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?



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.

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.


100 ideas for teaching mathematics (book tip)

4 November , 2011


One of my favorite authors when it comes to books about mathematics, must be Mike Ollerton. We use some of his books as curriculum on our courses within teacher education. For example, we used Inclusive mathematics on one of our master courses in mathematics education.
Mike Ollerton has written several books, and you can find most of them (I guess) on Amazon and other sites.

This book is just what the title says – it contains 100 starters for mathematics classes. They are more or less grouped by topic, although some activities might fit in everywhere. I have just read through this book, and I must say I found several new tips, activities and tasks that I could and will incorporate into my own lectures at the mathematics education department. I wasn’t able to find the solutions to all the activities as I read along, but I did some, and some where also what I would call classics of mathematics.

You must have a very bad imagination if you don’t find many activities to adopt to your classroom in this book! 🙂

Ollerton’s pedagogical way of thinking is quite clear from seeing these activities. It’s not about givint the students questions and tasks, but rather activities and problems. Some of the ideas  might not even have a specific answer to be found. The activities are also expanded upon by providing hints for how the teacher could take the ideas even further.

I’d like to mention one little tip that my students liked very much. My students arrive by bus mostly, and there are always one or two buses that arrive late, and some students who have to wait a couple of minutes. I then gave each pair of students five die as they arrived, and instructions to throw them all once. The problem is to make use of the five numbers in order to arrive at 100 in one way or another. They can use plus, minus, division, multiplication and parentheses as they like. There appeared to be something within this activity that made them sit there thinking quite hard. Could all throws result in 100? (Of course not, five ones can not be made into 100). How many hundreds can be made? (Well, with 6 to the power of 5 possibilities I doubt that that is easy to find out). Perhaps if we also included powers…

I highly recommend this book for anyone teaching or learning or being interested in mathematics. You can order it quite cheap from Amazon og Play.com

Exploration with diagonal in rectangle

9 August , 2011

I read this in a book by Mike Ollerton some time ago, I think it was his book called “Mathematics teacher handbook”. One of the activities in the first chapters of the book was this one:

Imagine a rectangle on a square grid, say a 9 x 3 rectangle. Draw the diagonal in the rectangle. How many of the squares within the rectangle will be crossed by the diagonal?

It turned out to be a very nice activity, and it can be attacked in different ways, as well as making a good arena for mathematizing and exploring. The aim being, of course, to see the connection between the size of the rectangle and the number of squares that the diagonal passes through. It is not too difficult, and not too easy either and everyone can understand the question.

I made this GeoGebra file to help explore the question. You can also see it embedded in a post on my Norwegian mathematics blog on blogger.com.

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).

Pythagorean spirals

3 June , 2011

I continue to post GeoGebra files, although I can’t embed them properly on WordPress. This time, I post a file I made on Pythagorean spirals.
Some explanation before you click the link to my Norwegian site (with GeoGebra files embedded): Two points are drawn on the canvas. Select the tool on the right side of the menu (the one with the spanner and screwdriver) and then click on the two points, black one first.
Another point is consctructed along with a triangle between the three points. What is the length of the hyoptenuse? You will need to know that the distance between the two original points is 1.

Continue to click two points, first the old black one, and then the new black one. You will have more and more triangles added. What are the hypothenuses?

Here’s the file: http://mattegreier.blogspot.com/2011/06/pytagoreisk-spiral.html


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.

Some links just to show I didn’t forget you…

24 May , 2011

Just a quick “I’m still here”-post…

Thought I’d share three cool posts I found today.

Wired.com doesn’t need any introduction I guess. Sometimes they have science related posts too good to miss, like this one: http://www.wired.com/wiredscience/2011/05/is-the-launch-speed-in-angry-birds-constant/

Lifehacker has grown to be one of my favorite Google Reader items, and I usually find interesting posts here every day. It’s noe necessarily science or math related, but they are often quite geeky, and well worth a look. Here’s a tip on using pencils if you find yourself playing yatzy without enough die. http://lifehacker.com/5804898/use-a-pencil-as-a-6+sided-die

Google sketchup is a cool freeware effort that also can be used in math education. I got this tip from the Math and Multimedia blog: http://mathandmultimedia.com/2011/05/25/intersection-of-planes-google-sketchup/

Microsoft Mathematics 4.0

16 February , 2011

Usually I am not too fond of Microsoft. I spent a lot of time as a student, trying to avoid using MS products. Internet Explorer was almost like foul language in the late nineties, so Netscape it was. There are still things uncomfortable regarding Microsoft software, for instance, a new windows PC is takes ages to prepare for proper use, removing malware and pieces of Windows. By the way, this can be automatized by paying a visit to sites like ninite or pcdecrapifier. Choose what to keep and what to remove, and hit the ground running.

Anyway, I have cleaned up my relationship with MS, I am even buying and paying for Office and Windows these days. Windows 7 is pretty good in my opinion, miles ahead of Windows Vista. But the latest reason I have come to like MS is their mathematics software. On my Norwegian blog, i wrote about Microsoft Office Math Add-in. Anders Sanne and I also wrote (a Norwegian) article on the mathematical writing on computers. Someone liked it, I guess :D. You can find it (what? You don’t understand Norwegian?) here: http://www.caspar.no/tangenten/2009/Gjovik-Sanne-409.pdf (pdf-format). This Word add-in makes it easy to make graphs and solve simple equations as well as simplify expressions. All inside of Word, which is handy. But the use is quite limited, wouldn’t it be nice to have the complete Mathematics package? Would you know it, Microsoft decided to hand out Microsoft Mathematics 4.0 for free. You can download the whole thing here.

You may have noted, I am a huge GeoGebrafan. The next version will include CAS and 3D, as mentioned in an earlier post. Does MSMath4 have anything to offer on top of that? Well, yes and no… The approach is certainly different. Plotting surfaces is easier than in the present beta of GeoGebra, for instance:

It seems MSMath4 is less picky regarding syntax. I just typed the expression in the picture above, without consulting any manual. The insertion field behaves much like the formula editor in Word. MSMath4 formated everything right, interpreted this as a function z(x,y) and offers to plot it.

Quite a few special treats are built in. Unit converter, triangle calculator and a book of formulas.

I tried to enter two equations in two unnowns. MSMath4 not only calculated the answer, but showed me step-by-step solutions. You can also choose which METHOD the software should use for solving the problem. Neat.

A few unfamiliar functions exist. I can choose to enter math by hand, (great for tablets or smartboards) and in the picture below my function is interpreted as mathematics instantly.

Press ENTER and I am offered the oportunity to plot the graph:

If this is my software to go to remains to be seen. I get by perfectly with wxMaxima and GeoGebra. In particular, GeoGebra has a lot of teacher-specific-functions like dynamic sliders and things like that make it perfect for teaching and exploring. Perhaps wxMaxima gets eaten by GeoGebra when the latter releases the CAS+3D version. MSMath4 has a great user interface, so many will probably get to like that prior to other software.