Although all four are well-known names, personally, I know the work of Ubi best, and found it particularly touching to see him thanking his wife for all her patience for 50 years...

The first two plenary lectures were on what has happened for the previous ten years and what should happen in the future. The venue for the plenary parts of the program is not very good - there is too much background noise, and it's difficult to see the screens from many positions. The solutions in the previous two ICMEs (which involved using actual buildings instead of a "tent") worked better. However, I thought that the first lecture (by Artigue and Kilpatrick) was quite inspiring and interesting. They tried to summarized the main trends in mathematics education over the previous ten years. Some keywords: more systemic approaches, a look at constraints, semiotic perspectives.

The panel on what we need to know, which was supposed to summarize the answers of educators around the globe, however, did not manage to inspire me. Well, Paul Kobb's interpretations of the (mainly US) answers from North America, were meaningful. The other three also discussed important questions, but in a way that made it look less like a panel than separate lectures. The format of plenary activities for thousands of listeners is a difficult one, indeed.

The highlight of the day was undoubtedly the workshop by one of my favorite colleagues, Peter Ransom. He and a colleague had a workshop on using sundials in the classroom. Peter is so enthusiastic that it's hard not to start loving sundials, and the activities were fun as well. I should try to find a way to include sundials in my teaching in Oslo...

That was the first day already. In between were coffee, lunches etcetera, of course. And lots of meetings with colleagues who I haven't met for 1, 4 or even 8 years. A nice start to the conference.

(As mentioned before, these posts are written during the conference, but posted afterwards.)

Mexico 2008 – and lovely randomness

2008-07-24T16:55:00.000+02:00

This summer, I will be staying in Mexico for 20 days. I will be attending both the ICME (International conference on mathematics education) in Monterrey, and the HPM (the meeting of The International Study Group on the Relations between History and Pedagogy of Mathematics) in Mexico City.

The trip here went well. However, I met two different approaches to randomness which were interesting. In Chicago, I was pulled aside for a routine random check of my luggage. Of course, I knew that nothing was wrong, so my only thought was “Why did they pick me?” I was stressed, since I didn’t have much time, and didn’t know exactly how much time all the procedures in transit in Chicago would take.

In Monterrey, much the same happened to my colleague – she was picked randomly to be checked. How? It was simple: every passenger picking the green line (Nothing to declare) had to press a big button. Then someone (probably a computer) decided if you should get a green or a red light – if you got a red light, you were chosen for random check.

The difference is not so large, but for the passenger, it does feel better, and the people working there will be able to defend themselves against the charges of racial profiling and all such charges that are usually impossible to defend yourself against. (However, they should still be able to check people who they found particularly suspicious…)

Then we were to take a taxi to town. We went to a counter and was presented by a screen. The woman behind the counter told us that we could pick any taxi company we wanted – or press the big “random” button. We chose to press “random”, paid our taxi ride and went out to find the company. An altogether better solution than to have every single taxi company having their own counter in the arrivals area – and the alternative would have been that we had walked out of the airport and chosen a taxi at random outside.

So already on the first day: two nice ways to use randomness to solve problems.

(Written 4th of June 2008.)

(The blog posts from Mexico will be posted with a delay of 20 days. This is simply because it is not considered advisable to advertise to the world that you are staying away from your apartment for weeks…)

Leonardo and Pacioli

2008-03-20T09:27:00.002+01:00

Who would you have chosen to illustrate your mathematical text if you could choose anyone - including people long dead? Maybe not Picasso, who might confuse the readers. What about Leonardo da Vinci?

It has long been known that Leonardo da Vinci collaborated with (and was a very good friend of) the mathematician Luca Pacioli. This week, newspapers and magazines (for instance National Geographic) have written about Pacioli's "De Ludo Schaccorum" ("Of the Game of Chess"), which apparently is an important treatise for chess historians and which was just rediscovered last year.

The interesting (if unsurprising) find is that the chess diagrams may have been drawn by Leonardo da Vinci. That's cool! I wonder if Leonardo was a good chess player - I would be very surprised if he wasn't...

ESU5 Day 5

2007-07-23T14:02:00.000+02:00

The first part of day 5 was a very interesting and engaging talk by Leo Corry on a bit more recent history of mathematics. He explored the connection between David Hilbert and the "new maths" in the US and "modern maths" in France. Hilbert has been portrayed as a proponent of "mathematics as a game" and axiomatics as the most important thing in mathematics. Leo Corry gave an important quote of Hilbert that suggests otherwise. I don't have the quote here, but the meaning was something like this: When mathematicians build their theories, they do not start by working for years on the foundations, before going on to build the theory on the foundations. Quite the contrary: mathematicians build beautiful spaces and corridors, and only when they start seeing signs that the foundations are not strong enough for further developments, they start to worry about them. Hilbert's opinion was never that axiomatics was the main thing in mathematics, but that axiomatics was important to put the beautiful theories already developed on a stronger foundation. Corry asked if this is maybe also what we should do in schools - develop (with our pupils) wonderful mathematics to let them see the beauty, and then only later worry about the details of the foundations. This is, of course, quite the contrary to the ideas of New Maths, which was so inspired by axiomatics.

The second thing I took part in today was my own workshop. It did turn out quite well, in my opinion - to the extent that the participants' activity is a measure of success. Most of the people there took part in the discussions, and some people told me that they appreciated a focus on primary and lower secondary pupils. What my workshop did was to look at some of the activities I do with my students, and to discuss whether they are meaningful. I added a subtitle to my talk: "If you can't do very much, can you still do something?", and I could actually also add: "If you don't KNOW very much, can you still do something?" After all, my knowledge of the history of mathematics is inferior to that of most people at this conference, but if I am to wait until I'm an expert before I start working on this with my students, they will never learn anything on history of mathematics. So my answer is clear: it is usually better to do something (and maybe make someone interested) than to wait until what you do is flawless...)

And of course, by presenting my stuff here, I've already removed some mistakes because of the comments of the participants...

Now, I've had a wonderful lunch, and will enjoy an afternoon of mathematics in the knowledge that I'm done with my presentation...

ESU5 Day 4

2007-07-23T13:50:00.000+02:00

Day 4 was a short one, mathematically speaking. After lunch, I joined a guided tour of Prague - on foot. It was an interesting walk, which lastet for four hours. The guide was quite a character, and she spoke several languages - English, French, German, Italian, Spanish, Russian as well as Czech, of course. She had strong opinions on many things, and while I don't take her word as the truth on every issue, it was quite entertaining. And of course, it was another chance to talk a little to a few of the conference participants.

The morning sessions were devoted to history of mathematics education, which I must admit is not my favorite subject. I do know that it is good for me to know a little about it, but it does not contribute very much to my main goals: to see how history of mathematics may be taught in primary and secondary school both to give students a better cultural understanding and a better mathematical understanding.

The first hour was by Gert Schubring and Helene Gispert (I'm sorry for not being able to write the French names and words correctly on these keyboards). They talked about the developments in Germany and France in the 20th century. For the next two hours they were joined by Livia Giacardi for a discussion on "The emergence of mathematics as a major teaching subject in secondary schools", with Nikos Kastanis' contribution on the Greek situation being read, as he could not be present. It is indeed interesting to see how wars and the change of borders do influence mathematics teaching in different regions. Maybe the most interesting part of the morning was to see illustration from Nazi era textbooks for childern. Anyone who thinks mathematics cannot have a political dimension, would certainly change their mind after seeing these illustrations...

