Archive for problem solving

Three Tips for Solving Physics Problems

Posted in Cute Problems, Education with tags , , , , , on November 2, 2012 by telescoper

I spent quite some time this morning going over some coursework problems with my second-year Physics class. It’s quite a big course – about 100 students take it – but I mark all the coursework myself so as to get a picture of what  the students are finding easy and what difficult. After returning the marked scripts I then go through general matters arising with them, as well as making the solutions available on our on-line system called Learning Central.

Anyway, this morning I decided to devote quite a bit of time to some tips about how to tackle physics problems, not only in terms of how to solve them but also how to present the answer in an appropriate way.

I began with the Feynman algorithm for solving physics problems:

  1. Write down the problem.
  2. Think very hard.
  3. Write down the answer.

That may seem either arrogant or facetious, or just a bit of a joke, but that’s really just the middle bit. Feynman’s advice on points 1 and 3 is absolutely spot on and worth repeating many times to an audience of physics students.

I’m a throwback to an older style of school education when the approach to solving unseen mathematical or scientific problems was emphasized much more than it is now. Nowadays much more detailed instructions are given in School examinations than in my day, often to the extent that students  are only required to fill in blanks in a solution that has already been mapped out.

I find that many, particularly first-year, students struggle when confronted with a problem with nothing but a blank sheet of paper to write the solution on. The biggest problem we face in physics education, in my view, is not the lack of mathematical skill or background scientific knowledge needed to perform calculations, but a lack of experience of how to set the problem up in the first place and a consequent uncertainty about, or even fear of, how to start. I call this “blank paper syndrome”.

In this context, Feynman’s advice is the key to the first step of solving a problem. When I give tips to students I usually make the first step a bit more general, however. It’s important to read the question too.

The middle step is more difficult and often relies on flair or the ability to engage in lateral thinking, which some people do more easily than others, but that does not mean it can’t be nurtured.  The key part is to look at what you wrote down in the first step, and then apply your little grey cells to teasing out – with the aid of your physics knowledge – things that can lead you to the answer, perhaps via some intermediate quantities not given directly in the question. This is the part where some students get stuck and what one often finds is an impenetrable jumble of mathematical symbols  swirling around randomly on the page.

Everyone gets stuck sometimes, but you can do yourself a big favour by at least putting some words in amongst the algebra to explain what it is you were attempting to do. That way, even if you get it wrong, you can be given some credit for having an idea of what direction you were thinking of travelling.

The last of Feynman’s steps  is also important. I lost count of the coursework attempts I marked this week in which the student got almost to the end, but didn’t finish with a clear statement of the answer to the question posed and just left a formula dangling.  Perhaps it’s because the students might have forgotten what they started out trying to do, but it seems very curious to me to get so far into a solution without making absolutely sure you score the points.  IHaving done all the hard work, you should learn to savour the finale in which you write “Therefore the answer is…” or “This proves the required result”. Scripts that don’t do this are like detective stories missing the last few pages in which the name of the murderer is finally revealed.

So, putting all these together, here are the three tips I gave to my undergraduate students this morning.

  1. Read the question! Some solutions were to problems other than that which was posed. Make sure you read the question carefully. A good habit to get into is first to translate everything given in the question into mathematical form and define any variables you need right at the outset. Also drawing a diagram helps a lot in visualizing the situation, especially helping to elucidate any relevant symmetries.
  2. Remember to explain your reasoning when doing a mathematical solution. Sometimes it is very difficult to understand what you’re trying to do from the maths alone, which makes it difficult to give partial credit if you are trying to the right thing but just make, e.g., a sign error.
  3.  Finish your solution appropriately by stating the answer clearly (and, where relevant, in correct units). Do not let your solution fizzle out – make sure the marker knows you have reached the end and that you have done what was requested.

There are other tips I might add – such as checking answers by doing the numerical parts at least twice on your calculator and thinking about whether the order-of-magnitude of the answer is physically reasonable – but these are minor compared to the overall strategy.

And another thing is not to be discouraged if you find physics problems difficult. Never give up without a fight. It’s only by trying difficult things that you can improve your ability by learning from your mistakes. It’s not the job of a physics lecturer to make physics seem easy but to encourage you to believe that you can do things that are difficult.

So anyway that’s my bit of “reflective practice” for the day. I’m sure there’ll be other folk reading this who have other tips for solving mathematical and scientific problems, in which case feel free to add them through the comments box.

More Order-of-Magnitude Physics

Posted in Cute Problems, Education with tags , , , on June 27, 2012 by telescoper

A very busy day today so I thought I’d just do a quick post to give you a chance to test your brains with some more order-of-magnitude physics problems. I like using these in classes because they get people thinking about the physics behind problems without getting too bogged down in or turned off by complicated mathematics. If there’s any information missing that you need to solve the problem, make an order-of-magnitude estimate!

