Mathematical Field Notes

Gromoll filtration

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In my latest preprint with Georgios Dimitroglou Rizell, we use the topology of diffeomorphism groups of high-dimensional spheres to produce interesting examples of nontrivial topology in symplectomorphism groups of cotangent bundles. Until we started thinking about this, I didn’t know much about the higher homotopy groups of Diff(S^n) so here is some interesting stuff I learned while we were writing this paper.

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Written by Jonny Evans

July 14, 2014 at 7:16 am

E-learning project report

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My final report on the e-learning project “Video lectures filmed by students” is now available to download in PDF form.

The purpose of this e-learning project was to test the effectiveness and viability of getting students to film mathematics lectures and the effect on student learning of making these videos available. The project was made possible by an E-Learning Development Grant (ELDG) and by the cooperation of a large number of people who I thank at the end.

Disclaimer. The project analysis is not scientific: there is no attempt made at comparison with a control group, the data sets are not large and the statistical methods used to analyse them are crude. This report is intended to be at best a rough guide to the UCL Mathematics Departmental Teaching Committee as to what action to take on filming of mathematics lectures.

Written by Jonny Evans

June 12, 2014 at 4:30 pm

Posted in E-learning, Teaching

Tagged with ,

December: video project update

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The video project has been progressing nicely: all of the videos have now been compressed and most have been uploaded to either Youtube or Lecturecast. Read on for some of the results.

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Written by Jonny Evans

December 14, 2013 at 2:34 pm

Posted in E-learning, Teaching

Geometry and undecidability

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These are the notes from a talk I gave to the UCL Undergraduate Mathematics Colloquium in early October and I would like to thank them for being such an attentive audience with so many good questions. The talk is a gentle introduction to the work of Nabutovsky and Weinberger, on how logical complexity gives rise to complexity for sublevel sets of functionals in geometry.

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Written by Jonny Evans

November 21, 2013 at 11:13 pm

Posted in Geometry, Teaching

Video-lecture project weeks 1 and 2

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The e-learning project for filming maths lectures is now underway (into its second week). You can see some of the initial results here and here. Read on for a summary of the early teething problems we have experienced and some of the solutions we have found.

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Written by Jonny Evans

October 10, 2013 at 2:49 pm

Posted in E-learning, Teaching

E-Learning: Video lectures filmed by students

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I recently received a grant from the UCL e-learning team to run a project for filming maths lectures.

The aim: The aim is to provide UCL mathematics students with high-quality video coverage of some of their core lectures. This would be particularly useful in mathematics where material is hard to absorb on a first hearing. We’d hope this would be particularly useful for our many overseas students with English as an nth language for n>1. Many of our courses are big ancillary courses for other departments and having lectures available online would help alleviate the possible clashes that might occur in timetabling.

The problem: Mathematical pedagogy focuses on board-based lectures, which are difficult to film under the current system used at UCL because the camera quality is not sufficient to capture extensive boardwork.

The proposal: Use a small, high-quality camera with a mount (borrowed from the e-learning team), operated by students from the front of the lecture theatre. We will train a group of students to act as camera-operators.

This is only a trial, so we will concentrate on two second-year courses (Mathematical Methods 3 and Complex Analysis), filmed by a team of four students. These students would preferably be from other classes so that the students in the target classes can focus on learning and not on filming. This would result in a workload of one or two hours/week for each student, which is hopefully not too much to distract from their own lectures, and they will be paid for their work (that’s where the grant is going).

The results will be posted to Lecturecast and the students will be able to give continuous feedback via Moodle forums, to that we can optimise our filming techniques. We will also assess the effectiveness of this method and the usefulness of the new resource using questionnaires.

From a lecturer’s perspective, I want my lectures to be videoed so that I can watch them back and know where I need to explain things better next time. While it is slightly terrifying for the unintentional verbal and notational errors one makes during lectures to be captured and viewable, it would surely be more useful for me to spot them so that I know why my students are getting confused.

While high-quality tracking cameras might be an optimal solution to this problem, they are expensive and we would first like to assess the usefulness of video lectures to our students. I also think that having a camera operated by someone who understands and appreciates the lectures, someone who can point the camera at what they think is most relevant, will lead to a more natural and useful video. I have seen the results of tracking cameras before, and I usually ended up standing on the far side of the board from the equation I was talking about. Having the camera placed at the front of the lecture theatre also seems like a more surefire way of being able to see what’s been written.

Written by Jonny Evans

July 2, 2013 at 5:52 pm

Posted in E-learning, Teaching

E-Learning: Spring 2013

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Henry Wilton, Bonita Carboo and I are the UCL Maths Department’s e-learning reps. In the interests of sharing ideas, here are a few things I have discovered this Spring about e-learning which may be useful to others.

Moodle.

Moodle is the online platform UCL uses to interact with its students. I am only just waking up to the possibilities it offers.

Most importantly for mathematicians, Moodle has LaTeX functionality. You may not discover it immediately, because the syntax is a little strange: you need to surround your LaTeX by double dollar signs, e.g. $$\sin(x)$$. This acts as if it were single dollar signs in ordinary LaTeX.

I was initially unconvinced of the utility of online forums for students, but that changed drastically. I had a feeling that students wouldn’t use them, but I decided to give it a go. I was being asked many questions by email, many of which were duplicates, so I created a forum for the students to ask questions during exam term. If I was emailed a question, I didn’t answer it and instead strongly encouraged the student to ask on the Moodle forum. The result was a big uptake of the forum with sixteen threads involving posts from sixteen different students in the month leading up to the exam. This is only 10% of the class, but I was able to point others who asked questions by email to the answers I had posted on the forum, saving me time. If I had realised earlier how effective this would be, I would have introduced a forum from the outset.

You can see stats on how many views your forum is getting by going to Navigation > Reports > Logs (thanks to Fiona Harkin for pointing this out). The revision forum had had 2491 views in just over a month. I would say this has been an extremely effective tool: while it was a lot of work to answer all the questions, that work counted for more.

Sage

Sage is an open-source computer algebra program. You can run it online using the Sage Notebook if you have an OpenID or using the Sage Cell Server which allows you to run small scripts without the bother of signing in. You can even embed it into webpages to create interactive content.

One of the biggest problems my students had was getting practice calculating Fourier series, so I have written the bare bones of a short (easily extendable) interactive webpage for next year which should give them the practice they need. I will add more examples to this (if and) when I have time. If you want to see how the page works, you can see the Sage code embedded in the HTML by viewing the page source. The important things are the script tags in the document head.

E-learning development grant

I am now the proud owner of an e-learning development grant to facilitate the filming of mathematics lectures and make them available online. I will expand on this in a blog post of its own.

Written by Jonny Evans

June 27, 2013 at 8:32 am

Posted in Code, E-learning, Teaching