ESU5 Day 3

2007-07-21T17:32:00.001+02:00

Day 3 started with a talk by Fritz Schweiger, entitled "The implicit grammar of mathematical symbolism". In a way, the talk was technical, so it will be difficult to discuss it here, in another way, it can be summed up quite effectively thus: It reminded me and made me more aware of the vast amount of implicit information there is in a mathematical text, which we (as educators) may at times not be good enough at discussing with our students. A simple example: 2x, 23 and 2 1/2 are read quite differently. Likewise, we have very clear ideas on what should be considered a good choice of letters: we will define a function f as f(x) = ax+b, but certainly not a function a as a(b) = fb+x. This is food for thought - what should we do to make the students see the importance and simplicity of such "rules"?

Thereafter, there was a panel discussion between Evelyne Barbin, Luis Radford, Fritz Schweiger and Frank Swetz. It is impossible to sum this up, but I'll try: Barbin talked about "perennial notions", how some notions include both epistemological depth, possibility of conceptual changes, links with other fields and historical and cultural interest. Her idea is that these notions are particularly suited for educational purposes.
Schweiger talked about "fundamental ideas", surely a concept of the same kind as "perennial notions". His definition is that fundamental ideas recur in the historical development of mathematics (a time dimension), recur in different areas of mathematics (horizontal dimension) and are anchored in everyday activities (human dimension).
Luis Radford, on the other hand, did not as much discuss the "How?" as the "Why?" of historical dimensions in mathematics teaching. His answer: no history means no understanding of reality.
Finally, Swetz said that we teach too much mathematics, and not enough ABOUT mathematics. His answer to this is: include the historical and cultural dimension and focus on problem solving. Historical problems give information about the society at the time, and are therefore a good tool.
The plenary discussion after this panel discussion was on "is mathematics universal?", "what is mathematics?" and "knowing=being?". No conclusion was reached.

In the afternoon, I took part in the workshop of Michael Fried and Bernard Alain. The title was "Reading and doing mathematics: Ancient and modern issues." The first part of the workshop was on Euclid, and then on some of Proclus' comments. To me, this served as a further reminder of whatever Euclid skills I may have had once. While useless for my own teaching in Norway, it is nonetheless important for me as a teacher educator to know a little about the work which more than any other have influenced mathematics teaching for the past 2000 year. The second part of the workshop was on the topic of Paideia (translated by Cicero as "Humanitas"). To me, it was interesting to see the way in which Euclid was considered a training of the mind - a point of view that fell out of favour some time ago. (I was also made aware of an article called "In Defence of Lecturing" by Mary Burgan. For me, who used to be a fanatical opponent of lecturing (even though I myself seem to have benefited from that kind of teaching - as well as others), it would be interesting to read this article.

After this, I decided to give my mind some rest. After writing this, I will go have a look at the town and prepare for the conference dinner which is this evening. I always get a bad feeling about missing a lot when I skip parts of the programme at such conferences, but on the other hand, it is important to have energy to actually take part in the workshops and listen attentively at the lectures - not just be there...

ESU5 Day 2 continues...

2007-07-21T17:17:00.000+02:00

After lunch on Friday, I was going to attend a workshop on "Histories on zeros", but as this talk suddenly switched its language into French (which I, sadly, do not master), I went instead to Maria Menghini's "The 'Elements de Geometrie' of A. M. Legendre: an analysis of some proofs from yesterday's and today's point of view". I always find Euclid a bit unsatisfying, as I always am presented with just parts of the work - which means I'm not aware of what is already proved. Not knowings which tools are available is frustrating. I guess the same phenomenon also explains part of Norwegian students' frustration with proofs in geometry - as Norwegian students never get the opportunity to follow the logical building from the foundations. Anyway, given these problems, I found this particular workshop enlightening, especially concerning the problems Legendre and his time had with what we call rational and irrational numbers.

Later in the evening, I heard Ferreira Eduardo Sebastiani's talk on "Ethnomathematic's use in Indian teacher's formation" - another talk suddenly switched to French. Here, however, the transparents were in English, and I got the general idea of the talk - how ethnomathematics is used in working with the Waimiri-Atroari Tribe.

Finally, I heard Robin Wilson's talk on "Lewis Carroll in Numberland". The talk has the same title as a book which is to be published by Penguin later, and which I definitely will want to read. The talk had several hilarious quotes from Lewis Carroll, but it also pointed me to the book "Suggestions as to the best method of taking votes", which I certainly will want to read as well.

Thus ended the second day of the conference - except that food and drink was consumed in the evening.

ESU5 Day 2

2007-07-20T13:33:00.000+02:00

The second day of the conference started with a talk by Ulrich Rebstock titled "Mathematics in the service of the Islamic community". I must admit that my knowledge of Islamic mathematics is even worse than my knowledge of for instance European mathematics, and every talk on the subject is therefore sure to give me new information. Rebstock showed how much of Arab mathematics focused on practical aspects of mathematics, and gave examples from a wealth of books on subjects like measuring, taxes and trade. But the more "theoretical" mathematics was never far from the surface, for instance, examples with made-up "practical interest" were many. I particularly liked this example: In a Turkish bath, one day 30 visitors had payed their entrance. The owner knew there had been three Jews, and that he had collected 30 dirhams. The prices were Muslims 1/2 dirham, Christians 2 dirhams, Jews 3 dirhams... Another interesting example of mathematics of the time were inheritance problems when the gender of one of the beneficiaries were uncertain (!)

Thereafter, I joined Chris Weeks´ talk on Condorcet´s paradox. Condorcet wrote a treatise (Essai sur l´application de l´analyse a la probabilité des décisions rendues a la pluralité des voix) on the problem of voting, showing that any voting systems has its problems (although it should be possible to create a voting system that doesn´t repeatedly elect Bush president...) The treatise is very interesting, and it is tempting to use it with my students this autumn - especially as there is a local election coming up.

Then there was lunch. 99 crowns for a three-course meal is great - maybe a fifth of what I would expect to pay in Norway...

ESU5 Day 1 continues

2007-07-20T11:54:00.000+02:00

After lunch, I enjoyed a three-hour workshop by Renaud Chorlay and Philippe Brin, both French high school teachers. Apparently, French kids start learning probability quite late, but when they do, at age 16-18, thez do it seriously. Chorlay and Brin showed three examples of mathematical original sources that thez use in their classrooms: a discussion on life expectancy between the Huygens brothers, a note on a game of dice by Leibniz and some familiar texts on the "problem of points" by Pacioli, Pascal, Fermat and others. In all, there were several texts here that I haven´t seen before, and that I would like to use in my own teaching.
After this, I heard five 10-minute presentations, on a mathematical exhibition (Oscar Joao Abdounur), on mathematics theater (Funda Gonulates), on error correcting codes (Uffe Thomas Jankvist), on Ramanujan (Jim Tattersall) and on area and volume (Luciana Zuccheri and Paola Gallopin). All of these were interesting, but frankly, 10-minute talks only serve as an introduction which should be followed bz personal communication whenever particularly interesting things occur. (Even though some of the presenters tried to put as much content into the presentations as physicallz possible.)

This finished off the official part of the first daz, but then some of us went out to enjoy Prague´s food and drinks, of course. The prices are ridiculously low by Norwegian standards (as in manz parts of the world...)

ESU5 Day 1

2007-07-20T11:38:00.000+02:00

I am attending the ESU5 - the 5th European Summer University On The History And Epistemology In Mathematics Education. This goes on in Prague this week, and I will report on it in this blog. More than 220 participants will choose from the 128 activities by presenters from 28 countries.

