Did you ever notice…

28 May , 2015
Reading "The Praise of Lectures" By Tom W. Körner

Reading “In Praise of Lectures” By Tom W. Körner

It may sound like something out of a stand-up-show by any comedian in the world. But it may also be a phrase very close to the heart of mathematics. Did you ever notice… that when you have been out travelling and returning home, suddenly the newpapers and TV shows are full of references to the place you just had been to? Where on earth were these news stories before you went there?

I just got home from a very nice travel back and forth to Bratislava, staying at a nice hotel called Ibis Bratislava Centrum Hotel (do I get a discount for promoting work here?). And on the train to work today I flipped through an article I have had lying on my desk for I don’t know how long, In praise of lectures, by T.W. Körner (click here for a pdf). I didn’t have much reason for having this paper around, other than the fact that I used a lot of Körner’s writing on Fourier Analysis 15 years ago for a thesis. And the point of departure was the Ibis, a sacred bird to the Egyptians.  I had read this before, but since Ibis made no recollection in my mind, I just scanned those lines quickly and moved on. This time, I googled the bird, read about it, made a review of the Ibis hotel and also got thrown back with some nice memories.

It seems that what we experience in life will dictate how we control our awareness. It could be people we meet, places we go to emotionally or geographically, things we perceive with our senses in one way or the other. This may not be very shocking, I mean, in what other ways could our inclinations for learning something be working with us?

I think all kinds of education work in much the same way. As a teacher of mathematics I can not jump into the minds of my students and twist their brains into what I want them to look like and how I want them to act and work (and I believe, metaphorically speaking, we have all tried to jump into the minds of our students!). If I tell a thirteen year old pupil that  the parameter in front of x will vary the slope of the graph of the linear function y=ax+b… then he might say “fine” and move on with his life. I know I would. Instead I could give him a very simple GeoGebra file to tinker with. For instance the standard one where you have gliders to control a and b in the mentioned expression. I could still do the mistake of telling him “Look, as I alter the parameter a, the graph slope changes accordingly”. I still don’t think this will stick to his brain, although a picture might do a better job than my words alone. And of course, the picture can also be improved, something I tried doing on a blog post on my Norwegian blog here.

Better still, I could ask him to alter a (and b) with the gliders, and have him tell me what happens. That would be the equivalent of my travel to Bratislava. The stay at Ibis Hotel pointed my awareness in that direction. I can also point the pupil’s awareness in the right direction and help him make sense of his discovery.

That’s basically all I can do, isn’t it?



A Mathematician’s Lament

5 July , 2011

I finally got around to reading this little book. I had previously enjoyed the PDF that has circulated the Internet and mathematics communities for some years. If you haven’t read that PDF file, you WILL like to do so. However, this post regards the entire book, 140 pages.

The book is mostly a piece of personal opinion regarding how mathematics learning happens, and how mathematics education, teaching and learning is really done these days. I am not reveiling too much if I say Lockheart is extremely critical of how schools present and teach mathematics to the children today. (The book was first published in 2009).

Picture from Amazon.com (Click to go to the books Amazon page)

He starts off the book comparing mathematics instruction with a nightmare, like if a musician goes to school, learns notes and partitures, and hardly ever gets to compose or play an instrument. He then goes on to show a simple example with a triangle inside a rectangle where a student is supposed to say something – anything! – about the areas of the figure inscribed in the other.  (Have a look at the GeoGebra files on http://mattegreier.blogspot.com/2009/10/areal-og-omkrets.html if you want to see this more clearly. Look for the triangle between two parallell lines and you will hopefully see what I mean). The point of the example is to make an argument about how big the triangle is in relation to the rectangle, and maybe bring forward a formula for computing the area of the triangle. (“Dissect it! Try many things! Try every way!”)

I love the way parts of the book are constructed as a dialogue between Simplicio and Salviati, it’s the first time I encountered this notion. We find the following explanation on Wikipedia, where Galileo Galieleis book, The Dialogue Concerning the Two Chief World Systems is covered:

  • Salviati argues for the Copernican position and presents some of Galileo’s views directly, calling him the “Academician” in honor of Galileo’s membership in the Accademia dei Lincei. He is named after Galileo’s friend Filippo Salviati (1582–1614).
  • Sagredo is an intelligent layman who is initially neutral. He is named after Galileo’s friend Giovanni Francesco Sagredo (1571–1620).
  • Simplicio, a dedicated follower of Ptolemy and Aristotle, presents the traditional views and the arguments against the Copernican position. He is supposedly named after Simplicius of Cilicia, a sixth-century commentator on Aristotle, but it was suspected the name was a double entendre, as the Italian for “simple” (as in “simple minded”) is “semplice”.[7] Simplicio is modeled on two contemporary conservative philosophers, Ludovico delle Colombe (1565-1616?), Galileo’s fiercest detractor, and Cesare Cremonini (1550–1631), a Paduan colleague who had refused to look through the telescope.[8] Colombe was the leader of a group of Florentine opponents of Galileo’s, which some of the latter’s friends referred to as “the pigeon league”.[9]

