Did you ever notice…

28 May , 2015

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.

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

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.

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

Believe

30 September , 2009

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

The problem with PowerPoint

19 August , 2009

From BBC News (shouldn’t this be rather old news?) comes a nice look into the 25th anniversary of PowerPoint.

The semester just started, and soon the freshmen and later-hopefully-to-be teachers are pouring in by the hundreds. Sitting on the lawn, knocking on our office door, asking good and dumb questions (they are human beings, almost like us!) and equipped with volleyballs and beer cans. Ready to take on the world.

These students are approx. 19 yrs old. They are to become teachers in four yrs time. They haven’t lived a day of their lives without PowerPoint existing in their world. And still we find teachers trying to pretend the whole PowerPoint thing never really caught on. (Hard to argue with the number of presentations held every minute…)

Personally, as a mathematics lecturer, I tend to appreciate that linear thinking is reeeeeeeeeeeally not the way stuff happens inside our brains. Textbooks and teachers try to persuade us to think this is the way mathematics came about. Not so. Hence, I have gotten into the habbit of having a slide no.1 stating just that “Where do we go from here”, equipped with a mindmap or something similar.

So Happy Birthday PowerPoint. May you NOT try to change our thinking in linear, often bad, ways, but rather help us convey ideas in meaningful and creative ways.

Visions for the future

18 July , 2009

So, what’s in store for the future?

1. Free e-book readers for all students. The Kindle might not be the best piece of technology available, but it would be very nice for education.
First, we won’t have the forgotten books. We WILL of course get forgotten e-books, but the chances of remembering one gadget must be greater than remembering several books.
Second, these machines are more powerful than books. Animations in the texts is one example. I guess they will evolve more, perhaps including more note-taking and interactivity.

2. New skills in the curriculum. Blogs, portfolios and wikis will replace or at least supplement a lot of traditional writing, linear or otherwise. An extebded notion of “texts” will develop and surpass the now traditional, recently modern, hypertexts.

3. Now standard tools will vanish. The logarithm tables are long gone. Next will be calculators, compasses and rulers. We will see a few new tools in the mathematics classrooms. GPS, digital measurment tools and dynamic softwares are well known. Perhaps these will be more common in the near future mathematics classrooms.

I hope I can elaborate on this post later. For now I will publish it just to Get it out of my cumbersome iPod…