The first plenary speaker was Luis Puig. His title was "Researching the history of algebraic ideas from an educational point of view." He compared three approaches to algebra: the Babylonian, as interpreted bz Jens Hoyrup, al-Khwarizmi´s and Jordanus de Nemore´s. Especially interesting to me was Nemore "De Numeris Datis", 1225, who assigned letters to all quantities, with no distinction between known and unknown. If a times a was needed, it would be assigned a letter, say b. Puig noted parallells with how some pupils treat letters in the beginning, before they see the point of the letters.

Then I joined Jan van Maanen´s workshop on "The work of Euler and the current discussion on skills". We were given copies of part of Euler´s 1770 algebra textbook, and studied parts of it. Euler goes to great length to define + and -, for instance, so it is clear that Euler saw a great need to improve the "basic skills" of his fellow men. However, he did so while relating algebra to the real world. In the copies were also Euler´s explanation of the algorithm for extracting square roots. As I have never learned that, I should do that as homework...
It is terribly hot in Prague now, but huge doses of refreshing mathematics and history helps!

Math concepts explained

2007-04-09T10:52:00.000+02:00

Just a short post to point to Math Concepts Explained, a blog with high ambitions. So far, the explanations included are on graphing only.

I'd also point to an extremely useful site: www.geogebra.at, where the free geometric software Geogebra has it's home. After using it in my teaching and in courses for teachers this past year, it will now be an important part of my main project for the time to come, in making it easier for teachers to show the dynamic sides of mathematics. A lot more on Geogebra later, I'm sure...

Beauty of Maths!

2007-03-28T08:40:00.001+02:00

clipped from www.frogview.com

Beauty of Maths!

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432

123456789 x 8 + 9 = 987654321

1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111

123456789 x 9 +10= 1111111111

9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888

98765432 x 9 + 0 = 888888888

Brilliant, isn't it?

And finally, take a look at this symmetry:

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111=12345678987654321

Maths problem at Verizon Wireless

2007-03-03T16:13:00.001+01:00

This is an absolutely absurd conversation, in which representatives of Verizon Wireless are making fools of themselves.

The problem is this: The caller has been quoted a rate of 0.002 cents/kb for internet traffic in Canada (and this is readily accepted by Verizon Wireless), but when he gets the bill, he is charged 0.002 dollars/kb. The really amazing part of the conversation is when Verizon Wireless makes this calculation:
0.002 c/k * 35,893 k = 71.79 dollars
without understanding that the caller has a problem with the sudden change from cents to dollars...

Would you buy shares in a company that don't know the difference between cents and dollars?

I must add that the tape may be a hoax, as it seems incredible that Verizon Wireless could be this stupid.

I have contacted Verizon Wireless for their comments, but I have not had an answer so far. I will update this story if I receive a comment.

(For the record: the "stupid" part is not the part of not knowing math - it's setting people who don't know math to be the company's representatives to discuss maths problems.)

giving 110 %

2006-10-06T19:48:00.000+02:00

Yesterday, I had the strange experience of being interviewed on national radio on the subject of "giving 110 percent". A journalist in NRK (Norwegian Broadcasting) had evidently got fed up with people (and in particular Norwegian soccer star John Arne Riise) claiming that they would "give 110 percent"), so he wanted a mathematician to say something on the topic.

I'm not sure if I said anything enlightening, except that "giving 110 percent" is nonsensical (as long as the 100 percent is "maximum effort" or something like that"), and that I did not really believe anyone would think that John Arne Riise would sound more academic just because he used numbers and the percentage concept to explain that he would do his best...

This is the second time I've been interviewed on national radio, but I'm not sure I want it to become a habit - at least not as long as I don't really have anything that I want to say...

The programme was "Osenbanden" on NRK-P3, by the way, and it was sent today (Friday) - in case anyone wants to listen to it...

Math Lesson

2006-09-30T11:43:00.000+02:00


Many have obviously seen this video before, but even so, I thought I'd post a reminder of it, to point out the importance of understanding the number system properly and not just learn the algorithms by heart.

Stair-step fractal

2006-09-24T17:13:00.000+02:00

I've come across a "fractal" that have turned out to be popular with teachers on in-service training courses. It is called the "stair-step fractal", and is discussed in detail on the webpage Studying Dimensionality Through a Stair-Step Fractal.

While it is interesting to let children meet a fractal, I have actually used this as the end of some work on the A-series of paper formats (most notably the A4). There are lots of mathematics included there, of course, most importantly the point that A3 is equivalent to two pieces of A4, A2 is two pieces of A3, A1 is two pieces of A2 and A0 both is two pieces of A1 and has an area of one square meter. The square root of two figures prominently.

The reason the stair-step fractal is fun to use here, is that the way of making it gives rise to small "rectangles" which are A5, A7, A9, A11, A13 etc. Teachers tend to be quite amazed when they realize this (even though they also think it is fairly obvious once they notice it).

The stair-step fractal can then be used to show quite persuasively that when you double all the sides, you do not also double the area and the volume, they are actually 4x and 8x as large, respectively. While there are of course lots of ways of showing that, this is a fun way of doing it.

This was just a little glimpse into what you may see if you join one small workshop with teachers for 10-13-year olds.

Studying original sources in mathematics education Day 5

2006-05-05T20:25:00.000+02:00

The week in Oberwolfach is drawing to a close. Today was the last day of the academic programme, and there is just one breakfast left of the great food here. It will soon be the time to go home and try to keep up the motivation which has been building this week.

Today's first talk was by Hans Niels Jahnke. He talked about working with Bernoulli's lectures on the differentual calculus (1692) with 16/17-year olds. Halfway through the talk we were ourselves faced with trying to understand the mathematical ideas. Remember that an important idea was that of infinitely small quantities - and then even infinitely smaller quantities. This was the "foundation" for calculus for lots of years, even though a shaky one. (I wrote a Master thesis on part of the British side of this story.) It is always funny to try to understand the way these people were thinking on these things, and the students seem to have though so as well. I would certainly have tried to copy Jahnke in this if I had students in calculus.

The second talk was by Jan van Maanen. He was, with Jahnke and Fulvia Furinghetti, the organizers of this conference. His talk was titled "Original sources, projects, lessons and lectures". He has (as I knew before) a soft spot for original sources as in the real thing - he thinks the experience of actually handling a 1700th century book is an important part of the whole thing. That is of course good as far as it goes, but if studying original sources is to become something that the average pupil in Norway does, we will have to loosen the definition of "original sources" a bit (as van Maanen is fully aware of, of course).

van Maanen first gave a couple of examples of his own teaching through the years. Then he went on to talk about Iris van Gulik-Gulikers' PhD thesis, consisting of (among other things) teaching experiments on the history of similarity (similar triangles) and on non-Euclidean geometry. Both sounded fascinating, and I promptly mentioned that I would really want a copy of the thesis... The study was a fairly large one, being tested on about 800 pupils. I find it good to see that such solid, large-scale studies are now being done.

The rest of the day consisted of some round-table discussions and of some participants presenting things that they had not had time for earlier in the week. Among other things, I was reminded of a 1654 letter from Pascal to Fermat about a dice problem that I should really try to incorporate in my own teaching, and I was also forced to try to understand the algorithm for extracting cube roots. I think I got the main point of it in the end.

I think that I now should end these writings from Oberwolfach. The first four days are treated here:
Day 1
Day 2
Day 3
Day 4

Studying original sources in mathematics education Day 4

2006-05-04T20:53:00.000+02:00

A new day, new possibilities for learning!

