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December 2014 Archives

In the last couple of weeks, I've been using Hermite Polynomials in my work. I won't go into what they are (look them up here if you like) suffice to say that they are one of many contributions to mathematics from Charles Hermite (1822-1901), who was himself one of  many french mathematicians whose work has laid a foundation for much of modern theoretical physics. A physicist would generally know these polynomials (when modulated by a gaussian function ) as the solutions to the 1-dimensional quantum harmonic oscillator, although that's not why I'm using them. 

The 1-dimensional quantum harmonic oscillator problem is a textbook problem that gets inflicted on generations of students. I remembering suffering the algebra that went with it. At the University of Waikato, we save our second year students the algebra by just talking about the solutions, but then spring it on them in third year. For those who like that kind of thing, it's an interesting analysis, but for those that don't, it really is quite horrible. 

Perhaps that is what motivated Paul Dirac to come up with (in my opinion) a really elegant complementary approach to solving the 1-dimensional quantum harmonic oscillator problem. While his approach is easily found in text-books, what I haven't been able to track down is a description of how he came up with it. The same seems to be true of many of the analyses that get wheeled out to students. While they look clean and tidy when presented now, I'm left with the question "How did they come up with this?". That tends to be overlooked in favour of the end product. Did Dirac spend weeks pondering over this, thinking "there must be a better approach - the symmetry between p and x in this equation should surely be exploitable somehow...", was it a sudden revelation, did he try twenty different approaches till something worked, or what? My text books don't say. 

What Dirac did was to reformulate the problem in terms of 'raising' and 'lowering' operators. He realized the problem as a ladder of energy-levels, and showed rather elegantly that these energy levels were equally spaced. Moreover, some rather neat operators, that he defined, could move a quantum state 'up' or 'down' the ladder. That's a very creative way of looking at the problem, and has been taken much further since then. For example, when analyzing problems with many electrons (which generally means just about anything electronic) we can now formulate the problem in terms of operators that create and destroy electrons. Whether electrons really are being created and destroyed is a moot point, but the formulation is a neat one that helps us to analyze what is going on. Theoretical physicists consider it a really useful 'tool' of the trade, even though the history behind its construction tends to be overlooked when we teach it. 

So what is the point of me telling you this? Well, it's about teaching. Just how do you teach creativity, especially in something that is, on the face of it, as tedious as physics. Physics isn't actually tedious (if it were I wouldn't be sitting here writing this) but we do tend to make it unnecessarily so at times. I wonder whether that's because that's the easiest path to take for undergraduate teaching. At PhD level and beyond, there's some really creative research going on, but do our undergraduates really see this? Likewise, from what I've seen at school science fairs, there's some great creativity at primary and intermediate school level, but that then vanishes late in secondary school in favour of 'content'. Somehow, we tend to smother out creativity and elegance in favour of 'something-that-gets-the-job-done.'  But truly great physicists, Dirac included, have never 'just-got-the-job-done'. 

Open-ended projects are a way to go (and we manage to some extent to do this with our engineering students), but, as many readers know, we run into trouble with time, the need to prepare students for exams, fitting in with timetabling requirements, and so forth. The problem may go much deeper than we think - indeed, does the whole secondary and tertiary education structure smother-out creativity from students (at least in physics)? 

And with that, have a creative Christmas, and Happy New Year to you all!  I'll be heading southward next week to the Canterbury hills - a part of the country I haven't been to before. 



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I've been reading a paper by Majorie Darrah and others (full reference below) on the use of 'virtual labs' in Undergraduate Physics. At Waikato (along with lots of other universities) our first year physics students carry out laboratory sessions to help them learn physics concepts and practical skills. If you are someone who has run a first-year laboratory class, you'll be well aware that these things are costly and time-consuming. If they're not done well, they become an expensive way of wasting everyone's time. 

Recently, there's been a lot of work on 'virtual' laboratories. These are laboratory sessions that aren't 'hands-on', but simulated on a computer. There are some pretty sophisticated ones out there. At our last NZ Institute of Physics conference, David Sokoloff, one of our keynote speakers, talked about some of these. The computer software allows a student to do pretty-much whatever would be done in a laboratory, but without the university having to purchase, set-up and maintain the expensive equipment. (And, from the student's perspective, they are not constrained as to when they carry out the 'lab'). 

So, do they work? I don't mean does the software work, but does the virtual lab give the same benefit to the student as undertaking a real lab. In other words, do the students achieve the same learning outcomes? To test this, Darrah and her colleagues worked with 224 students at two universities. They were put into three groups - one group did the traditional hands-on labs in a laboratory, one group did the hands-on labs AND the virtual labs, and the third group did just the virtual labs. Their learning was tested with a quiz after the lab , an assessment of the student's written lab report, and  tests. 

So what was the result? They found no difference between the groups. One of the universities conveniently carried out a test assessment both before and after the lab sessions and found that all groups improved as a result of the labs, be they real or virtual. That is certainly an encouraging result for the likes of Sokoloff, and those budget-pressed universities with lots of students to push through first year university physics. The problem of doing laboratory work has been one of the reasons why MOOCs for science and engineering have been fairly slow to get going, However, it may be reasonable to do away with this, if good virtual labs can be prodcued. 

But, there is a but. It's a big but in my opinion, and one that, surprisingly, the authors fail to comment on. Their post-lab assessment of learning was based on a written test of the physics theory concepts that were covered in the lab. In other words, they were testing how well the laboratory (real or virtual) supplemented the teaching of physics theory done in lectures and elsewhere. What they weren't testing were practical laboratory skills (e.g. how to wire up a circuit, track down problems with the apparatus, carry out experiments in a controlled manner, etc.) These are all important skills for a physicist. If universities as a whole shifted towards virutal labs in first year, where does that leave students in learning these other skills? The paper doesn't comment. What I'd like to see is the same study done, but the students afterwards given a laboratory test - put them in a real lab and get them to do a real experiment, assessing some practical learning outcomes. Then what happens? It would be nice to try it out - but it will take a bit of organizing (not least acquiring some virtual labs and convincing my colleagues that it is a good idea.) So don't expect a response from me soon.

Darrah, M., Humbert, R., Finstein, J. Simon, M. & Hopkins, J. (2014). Are virtual labs as effective as hands-on labs for undergraduate physics? A comparitive study at two major universities. Journal of Science Education Technology 23:803-814. doi 10.1007/s10956-014-9513-9 


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