Mathematics Teacher Article

7 September , 2012

It was quite fun for me, for the first time reading my writings in an American journal. I have written about the bird tetrahedron in Norwegian in Tangenten (2005) and I recently elaborated on this topic and expanded the original article into a new one for MT. You can read the English version on jstor or the august issue of Mathematics Teacher from NCTM. I loved the way they made those nice graphics and photos for the front and illustrations! There are also some templates you download in PDF from the Mathematics Teacher website.

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 (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 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:


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


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 – and cheap! Just what the cheap researcher ordered.

Reference software

14 March , 2009

This is so hard. I am talking about committing yourself to a database of some sort, and this time it’s the reference database I am worrying about.
If you collect a lot of music or movies and catalog them extensively, you know what a drag it is when you figure you have chosen the wrong database or cataloging software. (So far I have been pleased with Mediamonkey ( for the music and it seems XBMC ( will do the trick for flipping through your movies with style. But I digress.
Now I want to catalog, sort and search my references, so what software should I commit to? I used to be an EndNote user as long as I got it for free, but no more. I had to look for free options (I AM the cheap researcher). First to come along was zotero (, and believe me, it is still excellent. It is a Firefox add-on which allows you to add references from bookstores and article bases with ease. It then integrates with Word for easy use when referencing in your writing. The coming (now in beta) version includes sync to web, so that you have your references at hand at all times. Great. Converting from EndNote was a nightmare, but most of my thousands of references went along with the swap after some tinkering.
Then I got Word 2007, which had a built-in reference manager. This might have been the best option, but I found the managing capabilities a bit wanting.
Finally I have tried Mendeley (, a manager with a sync’ed software and web version, with capabilities to store and extract information from pdf-files.

But what to choose? This is almost a lifetime commitment (ok, it is not, but it would be nice if it was) so it’s important to chose right! Heeeelp….

Zotero: The Next-Generation Research Tool

6 January , 2009

Zotero: The Next-Generation Research Tool

One of the tools I have began to love the most is this free add-on for FireFox. I used to be a fan of EndNote, which is a great program, but it wasn’t free… And I AM the CHEAP researcher after all.

Zotero allows you to store references in a database, and later on insert the references into, say, Microsoft Word. Inside Word, you look up the reference you want and automagically a bibliography is created at the back. There’s a great little feature that you can store files with it, so that the pdf’s you refer to are in the right place. Another extremly handy function is the way you can get references into the database. Whenever you are on a library site or a bookstore site, you get a zotero icon next to the URL. ‘Click’, and the reference is added to your database. Great. The coming version of zotero will support external cloud storage of references, so you can reach your references from whatever computer you are on. At present the best way might be to keep the zotero files in a folder that you sync with DropBox, mentioned earlier on my blog.

Try this add-on for yourself, and see if you are satisfied!

A practical approach to GeoGebra

13 September , 2008

Just attended the 4th European Workshop on Mathematical & Scientific e-Contents here at NTNU in Trondheim. We had a short overview on our Practical pedagogy for GeoGebra. The abstract can be found in

Amdal, A., Gjøvik, Ø. (2008). A practical pedagogy for GeoGebra. In Amdal, A., et. al., Book of Abstracts, 4th European Workshop on Mathematical & Scientific e-Contents, PPU-serien, Programme for Teacher Education, NTNU