The first lecture of the day was Katja Peters' talk "Perceiving history of mathematics". She has done a project where 18 year old students from a German intensive course visited a library for three consequetive days to study original sources. This was a way for the library to teach students about old books. Sadly (but understandably) most libraries do not want schoolstudents anywhere near their old books, so such projects are not possible anywhere. On the other hand, more and more old books are available online, which means that variants of this project may be easier to do in future. (Anyone who knows sites where old mathematical texts suitable for 6-16 year olds are available, are urged to tell me...)

Peters also talked about her way of choosing the most suitable books, and this was maybe the most valuable part of the talk for me. That is because the "upscaling problem" is very much a problem in this case - it is not clear how this experience can be brought into anything like the average classroom.

Kathy Clark talked about "Use of original sources: One Teacher's Experience with Personal Study and Curricular Inclusion". The design of her research was very interesting. She gave five teachers professional development sessions on the historical development of logarithms, and then studied how the teachers taught logarithms thereafter. Obviously, not all five changed their way of teaching significantly, but more than one did. That is actually quite good news - at times, I get the feeling that what teachers learn in professional development courses are left behind before they go back into their classrooms.

Evelyn Barbin's second talk of the workshop was titled "Reading in a historical context: "depaysement". The example of The Geometry of Descartes" (sorry for leaving out at least one French point on an "e" in there...) The main point was to show one example of students interpreting a historical source in its historical context, and one example of them comparing different texts. The importance of The Geometry as a step towards our way of using algebra for solving geometrical problems was underlined - today's students may not see the genius of Descartes because what he does has become so "normal" today...

Barbin also underlined that there has been a change in how source books are produced (at least by IREM). Before, source books were perhaps more source books for studying history of mathematics. Today, they are made to be used in teaching - the point is to understand mathematics by historical texts. She showed some wonderful new books which sadly are in a language I don't read well enough to appreciate them fully (French, of course).

What should be noted over and over again in these notes from this workshop is of course that all interpretations are mine - the speakers' own account of what they really meant to say is best found in their papers when they are published. While waiting for that, the Oberwolfach Report should give some idea, that should be available within a few months' time.

Michael R. Glaubitz also had a very interesting talk. His title was "Reading Al-Khwarizmi's Treatise On Quadratic Equations with 9th-graders. An Empirical Study". He had produced a wonderful workbook on this and two related subjects, and had done a teaching experiment in seven classes with a control group of an additional three classes. He did not do the teaching himself, which would obviously have interfered with the value of the project. The results of the project are not clear yet, but it is surely an interesting way of doing it. One interesting outcome is obviously an answer to the question "Do the students learn quadratic equations better?" As interesting are the questions "What else do they learn in the process?" and "What kind of knowledge of quadratic equations do they get?"

The last talk of the day was Wann-Sheng Horng who talked about different proofs of Heron's formula (for the area of a triangle. This talk focused on the dual role as both a teacher and a historian. Certainly, that is a very interesting combination, but a question I always come back to is this: to what degree does the teacher have to know the history of mathematics to be able to enrich his teaching with history of mathematics. It is obviously a question in the same family as lots of other questions of the same sort: Do you have to know the name of all flowers to talk about flowers to your students? I believe (or at least hope) that it is possible to start including history of mathematics in the mathematics lessons without being a historian. No, that was too weak - I know that this is possible.

Wann-Sheng Horng also mentioned another interesting point: by looking at the history of Heron's formula in China in the (a bit) broader context of the history of formulas for the area of a triangle (still in China), there is a "vertical integration" pointing to things from primary, secondary and high school mathematics.

After dinner, we had an "extra session" with a more free-flowing discussion on several issues, although the main concern of everyone was how we should improve the situation concerning the topic of the workshop: "Studying original sources in mathematics education".

I have long pages with ideas of what I should do when I get back home - I don't think I will keep mentioning them here. But such a workshop is certainly inspiring, and at the moment I feel it is likely that I will both go to a conference in Prague next summer and two others in Mexico and thereabout the summer after that. I just hope I won't work too much, and that I will also have enough time for my boyfriend back home... (I miss him...)

Studying original sources in mathematics education Day 3

2006-05-04T07:35:00.000+02:00

Day 3 was in one way a very short day, in another a very long one - depending on how you look at it. There were only two talks, but there were a long walk in addition - two hours the one way and one and a half hour back. It was pleasant and the weather was great, and now everybody is probably ready for a new full day of work.

The first talk was by Costas Tzanakis, who is also the chair of the HPM group. He had three examples. The first one showed an example of using ancient Greek mathematical texts in the teaching of Euclidean geometry in the Greek Lyceum. This showed many aspects of the work they had done, but for me the most interesting part was the discussion of "why do we have to prove this" which always comes up in the classroom. This was treated by looking at the Epikureans' criticism of Euclid, and seems a fertile starting point for discussions.

His second example was on the concept of (instantaneous) speed. Here, a quote from Newton showed with all necessary clarity how confused this concept was at the time. A discussion on Newton and his immediate followers (and the criticism of him) could be very interesting. Working on Zeno's paradox is another option.

His third example was on Hamilton's quaternions, and Karin Reich followed up with a talk on "The historical roots of vector calculus: J. W. Gibbs (1839-1902)". Sadly, vector calculus is not in the Norwegian curriculum for 6-16-year-olds, so there was little in these two talks that I can take directly into my classroom. However, the point that we should some of the time look at "original sources" not as the ones that does neccessarily contain the most "original" ideas, but as some "old", authentic sources where mathematical ideas are treated, seems worth making.

The days here in Oberwolfach is making me start thinking of what I should do when I get back. One simple idea is just to use history of mathematics in my teaching at about the same level as before (or a bit more), and at the same time "tracking" the students' idea of the "value" of history of mathematics through the year. That may be an interesting starting point for a discussion. (On the other hand, I already have an ongoing interview study, and I will also do two studies on Norwegian mathematics textbooks, one on the historical contents and the other on the "family concept" in them, so perhaps I have enough to do for the next year...)

Studying original sources in mathematics education Day 2

2006-05-02T19:51:00.000+02:00

Another interesting day at the mini-workshop at the MFO. The day started with a talk by Peter Rasfeld, where he told us about a project he has done on the problem of points (the famous problem worked on by lots of mathematicians, for instance Luca Pacioli, Girolamo Cardano, Niccolo Tartaglia, Blaise Pascal and Pierre de Fermat). He described a four-week project with his pupils aged 16, where the pupils worked on original sources much of the time - after initially trying to solve the problem themselves.

I am very happy to be able to hear this kind of presentations, as I have on previous conferences and meetings of this kind had an impression that most people have more interest in things for older students. For me, who is teaching teachers-to-be for 6-16-year-olds, it is important that the mathematics is at a reasonably low level...

The second talk was my talk. I would have liked to be able to say that it was absolutely brilliant, but I am in no position to know that. However, I think I managed to get the point across that including history of mathematics in the average classroom takes a lot more than just putting it into the curriculum. I used a study on the TIMSS 1999 Video Study material, a study on the Norwegian textbooks in elementary and secondary school, and a study that I am doing at the moment where I interview teachers about their attitudes to history of mathematics.

The third talk was by Adriano Dematte (with a sign over the last "e" that I am unable to conjure up on this computer). He talked on a new collection of materials for secondary school students (12 to 18) that is coming out in Italy in a few months. I'm looking forward to getting hold of a copy and considering what I could do to get this kind of material available in Norwegian for Norwegian teachers. Together with the Historical Modules from the US and with a book I just heard of today, this new book certainly will give me some ideas (although things will certainly have to be adjusted to fit the (not too impressive) level of the Norwegian mathematical curriculum.

