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March 2013 Archives

 On the door of her office, Alison Campbell has a sign that says "the biggest factor in learning is what the learner already knows". Or something like that. In other words, students build upon an existing foundation when they make sense of the world. This can be very helpful, or very unhelpful, depending on whether that foundation is correctly laid. One of the roles of a teacher (I would say one of the hardest roles of a teacher) is to identify where there are cracks in those foundations before the student starts building too much upon it. If that doesn't happen, the student is likely to run into something that just doesn't fit with what he or she already knows, or thinks she already knows, and it's going to cause them problems. 

It's not what you don't know that hurts you, it's what you know that ain't so

as the saying goes. (I haven't been able to track down the origin of this quote - some attribute it to Mark Twain but I think others disagree. Can anyone help?)

I ran into an example of this while marking some third year assignments this week. Students were tackling a problem in which a mechanism was rotating. Now, to the large majority of the class, it is clearly evident that something of mass m moving in a circle of radius r at speed v has a  force on it of m times v squared divided by r. They did it till they were sick of it at school, and it has stuck. The problem is, that they are WRONG. What!, you cry, but that's correct isn't it? F=mv2/r for circular motion. Yes, but ONLY when the speed v is CONSTANT. If it isn't, there's another term to consider. 

Now, the interesting thing from my point of view is that I though that the students were over this misconception. I'd talked about it in class, and even done a couple of formative multiple choice questions with them in lecture time that showed me (so I thought) that they appreciated this subtlety. But, when the pressure was on with an assignment, a great many students abandoned what we'd talked about in class and reverted back to what they perceived as their foundation knowledge. 

So what went wrong with my formative assessment? Did it come too early after the discussions on the subject? Were my multiple choice questions just poorly written. I need to go back and have a rethink on this. At least I've identified that a problem still remains, via the assignment, which is a good thing as it gives some time to talk about the issue again before the exam (and before the students leave and try applying F=mv2/r inappropriately in the design of some safety-critical piece of machinery.)

 

 

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 A friend has just started a Bachelor of Arts degree here at Waikato. As part of her first year study, she's chosen to do a Philosophy paper. Apparently, one of the questions that has been posed, is "Is maths real?". 

Well, what is real? You certainly can't put 'maths' in a box and give it to someone. like you could with a chair or a chicken. But does it have more substance than just some made-up statements about how to add things or describing how large angles are?  I've often wondered, for example, whether it would be possible to have a universe in which the value of pi was four. In our universe, it isn't. But is pi, the ratio of the circumference of a circle to its diameter, necessarily 3.14159265...in all universes?  I don't know. I guess it depends on what a circle is. 

So is physics real?  I would think that it is more real than maths. I mean, physics is supposed to describe reality. Gravity is gravity. Objects attract each other proportional to their masses and inversely proportional to the distance between them. That's what happens. It's hard to be a physicist if you're not a positivist, or at least have strong positivist leanings. In other words, if you don't believe that there is a real world out there that we can know about, and that we can find out about this world objectively (e.g. by doing experiments), you are going to struggle to be a physicist. (It is true that quantum theory throws a spanner in the works at this point. The quantum world is weird indeed -an example here - and raises big issues about what is real.) 

Recently, I was at a teaching seminar that seemed to be populated mostly be social scientists. In social science, a common paradigm is social constructivism, or namby-pamby waffle as it is known by positivists. In social constructivism, what the world is is constructed in one's mind, and that what you can find out about it inevitably depends upon how go about finding out about it. In other words, everything is relative. 

So, back to the point. Where does maths sit in all of this? I'm not sure it does. It's hard to believe that maths depends on your point of view. It doesn't matter how you look at it, 1 + 2 doesn't equal 4, and it never will. But neither does it fit well with reality, either. 1 + 2 would be 4 in any universe, wouldn't it?  Maybe maths sits in some strange space of its own, separate from ties with this universe but not 'made up' in anyone's head. So what is it? Er, that's getting too close to philosophy for my liking.  I await my friend's response with interest. 

Perhaps the best answer is to say that maths is as real as philosophy. 

 

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 A few weeks ago we had a small, informal competition in the department - guess the maximum gradient on one of the roads on campus.  I think the motivation for this this small stretch of hill (or what passes for a hill here in Hamilton) was going to be used as part of a dynamics experiment, and so one of our technicians was about to go out and measure the gradient.

