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?

 


Oral Exams

15 June , 2012

Or, the art of writing so ugly that the student doesn’t understand what is written about him.

Hundred students, thousand study credits and a pile of wood from four days of writing.

Bilete


Blog

25 August , 2011

Just thought I’d recommend a blog I keep coming back to. Larry Cuban is an American (I think!) author and researcher. I have only read one of his books, Oversold and underused: Computers in the classroom. He writes quite a lot on his blog, and it is interesting stuff indeed. Have a look yourself: http://larrycuban.wordpress.com/


Learning – one of our favorite activities.

11 August , 2011

LEARN from Rick Mereki on Vimeo.


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).


Logicomix

27 May , 2011

I could be better at writing about, or ranting about, books I read, especially when they are connected to science, teaching or research.
The other day I read a comic book called Logicomix. This may seem unserious, and no doubt, comics and comic books are considered more childish in Norway, than in, say, UK or US. At least that is my impression. As a young collector I have always been into comics and even in this country there have been, at least a few, to share the interest with.

Picture from the Logicomix website

Picture from the Logicomix website

This book is a kind of drawn version of a lecture given by Bertrand Russell, about his life, philosophies and pursue of absolute truth. What on earth (or rather, elsewhere..) could we know, and how can we be sure? What can we be absolute sure of? The excuse for the lecture is a contemporary debate on whether US should join in the war on the English side. How can we be sure we do the absolute correct thing? In this book, we get to know Russell, his life and story, his more or less strange love stories, some of his paradoxes and contradictions, as well as a number of other mathematicians and philosophers. Frege, Wittgenstein, Cantor and Hilbert all show their faces.
This is a reading experience a little bit on the side, and I wholeheartedly recommend it to anyone with the slightest of interest for either mathematics, history, logics or philosophy. Or comic books 🙂

The book has its own website, and you can get some tastes of it there. It can also be ordered (free shipping) on play.com – and cheap! Just what the cheap researcher ordered.


Panoramas

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!


Toilet Thoughts on Learning

5 September , 2009
https://i2.wp.com/farm2.static.flickr.com/1440/984517607_4f8487dde3.jpg

Toilet thinking

I remember once, when attending confirmation training, we were forced (mildly, I should add) to learn the ten commandments by heart. Me being godless already at the age of thirteen, I thought this was a rather meaningless activity, but played along just to please grandparents and others. I digress – the point not being my own attitude towards the ten laws noone is capable of living by, but rather how the priest wanted us to learn them.

Take this cheat sheet with you, and sit in the bathroom, preferably in the toilet.

As absurd as the ten commandments appeared to me, this last statement proved to be much more vital to me. The priest’s words making a deeper impact than any god’s.

And it works. I can’t think of any better quality study time than the lonesome toilet scenario.

There are, of course, a lot of authors who have appreciated the toilet serenity. Even the guys at MAD magazine have their own Bathroom Companion (the turd in the series). Another favorite of mine is the Great American Bathroom Book, or GABB. In three volumes, they gather single-page (single sitting) summaries of all time best selling books.

At work, I have started the secret toilet-exercise-tournament. I print out A4-sized pages with a mathematics problem printed in large lettering on it. PowerPoint is a nice and easy way of making these poster pages. I am thinking of laminating them, in order to… you know, avoid incidents.
The current problem is this one (I think I read this in a lovely little book by Mike Ollerton, perhaps it was 100 ideas for teaching mathematics):

On a 2×2 grid of dots, you can draw one quadrilateral only. The square. How many quadrilaterals can be drawn on a 3×3 grid of dots?

I have so far just started to deploy these sheets on the toilets, so the ideas keep coming. Perhaps the exercises or problems could be more toilet-oriented (“How many sheets of paper…”,  “what will the radius of the paper holder be…”, “How big is the proportion of people who prefer the toilet paper end to hang on the inside instead of on the outside” etc…)

To be kind to the toilet-goers, you could consider leaving a stash of post-it notes and a pencil available.  Or make a bigger competition out of it; Stick problems on ten toilets in the school, who will be the first one to solve them all…

Suddenly the character “Shitbreak” from American Pie sprung to mind, so perhaps all these toilet exercises will be too weird for a lot of people, I don’t know. Right now it seems like a fun thing to do. If not THE right thing to do.

I will appreciate any suggestions for toilet exercises in the comments. (Pictures are from the flickrCC site). Have a nice weekend!


How much is one million (giraffes)?

12 June , 2009

I just flipped through a book called “How much is a million” (and later found out- others exist with a similar topic).
The points of the book(s) are to visualize to children how they can picture the size of a million objects of some sort.

My previous and current students and pupils more often than not amuse me with creativity and humor. Yesterday, a previous mathematics pupil of mine, Ola, seem to have fallen right into the trap of NOT being able to picture one million objects! 🙂 (…and who can blame him) I really hope he can prove me wrong so that all those mathematics lessons don’t turn out to have been useless, and perhaps he will! He made a bet with a friend, claiming he can collect a million hand drawn pictures of giraffes in around two years time. At the moment of writing he has gotten a few thousand drawings, have published a few hundred of them and has almost six hundred days to collect the rest of them. Being the kind and helpful teacher that I am (or was) I have to help trying to promote his project in several ways:
– I force all my family and friends to send him giraffes
– I promote him on my twitter feed
– I write this blog and cross-post it to my other blogs and facebook

This is the giraffe I sent Ola

This is the giraffe I sent Ola

Other helpful tips for Ola:
– use the next year’s Earth hour for all that it’s worth. The earth needs a giraffe hour.
– I will also try to make all my new students next year draw giraffes of course… This year students faced a similar task, trying to draw a mathematics teacher the way they picture him (I say him, because around 95 % are male. The class thought was pretty strange, because in teacher education, it seems around 90 % are female!)
– How about trying to have giraffe drawing as an exercise in kindergardens, pre-school, primary school (and upper secondary schools and higher education!) all over the world?
– Publishing on the Internet is one thing (or rather, one TEDIOUS thing…), why not make a “Best of the giraffes” and turn it into a book? Or t-shirts… (Next project onemillionmousemats.com ?) I’d buy a handpicked-by-Ola giraffe t-shirt anytime.
– Let people comment on each others drawings on the web site. Maybe vote for the best giraffes.
– Get interviews from those who made the bet
– Visit the Kristiansand Zoo for inspiration and giraffe info (this reminds me of Monkey news, from the Ricky Gervais show)
– Schedule interviews with Norwegian TV for children (Portveien 2, a famous children program from the 80-90’s?)
– Get permalinks, so people proudly can display “I’m a part of the onemilliongiraffes project!” on their blogs…
– Ask celebrities to send you their giraffes (some have already done that!)

 

So, head over to Onemilliongiraffes.com and upload/mms/send/mail/e-mail Ola your giraffe pictures! You can also follow the project on twitter. And while you are at it, you can listen to giraffe music on Spotify, or draw giraffes on Shidonni, that also will walk about and eat the food you draw for them…

*EDIT* Here’s one of several newspaper articles on his project: Aftenbladet. And I have lost count of his television appearances and other spots…