The fourth talk (after lunch) was by Caroline Bardini and Luis Radford, with the title "Unknown, Variables and Parameters". I think I have tended to think that the difference between these should not be stressed too much to the students, but I believe that I should think it over more thoroughly. It is obvious from the discussion in the lecture hall and outside that there are important questions to be answered here. The video segments from classrooms were also interesting.

The fifth talk was Evelyn Barbin's talk titled "The different readings of original sources: an experience in pre-service teaching". The talk included an amazing number of ideas for historical sources that can be used in a mathematics course - her own present experience was from a course for pre-service teachers who were aiming for primary school. (French primary schools, that is, which is at another level altogether than Norwegian ones.) Maybe the most important part of her lecture, however (which was exemplified throughout the rest of the lecture), was an attempt to cathegorize the different ways students can engage with a historical source - from "interpret in our mathematical language" at the one extreme to "interpret in its own historical context" at the other.

My main decisions coming out of this day were these:
- I will put more effort into making available for Norwegian teachers materials on history of mathematics (for use in the classroom) in Norwegian.
- I will look again at my own teaching to think of how history of mathematics could enhance it more than presently, and think of the different ways students could engage with the materials.

In addition to this, there has been wonderful weather, good food, a nice walk and lots of friendly people. I'm looking forward to the next three days. And I'm glad I've had my presentation, so I don't have to think of that any more...

Studying original sources in mathematics education

2006-05-01T18:51:00.000+02:00

I have been terrible at posting in this blog, but I'll try to improve. I am currently at a mini-workshop at the Mathematisches Forschungsinstitut in Oberwolfach on the topic of "Studying original sources in mathematics education". The place is wonderful - the place is so green and beautiful, and there are all neccessary amenities to have a great, productive time.

On this first day of the mini-workshop, I have heard two great talks. The first one was by Abraham Arcavi. He told us about a teacher workshop he conducted with Masami Isoda, where teacher students were confronted with problems from the Rhind Papyrus, on multiplication and on solving an equation (as we will see it). The starting point of the work was that whether you are learning history or you are learning to teach, you have to learn to interpret. And maybe it is easier to learn to interpret by working on historical sources than by working on student answers, as it is harder to dismiss a historical source as being "wrong".

Arcavi gave some details on this particular workshop, not least on how "scaffolding" was necessary to ensure that the teacher students managed to work through the text in a meaningful way. The point noted in the end of the talk, however, seems to be one of the most important - how can you "upscale" such an experiment? This particular workshop was conducted by two prominent researchers in the field of "history and pedagogy of mathematics" (HPM) - how can we make "ordinary" teachers do the same (or similar things)?

One small exercise: Ron is given the following exercise: "3/5 of a number is 12, what is the number?" Ron writes: "12*2 = 24 24:6 = 4 24-4 = 20" Is Ron correct? How would you as a teacher react to his answer?
(This should show the need of a teacher to listen attentively, not just "evaluative".

The other talk was by David Pengelley, who showed us how he works with Pascal's "Treatise on the Arithmetical Triangle" with his students. I did not know this text before (although I have referred to it many times), and it was interesting to look at it a bit closer. As mentioned in an earlier posting, I tend to use exercises on Pascal's triangle in my teaching, but surely I learned something new today - even though some of the more advanced stuff, such as Pascal's use of inductive proofs, is probably not suitable for our first course at my institution...

Obviously, people are speaking to the converts at such a workshop. As mentioned already, the important thing is to (to use Pengelley's words) "convince teachers that they want to do it". Developing materials seem to be the way to do that, and I'm sad to say that materials in Norwegian are few and far between. I should do something to remedy that at some point.

Mathematics day

2006-01-22T09:42:00.000+01:00

The Norwegian society for mathematics teachers (LAMIS) every year asks schools to have a dedicated "mathematics day" in February. They publish a booklet with activities every year as well.

This year, they are also collecting different kinds of online games and other activities on its own page. Many of them are taken from The Freudenthal Institute.

I hope this work by LAMIS makes more pupils see the fun of mathematics!

Easy solution

2005-11-28T15:08:00.000+01:00

I liked this solution to the eternal "Find x"-exercise a lot...

(I don't know the source - it just fell into my mailbox...)

Diary of a Black Mathematician

2005-11-26T23:42:00.000+01:00

I'm spending a little time seeing a little of what is out there in the mathematical blog sphere. One of the interesting ones are Diary of a Black Mathematician, with several interesting postings. (Among others, the comic strips added at the end of this posting.)

He mentions that he has kept a mathematical diary for years. That reminds me of the usefulness of keeping a diary for other purposes as well. For more than a year some years ago, I kept a diary of what happened at work. All good ideas of what to change in the way we taught and so on were recorded there. At the end of each month, I made a summary. And at the end of the academic year, I was able to write a detailed evaluation of the course - based on the diary. (After writing it, I let the students read it and react to it - they agreed that I had caught the main points.) Maybe I should try to get the time to take up this habit again...

(Okay, I know they turn out quite small. Right-click and choose "Show image" (or whatever is the way of doing it in your browser) to see a bigger version.)

Pascal's triangle (in Chinese)

2005-11-26T22:50:00.000+01:00

I like to give my students the following illustration and ask them for the pattern:

The illustration "dates from a book of 1303 CE written by Chu Shï-kié. The earliest known use of the pattern was by Yang Hui, whose books date from 1261 & 1275 CE. Chu Shï-kié refers to the triangle as already being old. Jamshid Al-Kashi, who died around 1436 CE, was an astronomer at the court of Ulugh Beg in Samarkand in the 15th Century. Al-Kashi was the first known Arabic author to consider 'Pascal's' Triangle.", to quote the BBC.

They quickly come up with the pattern (every number is the sum of the two above), and thereby can also agree on the interpretation of the chinese numerals. Then they are ready to start exploring some of the wonderful "hidden" patterns in the triangle. These are some of the ones they may find - an additional exercise is of course to explain the pattern:

- adding the numbers in a row gives a power of 2.
- start from the top row, and move downwards - every step either down to the left or down to the right. In how many ways can you get to the number 20 (in the middle of the seventh row)? (20)
- coloring the even numbers gives us some triangles. Why? And what about coloring the numbers divisible by three? Or the ones that give a remainder of 1 when divided by 3?
- the rows with prime numbers in have a special property - which?
- when ordering a pizza, you get a choice of 8 toppings. In how many ways can you choose two toppings? Three? Four? Why are these numbers found in one and the same row of Pascal's triangle?
- choose any number in the triangle (though not on the edge). Color the six numbers circling your chosen number red-blue-red-blue-red-blue. Multiply the red numbers. Then multiply the blue numbers. What do you see? Why is this so?
- calculate and so on (if you have the patience). Do you see a connection with Pascal's triangle?
- n people meet and everybody wants to shake hands with everybody else. How many handshakes does that make? Where do you find those numbers in the triangle?
- choose a row. Let the number to the far right be ones, the next tens, the next hundreds and so on. Factorize the combined number. What do you see?
- draw a circle and place five dots on the periphery of the circle. How many lines can you make between the dots? How many triangles can you make with corners in the dots? How many quadrilaterals? And so on...

These small exercises show some connections in mathematics between geometry, algebra, combinatorics and numbers - and most of them could easily be used in primary or lower secondary school. Given in the right way, they may also work as a "landscape of investigation" (Skovsmose).