(Sagredo does not enter Lockheart’s book, though.) The point is, Simplicio defends the traditional world view with a flat earth, whilst Salviati defends the heliocentric world view that Galilei proposes. The comparison to views about schooling is apparent and good fun. I like the part where Salviati replies that he doesn’t think the society benefits much from a lot of people walking around with vague memories of something about b square and the square root of minus 4ac or something like that. I remember myself how much – VERY much – time was spent trying to understand, use and remember the formula for solution to a square equation. I don’t think now that I understood it very well back then, and I can also tell from my students starting their teacher education that this formula only sticks for so long – unless you spent more time on building arguments and proofs for it, than you did inserting numbers into a,b and c. And when were you gonna use it anyway? Well, never, of course, even if the books meant to fool you into thinking the reasong for learning it was because you could use it to determine where a cannon ball hits the ground.

There’s one very important point that always comes up in discussions like this:

Simplicio: But we don’t have time for every student to invent mathematics for themselves! (…)

Of course, nobody has ever meant the children should INVENT ALL mathematics. It took mankind hundreds of years, for crying out loud. I know that a lot of researcher claims that ALL mathematics COULD be taught by starting with a phenomenon and then doing investigations. And they are probably right. But what is meant is that maybe not all mathematics in the present curriculum is necessary to carry about as mental baggage the rest of our lives. REMEMBERING a formula you never will use does seem completely irrelevant to me. Working with it to understand it, making up notation as needed, comparing things to established practices, discussing how to solve problemes, that is another thing.

Most of us don’t need the cosine rule, but if you venture into mathematics it will be necessary to understand it. (And you can, just take a look at the Proof without words series (Roger B. Nelsen)).  But why learn it if not to develop your thinking in the process of coming to understand it? For the hell of it, I can’t even think of a sound reason or a good context to use simple things as the Pythagorean theorem or the area of a triangle.  Making a corner on a football field, with a rope at a triangular shape, with three knots one side, four on the second and five on the third? When did you see anyone do that? If you really need 90 degrees, it is not accurate enough, if you don’t need it accurate, just make something that is almost accurate! But don’t forget to let your pupils play around with the IDEA of why this rope would make a triangle with one right angle. And in theory, the right angle will be perfect.

I remember in my first year of teaching, when we started on the triangle area formula. I argued that this was a smart thing to learn and understand, because you never know – one day you might need to calculate the area of a …errr… triangle garden in order to buy enough grass seeds!

Who was I fooling? Mostly myself I guess. And the poor kids, too.

Anyway, READ this book. You and your pupils will benefit from it. Maybe you won’t change the world, but perhaps you can change a little bit of yourself. And then another bit…and another. And perhaps, in the end, one of your pupils will have a different view on mathematics than kids in schools today have.

Others have blogged about this book too:

Squarecircles: http://www.squarecirclez.com/blog/a-mathematicians-lament-how-math-is-being-taught-all-wrong/2828

maa.org: http://www.maa.org/devlin/devlin_03_08.html

(written by Keith Devlin, who also wrote the foreword to the book).


2 March , 2011

Forgive the slightly personal post, but I thought the service zoom.it was too good to miss. It allows you to post a huge image file into a nice panoramic display. Probably useful in all sorts of situations. Complex graphs… maps of the world… huge tables where one wants to navigate freely… perhaps even the periodic table of the elements could be made into a nice panorama.

Here is the view from my childhood’s bedroom window. It always brings me back… (I wanted to embed the picture, but I couldn’t do it on WordPress…)

Blog recommendations

10 December , 2010

Former colleague, present mathematician (and part wizard) has his blog at http://naylors-in-norway.blogspot.com/. Mike Naylor is an American who are living in Norway with his family for a couple of years. He works at the National Mathematics Centre here in Trondheim, and writes about all things Norwegian, Mathy and creativity in his blog. So take a look!

Fun with words

28 September , 2010

OK, I was actually going to write a post entitled “All the things I did and do wrong with PowerPoint”, but instead I have to write about the name for the university I took my masters degree (or rather “major”, which in Norway was a slightly longer study than a master’s degree) at. I am not going to illustrate this post.