Give  order of magnitude answers to the following questions:

  1. What is the tension in a violin string?
  2. By how much would the temperature of the Earth increase if all its rotational energy were converted to heat?
  3. What fraction of the Earth’s water is in its atmosphere?
  4. How much brighter is sunlight than moonlight?
  5. What is the mass of water in a soap bubble?

There’s no prize involved, but feel free to post answers through the comments box. It would be helpful if you explained a  bit about how you arrived at your answer!

Order-of-magnitude Physics

Posted in Cute Problems, Education with tags , , , on November 14, 2011 by telescoper

A very busy day today so I thought I’d wind down by giving you a chance to test your brains with some order-of-magnitude physics problems. I like using these in classes because they get people thinking about the physics behind problems without getting too bogged down in or turned off by complicated mathematics. I’ve also kept some of these in archaic units just to annoy people who can only do things in the SI system. I think it’s good to practice swapping between systems, especially for us astro-types who use all kinds of bizarre units, so if you don’t know the units, look them up! And if there’s any information missing that you need to solve the problem, make an order-of-magnitude estimate!

Give  order of magnitude answers to the following questions:

  1. What is the mass of a body whose weight is equivalent to the total force exerted by a 40 mph gale on the side of a house 40 ft long and 20 ft high? Express your answer in tons.
  2. What is the power required to keep in the air a helicopter of mass 500 kg whose blades are 3m long? Express your answer in kilowatts.
  3. The base of the Great Pyramid  is 750 ft square and its  height is 500ft. How much work was done building it?  Express your answer in Joules.
  4. How high would the jet of a fountain reach if it were aimed vertically up and supplied by a water main in which the pressure is 3 atmospheres? Express your answer in feet.

There’s no prize involved, but feel free to post answers through the comments box. It would be helpful if you explained a  bit about how you arrived at your answer!

 

Whatever happened to Euclid?

Posted in Education, The Universe and Stuff with tags , , , , , , on May 24, 2011 by telescoper

An interesting article on the BBC website about the innate nature of our understanding of geometry reminded me that I have been meaning to post something about the importance of geometry in mathematics education – and, more accurately, the damaging consequences of the lack of geometry in the modern curriculum.

When I was a lad – yes, it’s one of those tedious posts about how things were better in the old days – we grammar school kids spent a disproportionate amount of time learning geometry in pretty much the way it has been taught since the days of Euclid. In fact, I still have a copy of the classic Hall & Stevens textbook based on Euclid’s Elements, from which I scanned the proof shown below (after checking that it’s now out of copyright).

This, Proposition 5 of Book I of the Elements, is in fact quite a famous proof known as the Pons Asinorum:

The old-fashioned way we learned geometry required us to prove all kinds of bizarre theorems concerning the shapes and sizes of triangles and parallelograms, properties of chords intersecting circles, angles subtended by various things, tangents to circles, and so on and so forth. Although I still remember various interesting results I had to prove way back then – such as the fact that the angle subtended by a chord at the centre of a circle is twice that subtended at the circumference (Book III, Proposition 20) – I haven’t actually used many of them since. The one notable exception I can think of is Pythagoras’ Theorem (Book I, Proposition 47), which is of course extremely useful in many branches of physics.

The apparent irrelevance of most of the theorems one was required to prove is no doubt the reason why “modern” high school mathematics syllabuses have ditched this formal approach to geometry. I think this was a big mistake. The bottom line in a geometrical proof is not what’s important – it’s how you get there. In particular, it’s learning how to structure a mathematical argument.

That goes not only for proving theorems, but also for solving problems; many of Euclid’s propositions are problems rather than theorems, in fact. I remember well being taught to end the proof of a theorem with QED (Quod Erat Demonstrandum; “which was to be proved”) but end the solution of a problem with QEF (Quod Erat Faciendum; “which was to be done”).

You can see what I mean by looking at the Pons Asinorum, which is a very simple theorem to prove but which illustrates the general structure:

  1. GIVEN
  2. TO PROVE
  3. CONSTRUCTION
  4. PROOF

When you have completed many geometrical proofs this way it becomes second nature to confront any  problem in mathematics (or physics) by first writing down what is given (or can be assumed), often including the drawing of a diagram. These are key ingredients of a successful problem solving strategy. Next you have to understand precisely what you need to prove, so write that down too. It seems trivial, but writing things down on paper really does help. Not all theorems require a “construction”, and that’s usually the bit where ingenuity comes in so is more difficult. However, the “proof” then follows as a series of logical deductions, with reference to earlier (proved) propositions given in the margin.

This structure carries over perfectly well to problems involving algebra or calculus (or even non-Euclidean geometry) but I think classical geometry provides the ideal context to learn it because it involves visual as well as symbolic logic – it’s not just abstract reasoning in that compasses, rulers and protractors can help you!

I don’t think it’s a particular problem for universites that relatively few students know how to prove the perpendicular bisector theorem, but it definitely is a problem that so many have no idea what a mathemetical proof should look like.

Come back Euclid, all is forgiven!

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