Please tell me if you know of more patterns in the triangle, by the way. (I know that the Fibonacci numbers are there, but I always find the "diagonals" where they occur too odd to point them out to my students...)

Foundations for fluxions

2005-11-25T13:58:00.000+01:00

I just got the idea that I should use this blog to promote (shamelessly) my Master Thesis. The title of it is "Foundations for fluxions", and the topic is Newton's several attempts to put his theory of fluxions on a sound footing, and some English (/Scottish) attempts to do the same in the period after his death (including Colin MacLaurin and Roger Paman).

The thesis was written in LaTeX, but the web version is only in ascii - sorry about that...

How to catch a lion in the Sahara desert

2005-11-25T07:11:00.000+01:00

Luckily, mathematics har good practical applications. Here are a few legendary ones - on how to catch a lion in the Sahara desert:

The method of inverse geometry: We place a spherical cage in the desert and enter it. We then perform an inverse operation with respect to the cage. The lion is then inside the cage and we are outside.

The set theoretic method: We observe that the desert is a separable space. It therefore contains an enumerable dense set of points from which can be extracted a sequence having the lion as the limit. We then approach the lion stealthily along this sequence bearing with us suitable equipment.

The Dirac method: We observe that wild lions are ipso facto not observable in the Sahara desert. Consequently if there are any lions in the Sahara, they are tame. The capture of a tame lion is left as an exercise for the reader.

The thermodynamic method: We construct a semi-permeable membrane which is permeable to everything except lions and sweep it across the Sahara.

The Kalra method: Make a list of the lion's whereabouts. Classify them into different fuzzy sets. The lion will get confused and fall into your trap.

Topological method: We observe that the lion has at least the connectivity of the torus. We transport the desert into four-space. It is then possible to carry out such a transformation that the lion can be returned to 3-space in a knotted condition. He is then helpless.

The Schrodinger method: At any given moment there is a positive probability that there is a lion in the cage. Sit down and wait.

The Heisenberg method: You will disturb the lion when you observe it before capturing. So keep your eyes closed.

The Einstein method: Run in the direction opposite to that of the lion. The relative velocity makes the lion run faster and hence he feels heavier and gets tired.

The Newtonian method: Let the lion catch you (let's assume you remain alive here). For every action there is an equal and opposite reaction. Therefore, you would have captured the lion.

The cartesian method: Take the origin as close as possible to the lion. Then perform rotation operation again and again. Initially, the lion will feel dizzy. Finally it will fall down.

The software method: Make a linked list of all objects in the desert. Then delete the pointers on either side of the lion. (Make sure you are not AFTER the lion.)

The automata method: Use a Non-Deterministic Finite Automaton with epsilon moves from all states to the final state, and no moves from the final state. The lion will soon enter the final state and be trapped.

The time-cop method: Use a time-machine and take the entire Sahara back a few years in time. The lion is just a cub now, and all you need is a mouse-trap.

The integro-differential method: Integrate the Sahara over its entire surface. The lion is now somewhere in the result. Differentiate the result w.r.t. the earth's rotation. The resulting value is zero, and the lion is no more.

The Shakespeare method: Hold the lion still for a moment (I don't care how you do it), and recite Shakespeare`s Hamlet to it. The lion will change from 'To be to Not-to-be'.

The Unix method: Put the lion on (the hard) disk. Do : ls -l | grep -i "lion" | cut $9 > trapfile. The “-i” option ensures that size of the lion does not matter. Now the lion is caught in a trapfile, and can be safely tar-ed or backed up into tape !

The Quantum Measurement Method: We assume that the sex of the lion is _ab initio_ indeterminate. The wave function for the lion is hence a superposition of the gender eigenstate for a lion and that for a lioness. We lay these eigenstates out flat on the ground and orthogonal to each other. Since the (male) lion has a distinctive mane, the measurement of sex can safely be made from a distance, using binoculars. The lion then collapses into one of the eigenstates, which is rolled up and placed inside the cage.

The nuclear physics method: Insert a tame lion into the cage and apply a Majorana exchange operator on it and a wild lion. As a variant let us assume that we would like to catch (for argument's sake) a male lion. We insert a tame female lion into the cage and apply the Heisenberg exchange operator, exchanging spins.

The Newton method (II): Cage and lion attract each other with the gravitation force. We neglect the friction. This way the lion will arrive sooner or later in the cage.

The Special relativistic method: One moves over the desert with light velocity. The relativistic length contraction makes the lion flat as paper. One takes it, rolls it up and puts a rubber band around the lion.

The general relativistic method: All over the desert we distribute lion bait containing large amounts of the companion star of Sirius. After enough of the bait has been eaten we send a beam of light through the desert. This will curl around the lion so it gets all confused and can be approached without danger.

The software method: Make a linked list of all objects in the desert. Then delete the pointers on either side of the lion. (Make sure you are not AFTER the lion.)

The Heisenberg method (II): Position and Velocity from a moving lion can not be measure at the same time. As moving lions have no physical meaningfull position in the desert, one can not catch them. The lion hunt can therefore be limited to resting lions. The catching of a resting, not moving lion is left as an exercise for the reader.

The atomic fission method: We irradiate the desert with slow neutrons. The lion becomes radioactive and starts to disintegrate. Once the disintegration process is progressed far enough the lion will be unable to resist.

The magneto-optical method: We plant a large, lense shaped field with cat mint (nepeta cataria) such that its axis is parallel to the direction of the horizontal component of the earth's magnetic field. We put the cage in one of the field's foci. Throughout the desert we distribute large amounts of magnetized spinach (spinacia oleracea) which has, as everybody knows, a high iron content. The spinach is eaten by vegetarian desert inhabitants which in turn are eaten by the lions. Afterwards the lions are oriented parallel to the earth's magnetic field and the resulting lion beam is focussed on the cage by the cat mint lense.

The Hilbert (axiomatic) method: We place a locked cage onto a given point in the desert. After that we introduce the following logical system: Axiom 1: The set of lions in the Sahara is not empty. Axiom 2: If there exists a lion in the Sahara, then there exists a lion in the cage. Procedure: If P is a theorem, and if the following is holds: "P implies Q", then Q is a theorem. Theorem 1: There exists a lion in the cage.

The projective geometry method: Without loss of generality, we can view the desert as a plane surface. We project the surface onto a line and afterwards the line onto an interiour point of the cage. Thereby the lion is mapped onto that same point.

The Bolzano-Weierstrass method: Divide the desert by a line running from north to south. The lion is then either in the eastern or in the western part. Let's assume it is in the eastern part. Divide this part by a line running from east to west. The lion is either in the northern or in the southern part. Let's assume it is in the northern part. We can continue this process arbitrarily and thereby constructing with each step an increasingly narrow fence around the selected area. The diameter of the chosen partitions converges to zero so that the lion is caged into a fence of arbitrarily small diameter.

The Peano method: In the usual way construct a curve containing every point in the desert. It has been proven [1] that such a curve can be traversed in arbitrarily short time. Now we traverse the curve, carrying a spear, in a time less than what it takes the lion to move a distance equal to its own length. ([1]: After Hilbert, cf. E. W. Hobson, "The Theory of Functions of a Real Variable and the Theory of Fourier's Series" (1927), vol. 1, pp 456-457)

The search method: We assume that the lion is most likely to be found in the direction to the north of the point where we are standing. Therefore the REAL problem we have is that of speed, since we are only using a PC to solve the problem.