Some of my English friends poked fun at me, since my position at the university was a student assistant. Abbreviated “stud.ass.” in Norwegian (*giggle*).

It got better. The university moved and expanded and changed names. The building we moved into was for those into “real fag”. It translates to science (real = real and fag = subject). The last equation sign got me banned on many serious web forums in the early days of web browsing. OK, it’s called realfag in Norwegian, but we usually dissect words wrong in Norway due to Internet influence.

It still got better. I was a stud.ass. for real fag, but the funniest part was when the university changed name from the previous two schools, NTH and AVH. The university was supposed to be called Norges teknisk-naturvitenskaplige universitet, in English: Norwegian University of Technology and Science. No wonder they would rather use NTNU for short.

So, working as a stud ass at the real fag for NUTS, I still managed to write a thesis on Fourier Analysis. (Yes, you could go on about half of analysis being … OK, good night now.


30 September , 2009

A really nice motivational video for teachers. By way of @ghveem on twitter.


9 July , 2009

Not that I publish a whole lot, but these days it’s even less. This is written on my iPod, in the sun, just before soccer time. Ahhhh…

Where is the share?

25 May , 2009

Anything with the words geek and/or chart in the title is bound to be great. On the Geek Chart website, you can make your own chart that visualizes your sharing around the web. Insert your username on twitter, youtube etc., and embed the chart that shows your acticity on the respective sites. Nice one. Now, why does that visual work on blogspot and not on wordpress?

Oisteing’s Geek Chart

The Childrens’ machine

21 March , 2009

A small one for the kids this weekend. My daughter has reached the age of two, and it’s impossible to keep her from hacking away at my keyboard. We have yet to buy one of those small kids’ computers for learning letters and stuff – but this weekend I am installing CrazyLittleFingers. Basically it allows you remap the keyboard and taylor it to your kiddo’s needs (and it’s fathers’ needs…).

You must provide the kid yourself and download CrazyLittleFingers here: http://www.donationcoder.com/Forums/bb/index.php?PHPSESSID=hon787djaqnnqrsthkjpflith5&topic=16131.0

Happy new year all!

4 January , 2009

Well, the academic life just started on Friday, and I love this period of a few days where you actually can prepare  yourself, read some of the stuff you should’ve read ages ago and perhaps even tidy up the office. Just a little bit.

I got this question about my profile photo (not the contrived serious one in the About Me page, but the small one on comments). The reason I love this photo (even if it is of myself) is that it shows myself learning some mathematical facts and connections on my Amstrad CPC 6128 computer. Also, it’s a rather amusing picture of myself around one of my favorite pass-times, with a lot of nostalgia on the walls… (A dog long gone, pop stars, a terrific hair cut, badges and medals, Bon Jovi, etc… ah.. the memories…). The Amstrad didn’t have a blue screen of death, it actually had a blue screen of life. With yellow text. Unfortunately blue (and red) was a colour not very suited for television sets, and the blue tended to blur so much it was hard to read blue text or text on a blue background…

Amstrad CPC 6128 - the wonder machine!

Amstrad CPC 6128 - the wonder machine!

I did not set out to learn mathematics on this computer, but it somehow forced itself into my motivation. I remember learning about sines and cosines in order to plot the circumference of a circle. If a teacher have told me this is what I should do, it wouldn’t have been half as fun. I remember learning about slopes in order to draw stars on the screen. This happened several years before sines and slopes entered my syllabus. I also subscribed to this magazine, named Amstrad Action, and there one could find listings of programs in Basic, which could be typed in and saved on floppies or cassettes. (Do you remember the sound of those tapes? You could listen to it, and after perfectioning your ear, you could say just by listening to the signal hiss whether the software was properly loaded or not.) Of course there was no hard drive, but the machine would ship with an enormous 128 Kb of memory. Not quite enough for everybody, according to Bill Gates, but nevertheless – endless possibilities in the eighties! One of the programs I typed in was a short program that would allow you to play with coefficients of quadratics. It would solve the equations and draw the graphs, and this was before we had ever heard of graphic calculators. I felt like I was on the edge of technical evolution… Anyhow, this “insight from within”, has been valuable to me when meeting the quadratics (and other functions) later on, and the Amstrad have also pointed me towards ways of treating my own students and pupils.

I later read Seymour Papert‘s “The Gears of My Childhood“, and things started to clear up a bit… I highly recommend the book “Mindstorms: Children, Computers and Powerful ideas” where I read the mentioned article in the foreword, to anyone interested in the teaching and learning, particularily of mathematics.