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?

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.

(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

Correlation coefficient on GeoGebra

26 January , 2011

I was asked a question about the correlation coefficient. On calculators, this is the r-value that may pop up when you have done a regression. If this value is close to -1 or 1, the data points in question provide a good fit to the proposed equation. Actually, this r-value only applies to linear regressions, it is only defined there.

Someone said that that was not correct, seeing as the r-value also occurs on logistic or exponential regression. Actually, these are also linear regressions, they are transposed into linear relationships before regression analysis is carried out.

Anyway, now that calculators are too old, and Excel is too expensive, how do we carry out such a regression with the correlation coefficient also calculated? The answer (again) is GeoGebra. OK, to be fair, OpenOffice could probably just as well be used for this, but since I am a GeoGebra fan…

In the video below I enter some data points, do the regression and find the correlation coefficient. The menus are in Norwegian, but I am sure you can find the right commands in whatever language you want. You can easily change language in GeoGebra from the settings menu.

Yet another GeoGebra

18 January , 2011

Several versions of GeoGebra are planned launched, and you can try some of them now in their beta stages. I have mentioned GeoGebra(prim) earlier on my Norwegian blog – a scaled down, simplified version intended for primary school use. I also suspect there’s a SMARTboard version in the making (although you can make your own version for the SMARTboard by making points and text much larger!) Yet another version is what will become GeoGebra 5.0 (ok, I know we haven’t reached 4.0 yet, but let’s not get into details…), GeoGebra 3D.

A lot seems familiar at first glance, but a couple of new entries can be seen on the toolbar.
CAS is mentioned in the Norwegian curriculum, and up until now, the only free and rather easy alternative is wxMaxima.

At least two new choices of “mode” has turned up. In the picture above you see the CAS mode. This is a part of GeoGebra that can be used for things like solving equations algebraically and simplifying expressions. I solved a quadratic in the screenshot, and I had no idea about the syntax beforehand. It turned out it was similar to most other CAS’s , for instance, wxMaxima, Texas Instruments-calculators and others.

Not many 3D software systems intended for school use exists to my knowledge. One alternative is Google Sketchup. One option in Google Sketchup is to “drag” areas upward to turn them into prisms. This is also an option in GeoGebra 3D, as you can see in the screenshot.
You can download the test version with 3D and CAS here: http://www.geogebra.org/forum/viewtopic.php?f=52&t=19846