The parallel search method: By using parallelism we will be able to search in the direction to the north much faster than earlier.

The Monte-Carlo method: We pick a random number indexing the space we search. By excluding neighboring points in the search, we can drastically reduce the number of points we need to consider. The lion will according to probability appear sooner or later.

The practical approach: We see a rabbit very close to us. Since it is already dead, it is particularly easy to catch. We therefore catch it and call it a lion.

The common language approach: If only everyone used ADA/Common Lisp/Prolog, this problem would be trivial to solve.

The standard approach: We know what a Lion is from ISO 4711/X.123. Since CCITT have specified a Lion to be a particular option of a cat we will have to wait for a harmonized standard to appear. $20,000,000 have been funded for initial investigations into this standard development.

Linear search: Stand in the top left hand corner of the Sahara Desert. Take one step east. Repeat until you have found the lion, or you reach the right hand edge. If you reach the right hand edge, take one step southwards, and proceed towards the left hand edge. When you finally reach the lion, put it the cage. If the lion should happen to eat you before you manage to get it in the cage, press the reset button, and try again.

The Dijkstra approach: The way the problem reached me was: catch a wild lion in the Sahara Desert. Another way of stating the problem is: Axiom 1: Sahara elem deserts. Axiom 2: Lion elem Sahara. Axiom 3: NOT(Lion elem cage).We observe the following invariant: P1: C(L) v Not(C(L)) where C(L) means: the value of "L" is in the cage. Establishing C initially is trivially accomplished with the statement ;cage := {}Note 0: This is easily implemented by opening the door to the cage and shaking out any lions that happen to be there initially. The obvious program structure is then: ;cage:={} ;do NOT (C(L)) -> ;"approach lion under invariance of P1" ;if P(L) -> ;"insert lion in cage" [ ] not P(L) ->;skip ;fi ;od where P(L) means: the value of L is within arm's reach. Note 1: Axiom 2 ensures that the loop terminates. Exercise 0: Refine the step "Approach lion under invariance of P1". Note 2: The program is robust in the sense that it will lead to abortion if the value of L is "lioness". Remark 0: This may be a new sense of the word "robust" for you. Note 3: From observation we can see that the above program leads to the desired goal. It goes without saying that we therefore do not have to run it.

Mathematicians hunt lions by going to Africa, throwing out everything that is not a lion, and catching one of whatever is left. Professors of mathematics prove the existence of at least one lion and leave the capture of an actual lion as an exercise for one of their graduate students.

Computer scientists hunt lions using algorithm A: 1. Go to Africa, 2. Start at the Cape of Good Hope, 3. Work northward in an orderly manner, traversing the continent alternately East and West. 4. During each traverse a. Catch each animal seen, b. Compare each animal caught to a known lion, c. Stop when a match is detected. Experienced computer programmers modify Algorithm A by placing a known lion in Cairo to ensure that the algorithm
will terminate.

Engineers hunt lions by going to Africa, catching yellow animals at random, and stopping when any one of them weighs within plus or minus 15 percent of any previously observed lion.

Economists don't hunt lions, but they believe that if lions are paid enough they will hunt themselves.

Statisticians hunt the first animal they see N times and call it an lion.

Consultants don't hunt lions, but they can be hired by the hour to advise those who do. Operations research consultants can measure the correlation of hat size and bullet color to the efficiency of lion hunting strategies, if someone else will identify the lions.

Politicians don't hunt lions, but they will share the lions you catch with the people who voted for them.

Lawyers don't hunt lions, but they do follow the herds around arguing about who owns the droppings. Software lawyers will claim that they own an entire herd based on the look and feel of one dropping.

When the Vice President of R&D tries to hunt lions, his staff will try to ensure that all lions are completely prehunted before he sees them. If the VP sees a nonprehunted lion, the staff will (1) Compliment the vice president's keen eyesight and (2) enlarge itself to prevent any recurrence.

Senior managers set broad lion hunting policy based on the assumption that lions are just like field mice, but with deeper voices.

Quality assurance inspectors ignore the lions and look for mistakes the other hunters made when they were packing the jeep.

Salespeople don't hunt lions but spend their time selling lions they haven't caught, for delivery two days before the season opens. Software salespeople ship the first thing they catch and write up an invoice for an lion. Hardware salespeople catch rabbits, paint them yellow and sell them as "desktop lions."

Good method: Let Q be the operator that encloses a word in quotation marks. Its square Q2 encloses a word in double quotes. The operator clearly satisfies the law of indices, Qm*Qn=Qm+n. Write down the word 'lion', without quotation marks. Apply to it the operator Q-1. Then a lion will appear on the page. It is advisable to enclose the page in a cage before applying the operator.

Roselius method: Let L denote the category whose objects are lions, with 'ancestor' as the only nontrivial morphism. Let l be the category of caged lions. The subcategory l is clearly complete, is nonempty (by inspection), and has both generator and cogenerator. Let F:l->L be the forgetful functor, which forgets the cage. By the Adjoint Functor Theorem, the functor F has a coadjoint C:L->l, which reflects each lion into a cage. We remark that this method is obviously superior to the Good method, which only guarantees the capture of one lion, and which requires an application of the Weierkäfig Preparation Theorem.

Surgical method: A lion may be regarded as an orientable three-manifold with a nonempty boundary. It is known that by means of a sequence of surgical operations (known as 'spherical modifications' in medical parlance) the lion can be rendered contractible. He may then be signed to a contract with Barnum and Bailey.

Logical method: A lion is a continuum. According to Cohen's theorem he is undecidable (especially when he must make choices). Let two men approach him simultaneously. The lion, unable to decide upon which man to attack, is then easily captured.

Functorial method: A lion is not dangerous unless he is somewhat gory. Thus the lion is a category. If he is a small category then he is a kittygory and certainly not to be feared. Thus we may assume, without loss of generality, that he is a proper class. But then he is not a member of the universe and is certainly not of any concern to us.

Method of differential topology: The lion is a three-manifold embedded in Euclidean 3-space. This implies that he is a handlebody. However, a lion which can be handled is tame and will enter the cage upon request.

Sheaf theoretic method: The lion is a cross-section of the sheaf of germs of lions on the Sahara Desert. Merely alter the topology of the Sahara, making it discrete. The stalks of the sheaf will then fall apart releasing the germs which attack the lion and kill it.

Method of transformation groups: Regard the lion as a surface. Represent each point of the lion as a coset of the group of homeomorphisms of the lion modulo the isotropy group of the nose (considered as a point). This represents the lion as a homogeneous space. That is, this
representation homogenizes the lion. A homogenized lion is in no shape to put up a fight.

Postlikov method: A male lion is quite hairy and may be regarded as being made up of fibres. Thus we may regard the lion as a fibre space. We may then construct a Postlikov decomposition of the lion. This being done, the lion, being decomposed, is dead and in bad need of burial.

Steenrod algebra method: Consider the mod p cohomology ring of the lion. We may regard this as a module over the mod p Steenrod algebra. Doing this requires the use of the table of Steenrod cohomology operations. Every element must be killed by some of these operations. Thus the lion will die on the operating table.

Homotopy method: The lion has the homotopy type of a one-dimensional complex and hence he is a K(Pi, 1) space. If Pi is noncommutative then the lion is not a member of the international commutist conspiracy and hence he must be friendly. If Pi is commutative then the lion has the homotopy type of the space of loops on a K(Pi, 2) space. We hire a stunt pilot to loop the loops, thereby hopelessly entangling the lion and rendering him helpless.