I'm happy to say that I won the competition, without even going out to the road and looking at it carefully. The prize was simply to feel smug. I predicted a maximum gradient of 9.5 degrees; I think from memory the measured gradient was 10.1. Being a physicist, I estimated rather than guessed. I simply thought "What is the average gradient?". This wasn't too difficult. Thinking about how the buildings are laid out on campus, I thought about how many floors that the road drops by. That gave me an estimate of the drop distance.  Then I compared it in my head to the length of the swimming pool to estimate the length of that stretch of road. Divide the former by the latter, take the inverse tangent, and I get the average gradient in terms of an angle.

Then came the bit that was rather more vague. I needed the maximum gradient, but had the average. Clearly the maximum is higher than the average. So to go from one to the other, I need to multiply by a number that's bigger than 1.  So I picked 2.  Though even that wasn't a wild guess. The road starts off level, and ends level, so if we assume it gains gradient uniformly then loses it uniformly, the maximum gradient will be about double the average. 

Perhaps I shouldn't have been all that surprised that I was very close. The point is that I estimated rather than guessed. It's a skill that is very important in physics, but it's one that often gets overlooked during teaching. The difference is that an estimate is based on what we do know about the situation - even if it's only approximate knowledge - rather than a guess which is simply a number plucked out of the air. 

Some fun things to get students to estimate include the number of carbon atoms that are worn off the soles of their shoes during a day's wear and the mass of the building they are sitting in. 

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 I've had my brother visiting from the UK, which has been a good excuse for doing some of the touristy things in the area. I wasn't taken by the prospect of zorbing, but we did give blackwater rafting a go in the Ruakuri cave at Waitomo. I've always wanted a go at that - and it's strange how you can have something a touch less than an hour's drive from you and you don't do it.  But that's been corrected now. Essentially, the concept isn't desperately complicated: you sit in a truck inner-tube and float down the river - the complicating factor being that the river is underground. 

I was surprised how much water was flowing through the cave, given the lack of rain in recent weeks. The guides were saying that it's low, but it was still plenty enough for a rafting exercise. One highlight, if it can be called that when in the dark, is the waterfall jump - when you jump backwards off the top of a small (2 metres?) waterfall and land in your ring in the pool below. Just what exactly happens on the impact of an adult-laden inner tube flat with the water I'm not sure - it being rather dark, but it certainly was wet. Having done the jump, we accumulated further down the cave and got very splashed by those still jumping, so I'd say there was some considerable water displacement  going on.

As a physicist, one is obliged to do some estimates of this. What is going on here? The falling person needs to be brought to a halt by the buoyancy force of the ring in the water. That involves working out how much water is being displaced by the ring and perhaps doing a bit of calculus. It's rather complicated because of the ring's shape - things would be rather simpler if the ring were a cube. With a bit more time - maybe in a later post, I'll have a go at estimating this, but I have other more pressing commitments like lectures to give.  

So for now I'll just comment that next time I'll be taking a tape measure and high speed low light camera into the cave.

 

 

 

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 I've been perusing YouTube looking for good videos illustrating wave packets - which are bunches of waves containing different wavelengths. I want to come up with a good illustration for a second year physics paper on introductory quantum theory. This contains a lot of 'wave' things.

Here's a nice one I've stumbled on. It shows dispersion in water waves and gives you a bit of a physics tutorial as well.  The key thing is that waves of different wavelengths (the distance from the peak of one wave to the peak of the next) move at different speeds. That means, when you throw a stone in a pond, the waves spread out. Some reach shore very quickly - these are long wavelength waves - some take much longer - the shorter wavelength ones. So, while the initial splash is always a bit of mess, wait a few seconds for the longer wavelengths to get ahead of the shorter ones, and things look rather tidier. 

In a larger-scale example, this is why ocean waves coming onshore have a very uniform nature to them - the gap between each successive wave hitting the beach is about the same if you watch over the course of only a few minutes. These waves have typically travelled a long way from a storm, and they have separated out into particular wavelengths. 

Next time you happen to throw something into a pond, watch the waves. Or watch the waves hitting a beach after a storm. 

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