Geogebra

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I’ve been trying to get a bit better with geogebra and have started putting some work on geogebra tube. Here is a link to one of my experiments:
Triangle Numbers

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Example vs Proof

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I recently read an old blog post by Kate Nowak.

She discussed a problem I have quite often with my students – getting them to understand the value of math proofs. In particular, I have trouble getting them to understand the difference between an example and a proof.

In the past I have used a silly “false pattern” to make my point. I start the class by writing the following conjecture on the board:

If I double any number, the result will always be less than 100

I then give a lot of examples. This generally gets a number of laughs, and they say it is silly. I then ask them why they do the same thing all the time in class if it is so silly. We discuss how it is actually possible to find infinite examples to support this conjecture, and yet it still isn’t true.

I like the example in Kate’s blog with the circles. However, I feel it is too complicated to really get the point across, just as I feel my example is too simple to really get the point across.

So, what I want to ask anyone who reads this is are there any good examples of the “right” level for 8th or 9th graders?

I want a false pattern where the result is not what it seems, where the first few examples you might try seem to support the idea, but at some point the pattern fails. More importantly, the “correct” pattern should be something accessible to good students with a strong grasp of basic algebra (but preferably not requiring quadratics).

I think these would be good examples to help students understand the value of a proof.

By the way, here is a related fun exercise. What is the next number in this sequence: 1, 2, 3, 4, 5, 6 ?

You can make it whatever you want, but here is a simple option to make it 8 (can you figure out how to make your own function like this? There is an easy trick):

Geogebra image

How much math should we teach?

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I recently read two articles in Swedish newspapers about how much math should be taught in school:

Matematik som huvudämne tillhör dåtiden …

DN Kultur har fel om matematik i skolan …

For those of you who can’t read Swedish, the first one says we should make very little math mandatory, while the next article strongly disagrees and claims that math is more and more important for the future.

I think this is an important topic, especially here in Sweden, and I have some very strong views about it. This circles back to my feelings that the way Sweden strives for fairness in schools has (strävan efter jämlikhet), in fact, the exact opposite result!

It seems very clear to me that as early as 7th grade (if not earlier) there should be two completely separate math tracks, with completely different requirements – one for those students who may actually need math in their future, and one for those students won’t. And just to make everyone happy, there should be made available additional classes later in life for anyone who later changes their minds (this is already available in Sweden through Komvux).

On the one hand, it is ridiculous that I have to try and teach algebra to students who will end up driving a tractor for a job. It is torture for them, torture for me, and a complete waste of everyone’s time, and only increases their feeling that ALL of math has nothing to do with the real world. When they ask me when they will use algebra, I tell them honestly that they will probably never ever find any use for it in the rest of their lives outside of math class, however by the law I have to teach it to them. They are welcome to try and change that law when they grow up and vote.

On the other hand, it is completely unfair that I have teach students who want to go into math or science fields at a snails pace guaranteeing that they will have little chance to compete internationally and knowing that they could be several years ahead of where they are if society was fair enough to give them the opportunity they deserve.

Putting everyone is the same math class based solely on age is just stupid and unproductive. Creating a theoretical and a practical math track would solve a lot of these issues.

And now comes the silly part – many Swedes would say it was not fair to be denying the practical students the chance to study math at a university, that it is maybe even elitist to separate them. However, in reality it is the ones making this stupid argument that are being elitist because they are implying that there is less value in driving a tractor than in working math or science or in some other technical field. That is the true elitist stance.

Although I am a strong believer in two different math tracks, I would not put a different value on one compared to the other. Different people have different interests in life and they are all equally valuable. There are a very large number of jobs that require no math skills at all without which society would fall apart, and I am very appreciative of all the men and women doing those jobs.

What do you think?