Covering space method: Cover the lion by his simply connected covering space. In effect this decks the lion. Grab him while he is down.

Game theoretic method: A lion is big game. Thus, a fortiori, he is a game. Therefore there exists an optimal strategy. Follow it.

Group theoretic method: If there are an even number of lions in the Sahara Desert we add a tame lion. Thus we may assume that the group of Sahara lions is of odd order. This renders the situation capable of solution according to the work of Thompson and Feit.

Biological method: Obtain a number of planarians and subject them to repeated recorded statements saying: 'You are a planarian'. The worms should shortly learn this fact since they must have some suspicions to this effect to start with. Now feed the worms to the lion in question. The knowledge of the planarians is then transferred to the lion. The lion, now thinking that he is a planarian, will proceed to subdivide. This process, while natural for the planarian, is disastrous to the lion.

Moore-Smith method: Letting A = Sahara Desert, one can construct a net in A converging to any point in the closure of A. Now lions are unable to resist tuna fish, on account of the charge atoms found therein (see Galileo Galilei, Dialogues Concerning Tuna's Ionses). Place a tuna fish in a tavern, thus attracting a lion. As noted above, one can construct a net converging to any point in a bar; in this net enmesh the lion.

Method of analytical mechanics: Since the lion has nonzero mass it has moments of inertia. Grab it during one of them.

Mittag-Leffler method: The number of lions in the Sahara Desert is finite, so the collection of such lions has no cluster point. Use Mittag-Leffler's theorem to construct a meromorphic function with a pole at each lion. Being a tropical animal a lion will freeze if placed at a pole, and may then be easily taken.

Method of natural functions: The lion, having spent his life under the Sahara sun, will surely have a tan. Induce him to lie on his back; he can then, by virtue of his reciprocal tan, be cot.

Boundary value method: As Dr. Morphy has pointed out, Brouwer's theorem on the invariance of domain makes the location of the hunt irrelevant. The present method is designed for use in North America. Assemble the requisite equipment in Kentucky, and await inclement weather. Catching the lion then readily becomes a Storm-Louisville problem.

Method of moral philosophy: Construct a corral in the Sahara and wait until autumn. At that time the corral will contain a large number of lions, for it is well known that a pride cometh before the fall.

The Ziplockian method: Assuming that the lion is wandering through the desert, do 1 of three things: 1. Follow closely behind the lion, stepping on it's heels, chanting loud mantras in it's ears, feeding it Italian Soda, giving it the finger, and farthing, and this attack on the senses will inevitably cause the lion to have a breakdown resulting in its inability to avoid capture. 2. Do not acknowledge it's authority. Place it, in your mind, at an equal or lower level of importance to yourself and it will appear less dangerous to ones self. By removing all fear, this will enable you to capture it. 3. Shoot it in the head.

The psychiatric method: Prescribe a drug such as Prozac to it. Upon takin this drug, it will have no emotional stimulus on which to react, thus causing it to return for further counselling.

The inductive method: Initial Condition: If you center a large cage on any one grain of sand, and a lion is on or close to the grain of sand. then he will be trapped by the cage. By close we mean within epsilon grains of sand. Given a cage the size of 2 * (size of lion * epsilon) it works. First Hypothesis: Given the first grain of sand in the desert, if the lion is standing on it you will trap him. Proof: Given by the initial condition. Induction Hypothesis: Assume that a lion is on a grain of sand n, and is trappable. Now, for grain n+1 (assume all grains of sand are ordered, inorder) n+1 is close to n, hence n is close to n+1. If the lion is on grain n, and is trappable; then he is close to n+1, and by the above condition, trappable. Hence, no matter where the lions are if you drop a cage centered on a piece of sand you will catch a lion.

The divide and conquer method (by recursion): Divide the desert in half. Repeat the process until you have the lion, a grain of sand, or some other object that cannot be divided without blood shed. You have the lion. The order of this method = O(insane). (Where sanity is anything reasonable.)

The square method: Square Sahara. Somewhere you will get a square lion. If the lion was negative, it will now be positive, which makes catching it a whole lot easier. (The actual catch is left as an exercise for the reader). Put lion in cage. Draw square root. If you prefer to have a negative lion, change signs. Warning: Do not draw square root of lions not already squared and put in a cage. If the lion was not square, it will become irrational. And if the lion was negative, you will start seeing imaginary lions.

The Darwin method: Put amoeba in a glass of water in a cage. Sooner or later, the amoeba will evolve into a lion.

The Bill Gates method: Convince Bill Gates that lions are important. Then he will put a lion in your next Windows. Therefore you will have a lion, even already framed.

The George W Bush method: Bomb (… somewhere). Then nobody will remember that you were supposed to catch a lion.

Sources:
http://www.telogy.com/~bkosanovic/fun/lion.htm
http://www.millennial.org/mail/talk/fmf-humor/hyper/0122.html
http://www.ags.uci.edu/~phillips/phhumor9.html
http://www.uni-karlsruhe.de/~un5p/fun/LionProblem.html
http://www.princeton.edu/~gopisety/elephant.html
http://info.ox.ac.uk:80/~invar/lions.html
http://members.xoom.com/ZiplocK/lion.htm
http://www.jk.math.usu.edu/cinnamon/jokes/gen_science/lion.txt
My own imagination

Mathematics music

2005-11-24T23:21:00.000+01:00

A student asked me this week (in an attempt to change the subject away from mathematics) what music I liked. I answered that at the moment I'm listening to for instance:

- Kate Bush: Pi
(Oh he love, he love, he love/He does love his numbers/And they run, they run, they run him/In a great big circle/In a circle of infinity/3.1415926535 897932 3846 264 338 3279)

- Tom Lehrer: New maths
(Hooray for New Math,/New-hoo-hoo Math!/It won't do you a bit of good to review math./It's so simple,/So very simple,/That only a child can do it!) - probably the only song in the world which includes the calculation of 342 minus 173, both in base 10 and in base 8...

- deLillos: Geometri
(Then you must tell yourself that you are in the centre, friend/it's only the circle around you that changes/and you can live friendly and reasonably in the middle of it/on the periphery of the others' (my translation))

- Tre små kinesere: Pi err i andre
(We all know the circle is round/Pi times r squared/The circle is eternal, it never breaks/That can't be changed (my translation))

There's lots of good music out there...

Edwin A. Abbott: Flatland: A romance of many dimensions

2005-11-24T21:43:00.000+01:00

This is one of many sites which has the full text of Flatland, Edwin A. Abbott's classic "romance of many dimensions" from 1884, wherein the "people" are geometrical shapes and live in two dimensions. The book is an interesting attempt at showing us how foolish we are when we assume three dimensions are all there is, and at the same time it is a satire of society in 1884.

I haven't yet found a good way to use this text with students, I'm afraid, but good ideas are always appreciated...

Geometrical look at tangens

2005-11-24T20:08:00.000+01:00

As mathematics is an important part of my life (I'm a teacher educator in mathematics), I thought it would be a good idea to start a blog where interesting small mathematical details could be posted...

Just a few weeks ago, I had a little revelation in trigonometry. I've been teaching trigonometry for years, and have given the definition of sin, cos and tan according to the triangle and the unit circle. I have never seen tangens clearly in the unit circle however - not before I came across this application. While cos u is the x-coordinate of the intersection of the one leg of the angle and the circle, and sin u is the y-coordinate of the same intersection, I have never before seen that tan u is the intersection of the leg and the vertical line x=1. A nice fact...

(And yes, I know I should have an illustration here, but for now the link to the Java-thing will do...)